tag:blogger.com,1999:blog-18762214303788078052024-03-19T06:38:44.203+01:00CrunchEconometrixEconometrics Resource for Beginners...and Data AnalysisBosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-1876221430378807805.post-78114347977000726842021-02-18T06:39:00.000+01:002021-02-18T06:39:00.845+01:00CrunchEconometrix-Teachable P.E.R.B.A. Launch<iframe frameborder="0" height="270" src="https://youtube.com/embed/OJOepyPLEHs" width="480"></iframe>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-67201718628629374892021-02-17T21:51:00.000+01:002021-02-17T21:51:33.084+01:00CrunchEconometrix Launches Practical Econometrics for Researchers, Beginners and Advanced-Level Users (P.E.R.B.A)<p> It is official and live on Teachable platform!!!</p>
<p>CrunchEconometrix Practical Econometrics for Researchers, Beginners and Advanced-Level Users (<strong>P.E.R.B.A</strong>) is launched today!!! A one-off payment is required for access to all videos published in the School. You will get VALUE for money paid. So, I encourage you to watch the PROMO and the 5 FREE-TO-WATCH videos to have a sneak peek on what is in store😄....I will see you in class🥰</p>
<p><span data-offset-key="40bp2-1-0">https://cruncheconometrix.teachable.com/p/practical-econometrics-for-researchers-beginners-and-advanced-level-users-perba/</span></p>
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<p><strong> </strong><strong>YouTube Custom URL:</strong> <a href="https://www.youtube.com/c/CrunchEconometrix">https://www.youtube.com/c/CrunchEconometrix</a></p>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-77598549304547769312020-07-14T16:49:00.000+01:002020-07-14T17:02:12.146+01:00Book Chapter: Access to Land and Food Security: Analysis of ‘Priority Crops’ Production in Ogun State, Nigeria<div style="background-color: white; border: 0px; box-sizing: border-box; color: #7a7a7a; font-family: Roboto, sans-serif; font-size: 14px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; text-align: justify; vertical-align: baseline;">
<div style="border: 0px; box-sizing: border-box; font-stretch: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; vertical-align: baseline;">
<h2>
<span style="background-color: #fcfcfc; color: #333333; font-family: "georgia" , serif; font-size: 17px; letter-spacing: 0.102px;"><b>Abstract:</b></span></h2>
</div>
<div style="border: 0px; box-sizing: border-box; font-stretch: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; vertical-align: baseline;">
<span style="background-color: #fcfcfc; color: #333333; font-family: "georgia" , serif; font-size: 17px; letter-spacing: 0.102px;">Using Ogun State located in South-western Nigeria, this chapter draws attention to the increase in output productivity of priority crops in the State from 2003 to 2015 due to the acquisitions of over 47,334 hectares of agricultural land across 28 communities in different Local Government Areas (LGAs). From Ogun State Agriculture Data, eight priority crops are analyzed: cassava, maize, rice, melon, yam, cocoyam, potato, and cowpea. Statistics reveal that the cultivation of cassava gives the highest average output of 4,515,620 metric tonnes and yield per hectare of 16.41 relative to other produce which affirms that Ogun State has the most comparative advantage in the cultivation of cassava followed by maize. The chapter further explores other pro-poor programmes directed at ensuring food security in the State.</span></div>
<div style="border: 0px; box-sizing: border-box; font-stretch: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; vertical-align: baseline;">
<span style="border: 0px; box-sizing: border-box; color: black; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"></span></div>
<div style="border: 0px; box-sizing: border-box; font-stretch: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; vertical-align: baseline;">
<span style="border: 0px; box-sizing: border-box; color: black; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Here is the link to the Book Chapter:</span></div>
<div style="border: 0px; box-sizing: border-box; font-stretch: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; vertical-align: baseline;">
<a href="https://link.springer.com/chapter/10.1007/978-3-030-41513-6_14" style="background-color: transparent;">https://link.springer.com/chapter/10.1007/978-3-030-41513-6_14</a></div>
<div style="border: 0px; box-sizing: border-box; font-stretch: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; vertical-align: baseline;">
<span style="color: black; font-family: inherit; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;"><br /></span>
<span style="color: black; font-family: inherit; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;">‘</span><span style="border: 0px; box-sizing: border-box; color: black; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: 600; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">The Palgrave Handbook on Agricultural and Rural Development in Africa’</span></div>
</div>
<div style="background-color: white; border: 0px; box-sizing: border-box; color: #7a7a7a; font-family: Roboto, sans-serif; font-size: 14px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; text-align: justify; vertical-align: baseline;">
<span style="border: 0px; box-sizing: border-box; color: black; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">In brief, the book examines agricultural and rural development in Africa from theoretical, empirical and policy perspectives. It presents a robust discourse on the developmental concerns needed to be addressed in rural communities through agricultural transformation. It also emphasises on the significance of the agricultural sector as it is closely related to the issues of food sustainability, poverty reduction, employment creation, and the attainment of the United Nations Sustainable Development Goals (SDGs) in Africa. </span><span style="color: black; font-family: inherit; font-style: inherit; font-weight: inherit;">Apart from the introduction and the conclusion chapters, the book contains 26 other chapters structured in five sections. The contributing authors provide the interconnections among the different aspects covered in the text, relating to agricultural and rural development in Africa. Hence, the book broadly recommends multiple evidence-based policies to develop the rural areas in Africa through the transformation of the agricultural sector that can benefit the continent.</span></div>
<div style="background-color: white; border: 0px; box-sizing: border-box; color: #7a7a7a; font-family: Roboto, sans-serif; font-size: 14px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; text-align: justify; vertical-align: baseline;">
<span style="border: 0px; box-sizing: border-box; color: black; font-family: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">Here is the link to the Handbook:</span></div>
<div style="background-color: white; border: 0px; box-sizing: border-box; color: #7a7a7a; font-family: Roboto, sans-serif; font-size: 14px; font-stretch: inherit; font-variant-east-asian: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 1.5em auto; max-width: 700px; outline: 0px; padding: 0px 20px; text-align: justify; vertical-align: baseline;">
<a href="https://link.springer.com/book/10.1007/978-3-030-41513-6#toc">https://link.springer.com/book/10.1007/978-3-030-41513-6#toc</a></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-89768633452743496382020-06-08T17:57:00.000+01:002020-06-08T17:57:01.808+01:00Teach Yourself Econometric Data Analysis with EViews: Step by Step Guide From Basic to Advance: Econometrics & Statistics in Practice<span style="background-color: white; font-family: "YouTube Noto", Roboto, arial, sans-serif; font-size: 13px; white-space: pre-wrap;">Fellow Researchers,
You may find this textbook relevant if you are using EViews: "Teach Yourself Econometric Data Analysis with EViews: Step by Step Guide From Basic to Advance: Econometrics & Statistics in Practice". Get it on Amazon </span><a class="yt-uix-sessionlink " data-sessionlink="itct=CAAQtnUiEwik5aGf1vLpAhV4FfEFHWXOBlY" data-target-new-window="True" data-url="/redirect?q=https%3A%2F%2Famzn.to%2F3dKeWwJ&event=backstage_event&redir_token=1K2c4zBDMR9vdPaD9PFt4aZWpb58MTU5MTcyMTM1MkAxNTkxNjM0OTUy" href="https://www.youtube.com/redirect?q=https%3A%2F%2Famzn.to%2F3dKeWwJ&event=backstage_event&redir_token=1K2c4zBDMR9vdPaD9PFt4aZWpb58MTU5MTcyMTM1MkAxNTkxNjM0OTUy" rel="nofollow noopener" style="background: rgb(255, 255, 255); border: 0px; color: #167ac6; cursor: pointer; font-family: "YouTube Noto", Roboto, arial, sans-serif; font-size: 13px; margin: 0px; padding: 0px; text-decoration-line: none; white-space: pre-wrap;" target="_blank">https://amzn.to/3dKeWwJ</a>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com1tag:blogger.com,1999:blog-1876221430378807805.post-58807358360738436262018-03-27T06:00:00.000+01:002018-03-27T06:00:10.834+01:00Time Series Analysis (Lecture 4 Part 2): Bounds Cointegration Test in EViews<div align="center" class="MsoNoSpacing" style="text-align: center;">
<h2 style="text-align: left;">
<b style="text-align: justify;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">After
unit root testing, what next?</span></b></h2>
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In the Part 1 of
this structured tutorials, we discussed <a href="https://cruncheconometrix.blogspot.com.ng/2018/03/time-series-analysis-lecture-4-part-1.html" target="_blank">Scenario 1</a>: when the series are
stationary in levels that is <i style="mso-bidi-font-style: normal;">I</i>(0)
series and Scenario 2: when they are stationary at <a href="https://cruncheconometrix.blogspot.com.ng/2018/03/time-series-analysis-lecture-4-part-1.html" target="_blank">first difference</a>. In the
first scenario, it implies that any shock to the system in the short run
quickly adjusts to the long run. Hence, only the long run model should be
estimated.<span style="mso-spacerun: yes;"> </span>While for the second
scenario, the relevance of the variables in the model is required, therefore there
is need to test for cointegration. If there is cointegration, specify the
long-run model and estimate VECM but if otherwise, specify only the short-run
model and apply the VAR estimation technique and not VECM. In today’s lecture
we consider the third scenario of when the variables are integrated of
different orders.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Scenario
3: The series are integrated of different orders?<span style="mso-spacerun: yes;"> </span><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Should in case the series are integrated
of different orders, like the second scenario, cointegration test is also
required but the use of <a href="https://cruncheconometrix.blogspot.com.ng/2018/03/time-series-analysis-lecture-4-part-1.html">Johansen cointegration</a> test is no longer valid. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The appropriate cointegration test is
the <b style="mso-bidi-font-weight: normal;"><span style="color: red;">Bounds test
for cointegration</span></b> proposed by Pesaran, Shin and Smith (2001)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<div class="separator" style="clear: both; text-align: center;">
</div>
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The estimation technique to apply is not
VAR but the autoregressive distributed lag (ARDL) model. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">4.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Similar to scenario 2, if series are not
cointegrated based on Bounds test, we are expected to estimate only the short
run. That is, run only the ARDL model (where variables are neither lagged nor
differenced). It is the <b style="mso-bidi-font-weight: normal;"><u><span style="color: red;">static form</span></u></b> of the model. </span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">5.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, both the long run and short run
models are valid if there is cointegration. That is, run both ARDL and ECM
models.</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; text-align: justify;">
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font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-bidi-font-family:Arial'><m:r>ω</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-bidi-font-family:Arial'><m:r>lnpcei</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-bidi-font-family:Arial'><m:r>∆</m:r><m:r>lnpce</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-bidi-font-family:Arial'><m:r>t</m:r><m:r>-</m:r><m:r>1</m:r></span></i></m:sub></m:sSub></m:e></m:nary><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>+</m:r></span></i></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:287.25pt;height:38.25pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; mso-fareast-font-family: "Times New Roman"; mso-fareast-theme-font: minor-fareast;"><span style="mso-spacerun: yes;">
</span></span><!--[if gte msEquation 12]><m:oMath><m:nary><m:naryPr><m:chr
m:val="∑"/><m:limLoc m:val="undOvr"/><span style='font-family:"Cambria Math","serif";
mso-ascii-font-family:"Cambria Math";mso-hansi-font-family:"Cambria Math";
mso-bidi-font-family:Arial;font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>i</m:r><m:r>=0</m:r></span></i></m:sub><m:sup><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>q</m:r><m:r>-</m:r><m:r>1</m:r></span></i></m:sup><m:e><m:sSub><m:sSubPr><span
style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:Arial;
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-bidi-font-family:Arial'><m:r>ω</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:
"Cambria Math","serif";mso-bidi-font-family:Arial'><m:r>Xi</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>∆</m:r></span></i></m:e></m:nary><m:sSub><m:sSubPr><span
style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:Arial;font-style:
italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><b
style='mso-bidi-font-weight:normal'><i style='mso-bidi-font-style:normal'><span
lang=EN-US style='font-family:"Cambria Math","serif";mso-bidi-font-family:
Arial'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="bi"/></m:rPr>X</m:r></span></i></b></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>t</m:r><m:r>-</m:r><m:r>i</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>+</m:r></span></i><m:sSub><m:sSubPr><span
style='font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:Arial;font-style:
italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><b
style='mso-bidi-font-weight:normal'><i style='mso-bidi-font-style:normal'><span
lang=EN-US style='font-family:"Cambria Math","serif";mso-bidi-font-family:
Arial'><m:r><m:rPr><m:scr m:val="sans-serif"/><m:sty m:val="bi"/></m:rPr>ε</m:r></span></i></b></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-US style='font-family:"Cambria Math","serif";
mso-bidi-font-family:Arial'><m:r>1</m:r><m:r>t</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 4.0pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:98.25pt;height:15.75pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Bounds
Cointegration Test in EViews<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=1876221430378807805" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">In this example,
we use the <i><a href="https://drive.google.com/drive/u/0/folders/1yh7t1htIV8hHAx9f6G9n1HLbxpuSxl5a">Dar.xlsx</a></i> data on Nigeria from 1981 to 2014 and the variables are the
log of manufacturing value-added (<i style="mso-bidi-font-style: normal;">lnmva</i>),
real exchange rate (<i style="mso-bidi-font-style: normal;">rexch</i>) and gross
domestic growth rate (<i style="mso-bidi-font-style: normal;">gdpgr</i>). The
model examines the effect of real exchange rate on manufacturing sector while
controlling for economic growth.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="background: lightgrey; font-family: "times new roman" , "serif"; font-size: 10.0pt;">Note:</span></b><span lang="EN-US" style="background: lightgrey; font-family: "times new roman" , "serif"; font-size: 10.0pt;"> Cointegration test should be performed on the level
form of the variables and not on their first difference. It is okay to also use
the log-transformation of the raw variables, as I have done in this example.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step
1: </span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Load
data into EViews (see video on how to do this)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step
2: </span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Open
variables as a Group data (see video on how to do this) and save under a new
name<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step
3:</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">
Go to <b style="mso-bidi-font-weight: normal;">Quick >> Estimate</b> <b style="mso-bidi-font-weight: normal;">Equation</b> <b style="mso-bidi-font-weight: normal;">>> </b>and specify the static form of the model which is stated
as: <i style="mso-bidi-font-style: normal;">lnmva<sub>t</sub> = b<sub>0</sub> </i>+<i style="mso-bidi-font-style: normal;"> b<sub>1</sub>rexch<sub>t</sub> </i>+<i style="mso-bidi-font-style: normal;"> b<sub>2</sub>gdpgr<sub>t</sub> </i>+<i style="mso-bidi-font-style: normal;"> u<sub>t</sub></i><sub> </sub>in the <b style="mso-bidi-font-weight: normal;">Equation Estimation Window<o:p></o:p></b></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTeVrZZV22FCa1ryYkACi9VZSXh3qgNn1b8vZGkdyIGU7pZ6tqTsgg0RV-87z2coDkRSD1RSiZ-iP8nh2rOdJhlmbZsBetqJuuGJF6qj-4iySybTAcntiPVA4JWAMY9EGSbTmYKcuLAzo/s1600/EViews+-+Equation+Estimation+Window.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Estimation Equation Dialog Box from cruncheconometrix.com.ng" border="0" data-original-height="430" data-original-width="476" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTeVrZZV22FCa1ryYkACi9VZSXh3qgNn1b8vZGkdyIGU7pZ6tqTsgg0RV-87z2coDkRSD1RSiZ-iP8nh2rOdJhlmbZsBetqJuuGJF6qj-4iySybTAcntiPVA4JWAMY9EGSbTmYKcuLAzo/s1600/EViews+-+Equation+Estimation+Window.png" title="EViews Estimation Equation Dialog Box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews Estimation Equation Dialog Box<br />Source: CrunchEconometrix</td></tr>
</tbody></table>
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><!--[if gte vml 1]><v:shape
id="Picture_x0020_2" o:spid="_x0000_i1030" type="#_x0000_t75" style='width:294pt;
height:265.5pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Step 4: Choose
the appropriate estimation technique<o:p></o:p></span></b></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click on the drop-down button in
front of <b style="mso-bidi-font-weight: normal;">Method</b> under <b style="mso-bidi-font-weight: normal;">Estimation settings</b> and select <b>ARDL
– Auto regressive Distributed Lag Models</b><o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkXQ1q6H458GUAO6xa0RAdrNHY3zhyphenhyphenybNEXBu6-ZCaINW2sKXIAb7kuK-SI57bsUZDYmnrlRZFQwvUHxR0gn109t31rZPpiBd1eZTJD2-ZKZosZiID9GBx1LbVkkt74-UTKz99BCYAhV8/s1600/EViews+-+ARDL+Method.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="http://cruncheconometrix.com.ng" border="0" data-original-height="558" data-original-width="475" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkXQ1q6H458GUAO6xa0RAdrNHY3zhyphenhyphenybNEXBu6-ZCaINW2sKXIAb7kuK-SI57bsUZDYmnrlRZFQwvUHxR0gn109t31rZPpiBd1eZTJD2-ZKZosZiID9GBx1LbVkkt74-UTKz99BCYAhV8/s1600/EViews+-+ARDL+Method.png" title="EViews: Estimation Technique Dialog Box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Source: CrunchEconometrix</td></tr>
</tbody></table>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_3" o:spid="_x0000_i1029"
type="#_x0000_t75" style='width:259.5pt;height:305.25pt;visibility:visible;
mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image009.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Step 5: Choose
the appropriate maximum lags and trend specification<o:p></o:p></span></b></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The lag length must be selected
such that the degrees of freedom (defined as <i>n - k</i>) must not be less
than 30. The <b>Constant </b>option under the <b style="mso-bidi-font-weight: normal;">Trend specification</b> is also selected.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-align: center; text-autospace: none;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxHA7Dg3AlqHlt3PVLGFGIe9CvnNF9ochw_tnSNhH-A29wRrW9zlNgZGkFBu0lqWwgrfpKgNYqzDmUnPDpfm3EzQ_bVVa7Ion9XPpP896JQf-BcThA7It17r_vlFGRBvKpWWRb3GJsgZk/s1600/EViews+-+Lags%252BTrend+Option.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Lags+ Trend Option from cruncheconometrix.com.ng" border="0" data-original-height="451" data-original-width="473" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxHA7Dg3AlqHlt3PVLGFGIe9CvnNF9ochw_tnSNhH-A29wRrW9zlNgZGkFBu0lqWwgrfpKgNYqzDmUnPDpfm3EzQ_bVVa7Ion9XPpP896JQf-BcThA7It17r_vlFGRBvKpWWRb3GJsgZk/s1600/EViews+-+Lags%252BTrend+Option.png" title="EViews: Lags+ Trend Option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Lags Option<br />Source: CrunchEconometrix</td></tr>
</tbody></table>
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_4"
o:spid="_x0000_i1028" type="#_x0000_t75" style='width:295.5pt;height:281.25pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Step 6: Choose
the appropriate lag selection criterion for optimal lag</span></b></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click on <b>Options </b>tab, then
click on the drop-down button under <b>Model Selection Criteria </b>and select
the <b>Akaike info Criterion (AIC)</b>, then click <b>Ok.<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyQrt0ky1titr_yHTk7FkJ0ahN4FslXdcLwTeQGWU9AG6SNOhPneN4byUfdWTBgxzNQogA1RLkL81wYcFhICvs_E0goHiwy6ZUMNYnGRXca8J7R65VLqQDraeTihD26Af-lZCkTcggiJU/s1600/EViews+-+Criterion+Selection.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Information Criterion Selection from cruncheconometrix.com.ng" border="0" data-original-height="447" data-original-width="472" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyQrt0ky1titr_yHTk7FkJ0ahN4FslXdcLwTeQGWU9AG6SNOhPneN4byUfdWTBgxzNQogA1RLkL81wYcFhICvs_E0goHiwy6ZUMNYnGRXca8J7R65VLqQDraeTihD26Af-lZCkTcggiJU/s1600/EViews+-+Criterion+Selection.png" title="EViews: Information Criterion Selection" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Information Criterion Selection<br />Source: CrunchEconometrix</td></tr>
</tbody></table>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1027"
type="#_x0000_t75" style='width:234.75pt;height:222pt;visibility:visible;
mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image013.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Step 7: Estimate
the model based on Steps 3 to 6</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrc_u8ssWB9Asl4QM6i-i-5hExTzJ6fRA9Wzo6bOJiWBN7W-JG0kFOjxuItPYeFz2XmaZVTlALtz0eaw5xsHaoMk10DgYJ6lFwOQ_dbVubqOZFIEDuGX34bg_Uiv0T9klyeZk_tJ2biyk/s1600/EViews+-+ARDL+Output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: ARDL Output from cruncheconometrix.com.ng" border="0" data-original-height="532" data-original-width="434" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrc_u8ssWB9Asl4QM6i-i-5hExTzJ6fRA9Wzo6bOJiWBN7W-JG0kFOjxuItPYeFz2XmaZVTlALtz0eaw5xsHaoMk10DgYJ6lFwOQ_dbVubqOZFIEDuGX34bg_Uiv0T9klyeZk_tJ2biyk/s1600/EViews+-+ARDL+Output.png" title="EViews: ARDL Output " /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: ARDL Output<br />Source: CrunchEconometerix</td></tr>
</tbody></table>
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1026"
type="#_x0000_t75" style='width:256.5pt;height:314.25pt;visibility:visible;
mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image015.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Step 8: Evaluate
the preferred model and conduct Bounds test<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The hypothesis
is stated as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">H<sub>0</sub>:
no cointegrating equation<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">H<sub>1</sub>: H<sub>0</sub>
is not true<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Rejection of the null hypothesis is at the relevant
statistical level, 10%, 5% level, 1%.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">a. Click on <b>View </b>on the
Menu Bar<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-layout-grid-align: none; text-autospace: none;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">b. Click on <b>Coefficient
Diagnostics<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">c. Select the <b>Bounds
Test </b>option<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The following
result is displayed below:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Here is the
EViews result on the </span><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">ARDL Bounds Test</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> of <i style="mso-bidi-font-style: normal;">lnmva, rexch </i>and<i style="mso-bidi-font-style: normal;"> gdpgr</i>:<o:p></o:p></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiw537ZwcZhdJ8_2MDDEo7RRcnS4gls1RyNGThctsFsb1hyphenhyphen4MHTyAbaCMRtSXFVQbsB-jcUhafXWhu8K50EKBtqLlHI-TgBPfLJ_c4AcM9OH5d4s2VTWj29ukWJAjWy9km4QIRH0_5o-s/s1600/EViews+-+Bounds+Test+Result.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: ARDL Bounds Test Result from cruncheconometrix.com.ng" border="0" data-original-height="503" data-original-width="467" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiw537ZwcZhdJ8_2MDDEo7RRcnS4gls1RyNGThctsFsb1hyphenhyphen4MHTyAbaCMRtSXFVQbsB-jcUhafXWhu8K50EKBtqLlHI-TgBPfLJ_c4AcM9OH5d4s2VTWj29ukWJAjWy9km4QIRH0_5o-s/s1600/EViews+-+Bounds+Test+Result.png" title="EViews: ARDL Bounds Test Result" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: ARDL Bounds Test Result<br />Source: CrunchEconometrix</td></tr>
</tbody></table>
<br />
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt;">Step
9: Interpret your result appropriately using the following decision criteria<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt;">The three
options of the decision criteria are as follows:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 0cm; mso-list: l0 level1 lfo1; tab-stops: 14.2pt; text-align: justify; text-indent: 0cm;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt;">If
the calculated <i style="mso-bidi-font-style: normal;">F</i>-statistic is greater
than the critical value for the upper bound <i>I(1)</i>, then we can conclude
that there is cointegration that is there is long-run relationship.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 0cm; mso-list: l0 level1 lfo1; tab-stops: 14.2pt; text-align: justify; text-indent: 0cm;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt;">If
the calculated <i style="mso-bidi-font-style: normal;">F</i>-statistic falls
below the critical value for the lower bound <i>I(0) </i>bound, then we
conclude that there is no cointegration, hence, no long-run relationship<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 0cm; mso-list: l0 level1 lfo1; tab-stops: 14.2pt; text-align: justify; text-indent: 0cm;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt;">The
test is considered inconclusive if the <i style="mso-bidi-font-style: normal;">F</i>-statistic
falls between the lower bound <i>I(0) </i>and the upper bound <i>I(1)</i>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="background: yellow; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-highlight: yellow;">Decision</span></b><span lang="EN-US" style="background: yellow; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-highlight: yellow;">: The obtained <i style="mso-bidi-font-style: normal;">F</i>-statistic
of <b style="mso-bidi-font-weight: normal;">0.6170</b> falls below the lower
bound <i>I(0), </i><span style="mso-bidi-font-style: italic;">h</span>ence, we will
consider only short run models since the variables show no evidence of a
long-run relationship as indicated by the results from the Bounds test.</span><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 13.0pt;">[Watch video on how to conduct Bounds
test for cointegration in EViews]</span></b><iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/xpBmXkz1jAg/0.jpg" src="https://www.youtube.com/embed/xpBmXkz1jAg?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "Times New Roman","serif"; font-size: 13.0pt;">If there are
comments or areas requiring further clarification, kindly post them below….<o:p></o:p></span></div>
<br /></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-32412428583850320532018-03-24T13:14:00.000+01:002018-03-24T13:25:37.640+01:00Time Series Analysis (Lecture 4 Part 1): Johansen Cointegration Test in EViews<div style="text-align: left;">
<b style="text-align: justify;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">After
unit root testing, what next?</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The outcome of
unit root testing matters for the empirical model to be estimated. The
following scenarios explain the implications of unit root testing for further
analysis. Still drawing on the previous tutorials
(see here for EViews, Stata and Excel) on unit root testing with the augmented
Dickey-Fuller procedure (see <a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos?view_as=subscriber" target="_blank">videos</a>), we are using the same data from <a href="https://drive.google.com/drive/u/0/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table 21.1 quarterly data of 1970q1 to 1991q4. The variables in
question are <i>pce</i>, <i>pdi</i> and <i>gdp</i> in natural logarithms.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Scenario
1: When series under scrutiny are
stationary in levels.<o:p></o:p></span></b></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In this scenario, it is assumed that <i>lnpce</i>, <i>lnpdi</i> and <i>lngdp </i>are
stationary in levels, that is, they are <i>I</i>(0)
series (integrated of order zero). In
this situation, performing a cointegration test is <b><i><u><span style="color: red;">not</span></u></i></b>
necessary. This is because any shock to the system in the short run quickly
adjusts to the long-run. <span style="color: red;">Consequently, only the long
run model should be estimated using OLS</span> (where variables are neither
lagged nor differenced). It is the <b><u><span style="color: red;">static form</span></u></b> of the model. </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">In essence, the
estimation of short run model is not necessary if series are </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">I</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">(0). </span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Scenario
2: When series are stationary in first differences.<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under this scenario, the series are
assumed to be non-stationary but became stationary after first difference<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">One special feature of this is that they
are of the same order of integration. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under this scenario, the model in
question is not entirely useless although the variables are unpredictable. To
verify further the relevance of the model, <b><span style="color: red;">there is need to test for cointegration.</span></b> That is, can we assume a long run
relationship in the model despite the fact that the series are drifting apart
or trending either upward or downward?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">There are however, two prominent
cointegration tests for <i>I</i>(I) series
in the literature. They are Engle-Granger cointegration test and <b><span style="color: red;">Johansen
Cointegration</span></b> test. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The Engle-Granger test is meant for
single equation model while Johansen cointegration test is considered when
dealing with multiple equations.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">If there is cointegration:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Implies that the series in question are
related and therefore can be combined in a linear fashion. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">That is, even if there are shocks in the
short run, which may affect movement in the individual series, they would
converge with time (in the long run). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Estimate <b><u><span style="color: red;">both</span></u></b> long-run and short-run
models.</span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">4.</span><span lang="EN-US"><span style="font-size: xx-small;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The estimation will require the use of
vector autoregressive (VAR) model and vector error correction model (VECM)
analysis. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></b></div>
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<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">If
there is no cointegration:<o:p></o:p></span></b></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Estimate only the short-run model, which
is VAR and not VECM.<o:p></o:p></span></div>
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<br /></div>
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<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Johansen
Cointegration Test in EViews</span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The hypothesis
is stated as:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">H<sub>0</sub>:
no cointegrating equation<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">H<sub>1</sub>: H<sub>0</sub>
is not true<o:p></o:p></span></div>
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<span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Rejection of the null hypothesis is at the 5% level.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
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<b><span lang="EN-US" style="background: lightgrey; font-family: "times new roman" , "serif"; font-size: 10.0pt;">Note:</span></b><span lang="EN-US" style="background: lightgrey; font-family: "times new roman" , "serif"; font-size: 10.0pt;"> Cointegration test should be performed on the level
form of the variables and not on their first difference. It is okay to also use
the log-transformation of the raw variables, as I have done in this example.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 10.0pt;"><o:p></o:p></span></div>
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<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Steps:<o:p></o:p></span></b></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Load data into EViews (see video on how
to do this)<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Open as Group data (see video on how to
do this)<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Go to <b>Quick >> Group Statistics >> Johansen Cointegration
>> dialog box opens >> list the variables >> Click OK
>> Select option 3 [Intercept (no trend)] >> Click OK<o:p></o:p></b></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Here is the
EViews result on the Johansen Cointegration test of <i>lnpce, lnpdi </i>and<i> lngdp</i>:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJItHa9dUjUryVKY4V1oJ7NK54eFJ_bjKW6SeQgf-L37nVzghyphenhyphenNYf5bT9ZdUb9OPw4IUgOGL0EW9XNAnSBZGOkclOHALdC_1V36rf8Cx3OhSH9l3E9Ph9YtBWA4B9yp26gcTJ_yonDFGM/s1600/EViews+-+Johansen+Cointegration+Test.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Johansen Cointegration Test from cruncheconometrix.com.ng" border="0" data-original-height="421" data-original-width="302" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJItHa9dUjUryVKY4V1oJ7NK54eFJ_bjKW6SeQgf-L37nVzghyphenhyphenNYf5bT9ZdUb9OPw4IUgOGL0EW9XNAnSBZGOkclOHALdC_1V36rf8Cx3OhSH9l3E9Ph9YtBWA4B9yp26gcTJ_yonDFGM/s640/EViews+-+Johansen+Cointegration+Test.png" title="EViews - Johansen Cointegration Test" width="459" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Johansen Cointegration Test<br />
Source: CrucnhEconometrix</td></tr>
</tbody></table>
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<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Interpreting
Johansen Cointegration Test Results</span></b></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The EViews output releases two
statistics, Trace Statistic and Max-Eigen Statistic<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Rejection criteria is at 0.05 level<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Rejection of the null hypothesis is
indicated by an asterisk sign (<b>*</b>)<o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Reject the null hypothesis if the
probability value is less than or equal to 0.05<b><o:p></o:p></b></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Reject the null hypothesis if the Trace
or Max-Eigen statistic is higher than the 0.05 critical value<b><o:p></o:p></b></span></div>
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<br /></div>
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<b><span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Decision: Given the results generated,
the null hypothesis of no cointegrating equation is rejected at the 5% level.
Hence, it is concluded that a long-run relationship exist among the three
variables.</span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></b></div>
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<b><span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> </span><b style="font-family: "Times New Roman", serif; font-size: 13pt; text-align: center;">[Watch video on how to conduct Johansen cointegration test in EViews]</b></div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/TB4m9M1sIJ0/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/TB4m9M1sIJ0?feature=player_embedded" width="320"></iframe></div>
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<span style="font-family: "times new roman" , serif; font-size: 13pt; text-align: justify;">However, if the
null hypothesis cannot be rejected, it evidences no cointegration and hence
there is no long-run relationship among the series. This implies that, if there
are shocks to the system, the model is not likely to converge in the long-run. In
addition, if there is no cointegration, only the short run model </span><b style="font-family: "Times New Roman", serif; font-size: 13pt; text-align: justify;"><u>should</u></b><span style="font-family: "times new roman" , serif; font-size: 13pt; text-align: justify;"> be estimated. That is, estimates
</span><b style="font-family: "Times New Roman", serif; font-size: 13pt; text-align: justify;"><u>only</u></b><span style="font-family: "times new roman" , serif; font-size: 13pt; text-align: justify;"> VAR do not estimate a VECM!</span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt; line-height: 107%;">If there are comments
or areas requiring further clarification, kindly post them below….</span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-29064764306586927982018-03-02T07:00:00.000+01:002018-03-02T07:00:25.601+01:00Panel Data Analysis (Lecture 2): How to Perform the Hausman Test in EViews<br />
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<b><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">Introduction to Panel Data Models</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<b><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;"><br /></span></b></div>
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<b><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">The panel data approach pools time series data with
cross-sectional data. Depending on the application, it can comprise a sample of
individuals, firms, countries, or regions over a specific time period. The
general structure of such a model could be expressed as follows:</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<b><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;"><br /></span></b></div>
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<i><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">Y</span></i><sub><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">it</span></sub><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;"> =
a<sub>o</sub> + b<i>X</i><sub>it</sub> + u<sub>it</sub></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">where u<sub>it</sub> ~ IID(0, </span><span style="color: black; font-family: "cambria math" , "serif"; font-size: 12.0pt;">𝜎</span><sup><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">2</span></sup><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">)
and <i>i</i> = 1, 2, ..., <i>N</i> individual-level
observations, and <i>t</i> = 1, 2, ...,<i>T</i> time series
observations.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">In this application, it is assumed
that <i>Y</i><sub>it</sub> is a continuous variable. In this model,
the observations of each individual, firm or country are simply stacked over
time on top of each another. This is the standard pooled model where intercepts
and slope coefficients are <i>homogeneous</i> across all <i>N</i> cross-sections
and through all <i>T</i> time periods. The application of OLS to this
model <i>ignores</i> the temporal and spatial dimension inherent in
the data and thus throws away useful information. It is important to note that
the temporal dimension captures the ‘within’ variation in the data while the
spatial dimension captures the ‘between’ variation in the data. The pooled OLS
estimator exploits both ‘between’ and ‘within’ dimensions of the data but does
not do so efficiently. Thus, in this procedure each observation is given equal
weight in estimation. In addition, the unbiasedness and consistency of the
estimator requires that the explanatory variables are uncorrelated with any
omitted factors. The limitations of OLS in such an application prompted
interest in alternative procedures. There are a number of different panel
estimators but the most popular is the fixed effects (or ‘within’) estimator.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<b><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">Fixed Effects or Random Effects?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">The question is usually asked which
econometric model an investigator should use when modelling with panel data.
The different models can generate considerably different results and this has
been documented in many empirical studies. In terms of a model where time
effects are assumed absent for simplicity, the model to be estimated may be
given by:</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<i><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">Y</span></i><sub><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">it</span></sub><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;"> = <i>a<sub>i</sub> </i>+ </span><b><span style="color: black; font-family: "" "symbol" "" , "serif"; font-size: 12.0pt;">b</span></b><b><i><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">X</span></i></b><b><sub><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">it</span></sub></b><span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;"> +
u<sub>it</sub></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "" serif "" , "serif"; font-size: 12.0pt;">The question, therefore, is do we
treat <i>a<sub>i</sub> </i>as fixed or random? The following points
are worth noting.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "" "symbol" "" , "serif"; font-size: 12.0pt;">·</span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">1) The
estimation of the fixed effects model is costly in terms of degrees of freedom.
This is a statistical and not a computing cost. It is particularly problematic
when <i>N</i> is large and <i>T</i> is small. The
occurrence of large <i>N</i> and small <i>T</i> currently
tends to characterize most panel data applications encountered.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<span style="font-size: 12pt; text-indent: -14.2pt;">·</span><span style="font-size: 12pt; text-indent: -14.2pt;">2) The <i>a<sub>i</sub> </i>terms
are taken to characterize (for want of a better expression) investigator
ignorance. In the fixed effects model does it make sense to treat one type of
investigator ignorance (<i>a<sub>i</sub></i>) as fixed but another as random (u<sub>it</sub>)?</span></div>
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<span style="color: black; font-family: "" "symbol" "" , "serif"; font-size: 12.0pt;">·</span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">3) The fixed effects formulation is viewed as one
where investigators make inferences conditional on the fixed effects in the
sample.</span></div>
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<span style="font-size: 12pt; text-indent: -14.2pt;"> 4)The
random effects formulation is viewed as one where investigators make
unconditional inferences with respect to the population of all effects.</span></div>
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<span style="color: black; font-family: "" "symbol" "" , "serif"; font-size: 12.0pt;">·</span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 7.0pt;"> </span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">5) The
random effects formulation treats the random effects as independent of the
explanatory variables (i.e. <i>E</i>(<i>a<sub>i</sub></i> <b>X</b><sub>it</sub>)
= 0). Violation of this assumption leads to bias and inconsistency in the </span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;"> vector.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">Advantage and disadvantage of the fixed
effects model</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">The main advantage of the fixed effects
model is its relative ease of estimation and the fact that it does not require
independence of the fixed effects from the other included explanatory
variables. The main disadvantage is that it requires estimation of <i>N</i> separate
intercepts. This causes problems because much of the variation that exists in
the data may be used up in estimating these different intercept terms. As a
consequence, the estimated effects (the </span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">s) for other explanatory variables in the
regression model may be imprecisely estimated. These might represent the more
important parameters of interest from the perspective of policy. As noted above
the fixed effects estimator is derived using the deviations between the cross-sectional
observations and the long-run average value for the cross-sectional unit. This
problem is most acute, therefore, when there is little variation or movement in
the characteristics over time, <i>that is when the variables are
rarely-changing or they are time-invariant</i>. In essence, the effects of
these variables are eliminated from the analysis.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">Advantage and disadvantage of the random effects model</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">The main advantage of the random
effects estimator is that it uses up fewer degrees of freedom in estimation
and <i>allows for the inclusion of time invariant covariates</i>. The main
disadvantage of the model is the assumption that the random effects are independent
of the included explanatory variables. It is fairly plausible that there may be
unobservable attributes not included in the regression model that are
correlated with the observable characteristics. This procedure, unlike fixed
effects, does not allow for the elimination of the omitted heterogeneous
effects.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">The Hausman Test</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">In determining which model is the more
appropriate to use, a statistical test can be implemented. The Hausman test
compares the random effects estimator to the ‘within’ estimator. If the null is
rejected, this favours the ‘within’ estimator’s treatment of the omitted
effects (i.e., it favours the fixed effects but only relative to the random
effects). The use of the test in this case is to discriminate between a model
where the omitted heterogeneity is treated as fixed and correlated with the
explanatory variables, and a model where the omitted heterogeneity is treated
as random and independent of the explanatory variables.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "" "symbol" "" , "serif"; font-size: 12.0pt;">·</span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 7.0pt;"> </span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">If the omitted effects are uncorrelated with the explanatory
variables, the random effects estimator is consistent and efficient. However,
the fixed effects estimator is consistent but not efficient given the
estimation of a large number of additional parameters (i.e., the fixed
effects).</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "" "symbol" "" , "serif"; font-size: 12.0pt;">·</span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 7.0pt;"> </span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">If the effects are correlated with the explanatory
variables, the fixed effects estimator is consistent but the random effects
estimator is inconsistent. The Hausman test provides the basis for
discriminating between these two models and the matrix version of the Hausman
test is expressed as:</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">[</span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">RE</span></sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">– </span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">FE</span></sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">][<b>V</b>(</span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">FE</span></sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">) – <b>V</b>(</span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">RE</span></sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">)]<sup>-1</sup>[</span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">RE</span></sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;"> – </span><b><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 107%;">b</span></b><sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">FE</span></sub><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">]′
~ </span><span style="color: black; font-family: "cambria math" , "serif"; font-size: 12.0pt;">𝝌</span><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">²</span><i><sub><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;">k</span></sub></i><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">where <i>k</i> is the number
of covariates (excluding the constant) in the specification. If the random
effects are correlated with the explanatory variables, then there will be a
statistically significant difference between the random effects and the fixed
effects estimates. Thus, the null and alternative hypotheses are expressed as:</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<h2 style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;"> H<sub>0</sub>: Random effects are
independent of explanatory variables</span></b></h2>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-indent: 36.0pt;">
<b><span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">H<sub>1</sub>: H<sub>0</sub> is
not true.</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">The null hypothesis is the random
effects model and if the test statistic exceeds the relevant critical value,
the random effects model is rejected in favour of the fixed effects model. In
finite samples the inversion of the matrix incorporating the difference in the
variance-covariance matrices may be negative-definite (or negative
semi-definite) thus yielding non-interpretable values for the chi-squared.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<span style="color: black; font-family: "" "times new roman" "" , "serif"; font-size: 12.0pt;">The selection of one model over the
other might be dictated by the nature of the application. For example, if the
cross-sectional units were countries and states, it may be plausible to assume
that the omitted effects are fixed in nature and not the outcome of a random
draw. However, if we are dealing with a sample of individuals or firms drawn
from a population, the assumption of a random effects model has greater appeal.
However, the choice of which model to choose is ultimately dictated
empirically. If it does not prove possible to discriminate between the two
models on the basis of the Hausman test, it may be safest to use the fixed
effects model, where the consequences of a correlation between the fixed effects
and the explanatory variables are less devastating than is the case with the
random effects model where the consequences of failure result in inconsistent
estimates. Of course, if the random effects are found to be independent of the
covariates, the random effects model is the most appropriate because it
provides a more <b><i><u>efficient</u></i></b> estimator than the
fixed effects estimator.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: justify;">
<i><span style="background: yellow; color: black; font-family: "" "times new roman" "" , "serif"; font-size: 10.0pt;">**This tutorial is
culled from my lecture note as given by Prof. Barry Reilly (Professor of
Econometrics, University of Sussex, UK).</span></i><span style="color: black; font-family: "times new roman" , "serif"; font-size: 13.5pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">How
to Perform the Hausman Test in EViews<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">First</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Load
file into EViews and create <b style="mso-bidi-font-weight: normal;">Group</b>
data (see <a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos" target="_blank">video</a> on how to do this)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Second</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Perform
fixed effects estimation: <b style="mso-bidi-font-weight: normal;">Quick >>
Estimate Equation >> Panel Options >> Fixed >> OK<o:p></o:p></b></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUGsPjRjKYEJIsfxlJM_xFSbsEQVBGmXG9ELp1p9M4vavItd0d7lxBtJj0sMi_RGA9syxZu9XQt3qFULz9uE4_aHIds3c8quhBU2S4YGWh2ZBoQdHBtiYKDu23O3HXmpJ3MJv0QK3sSFY/s1600/Panel+Dialog+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Equation Estimation Dialog Box from cruncheconometrix.com.ng" border="0" data-original-height="397" data-original-width="492" height="322" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUGsPjRjKYEJIsfxlJM_xFSbsEQVBGmXG9ELp1p9M4vavItd0d7lxBtJj0sMi_RGA9syxZu9XQt3qFULz9uE4_aHIds3c8quhBU2S4YGWh2ZBoQdHBtiYKDu23O3HXmpJ3MJv0QK3sSFY/s400/Panel+Dialog+Box.png" title="EViews: Equation Estimation Dialog Box" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Equation Estimation Dialog Box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><b style="mso-bidi-font-weight: normal;"><br /></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Third</span></b><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 12pt;">:
Perform random effects estimation: <b>Quick
>> Estimate Equation >> Panel Options >> Random >> OK</b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fourth</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Perform
the Hausman test: <b style="mso-bidi-font-weight: normal;">View >> Fixed/Random
Effects testing >> Correlated Random Effects – Hausman Test<i style="mso-bidi-font-style: normal;"><o:p></o:p></i></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fifth</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
Interpret results:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Reject the null
hypothesis if the prob-value is statistically significant at 5% level. It
implies that the individual effects (<i style="mso-bidi-font-style: normal;">a<sub>i</sub></i>)
correlate with the explanatory variables. Therefore use the fixed effect
estimator to run the analysis. Otherwise, use the random effects estimator.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">[Watch video tutorial on performing the
Hausman test in EViews]<o:p></o:p></span></b></div>
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/PnqmGQyJuqk/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/PnqmGQyJuqk?feature=player_embedded" width="320"></iframe></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">If you still
have comments or questions regarding how to perform the Hausman test, kindly
post them in the comments section below…..</span></div>
<br />Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-33507499165691636842018-02-28T06:00:00.000+01:002018-02-28T06:00:45.092+01:00Panel Data Analysis (Lecture 2): How to Perform the Hausman Test in Stata<h2 style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"> </span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Introduction
to Panel Data Models <o:p></o:p></span></b></div>
<h2 style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The panel data
approach pools time series data with cross-sectional data. Depending on the
application, it can comprise a sample of individuals, firms, countries, or
regions over a specific time period. The general structure of such a model
could be expressed as follows:</span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Y</span></i><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">it</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = a<sub>o</sub>
+ b<i>X</i><sub>it</sub> + u<sub>it</sub><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">where u<sub>it</sub>
~ IID(0, 𝜎<sup>2</sup>) and <i>i</i> = 1, 2,
..., <i>N</i> individual-level observations,
and <i>t</i> = 1, 2, ...,<i>T</i> time series observations. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In this
application, it is assumed that <i>Y</i><sub>it</sub>
is a continuous variable. In this model, the observations of each individual,
firm or country are simply stacked over time on top of each another. This is
the standard pooled model where intercepts and slope coefficients are <i>homogeneous</i> across all <i>N</i> cross-sections and through all <i>T</i> time periods. The application of OLS
to this model <i>ignores</i> the temporal
and spatial dimension inherent in the data and thus throws away useful
information. It is important to note that the temporal dimension captures the
‘within’ variation in the data while the spatial dimension captures the
‘between’ variation in the data. The pooled OLS estimator exploits both
‘between’ and ‘within’ dimensions of the data but does not do so efficiently.
Thus, in this procedure each observation is given equal weight in estimation.
In addition, the unbiasedness and consistency of the estimator requires that
the explanatory variables are uncorrelated with any omitted factors. The
limitations of OLS in such an application prompted interest in alternative
procedures. There are a number of different panel estimators but the most popular
is the fixed effects (or ‘within’) estimator.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fixed
Effects or Random Effects? <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The question is
usually asked which econometric model an investigator should use when modelling
with panel data. The different models can generate considerably different results
and this has been documented in many empirical studies. In terms of a model
where time effects are assumed absent for simplicity, the model to be estimated
may be given by: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Y</span></i><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">it</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = <i>a<sub>i</sub> </i>+ </span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><b><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">X</span></i></b><b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">it</span></sub></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> + u<sub>it<o:p></o:p></sub></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The question,
therefore, is do we treat <i>a<sub>i</sub> </i>as
fixed or random? The following points are worth noting. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
estimation of the fixed effects model is costly in terms of degrees of freedom.
This is a statistical and not a computing cost. It is particularly problematic
when <i>N</i> is large and <i>T</i> is small. The occurrence of large <i>N</i> and small <i>T</i> currently tends to characterize most panel data applications
encountered. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The <i>a<sub>i</sub> </i>terms are taken to
characterize (for want of a better expression) investigator ignorance. In the
fixed effects model does it make sense to treat one type of investigator
ignorance (<i>a<sub>i</sub></i>) as fixed
but another as random (u<sub>it</sub>)? <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
fixed effects formulation is viewed as one where investigators make inferences
conditional on the fixed effects in the sample. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
random effects formulation is viewed as one where investigators make
unconditional inferences with respect to the population of all effects. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
random effects formulation treats the random effects as independent of the
explanatory variables (i.e. <i>E</i>(<i>a<sub>i</sub></i> <b>X</b><sub>it</sub>) = 0). Violation of this assumption leads to bias
and inconsistency in the </span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">
vector. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Advantage
and disadvantage of the fixed effects model<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The main
advantage of the fixed effects model is its relative ease of estimation and the
fact that it does not require independence of the fixed effects from the other
included explanatory variables. The main disadvantage is that it requires
estimation of <i>N</i> separate intercepts.
This causes problems because much of the variation that exists in the data may
be used up in estimating these different intercept terms. As a consequence, the
estimated effects (the </span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">s)
for other explanatory variables in the regression model may be imprecisely
estimated. These might represent the more important parameters of interest from
the perspective of policy. As noted above the fixed effects estimator is
derived using the deviations between the cross-sectional observations and the
long-run average value for the cross-sectional unit. This problem is most
acute, therefore, when there is little variation or movement in the
characteristics over time, <i>that is when
the variables are rarely-changing or they are time-invariant</i>. In essence,
the effects of these variables are eliminated from the analysis. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Advantage
and disadvantage of the random effects model<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The main
advantage of the random effects estimator is that it uses up fewer degrees of
freedom in estimation and <i>allows for the inclusion
of time invariant covariates</i>. The main disadvantage of the model is the
assumption that the random effects are independent of the included explanatory
variables. It is fairly plausible that there may be unobservable attributes not
included in the regression model that are correlated with the observable
characteristics. This procedure, unlike fixed effects, does not allow for the
elimination of the omitted heterogeneous effects. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
Hausman Test<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In determining
which model is the more appropriate to use, a statistical test can be
implemented. The Hausman test compares the random effects estimator to the
‘within’ estimator. If the null is rejected, this favours the ‘within’
estimator’s treatment of the omitted effects (i.e., it favours the fixed effects
but only relative to the random effects). The use of the test in this case is
to discriminate between a model where the omitted heterogeneity is treated as
fixed and correlated with the explanatory variables, and a model where the
omitted heterogeneity is treated as random and independent of the explanatory
variables. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If
the omitted effects are uncorrelated with the explanatory variables, the random
effects estimator is consistent and efficient. However, the fixed effects
estimator is consistent but not efficient given the estimation of a large
number of additional parameters (i.e., the fixed effects). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If
the effects are correlated with the explanatory variables, the fixed effects
estimator is consistent but the random effects estimator is inconsistent. The
Hausman test provides the basis for discriminating between these two models and
the matrix version of the Hausman test is expressed as: <o:p></o:p></span></div>
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<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">[</span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">RE</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">– </span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">FE</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">][<b>V</b>(</span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">FE</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">) – <b>V</b>(</span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">RE</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">)]<sup>-1</sup>[</span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">RE</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> – </span><b><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">b</span></b><sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">FE</span></sub><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">]′ ~ </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">𝝌²</span><sub style="font-family: "Times New Roman", serif; text-align: justify; text-indent: -18.9333px;"><i>k</i></sub></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">where <i>k</i> is the number of covariates (excluding
the constant) in the specification. If the random effects are correlated with
the explanatory variables, then there will be a statistically significant
difference between the random effects and the fixed effects estimates. Thus,
the null and alternative hypotheses are expressed as: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 72.0pt;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">H<sub>0</sub>:
Random effects are independent of explanatory variables<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 36.0pt; text-indent: 36.0pt;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">H<sub>1</sub>: H<sub>0</sub> is not
true.<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The null
hypothesis is the random effects model and if the test statistic exceeds the
relevant critical value, the random effects model is rejected in favour of the
fixed effects model. In finite samples the inversion of the matrix
incorporating the difference in the variance-covariance matrices may be
negative-definite (or negative semi-definite) thus yielding non-interpretable
values for the chi-squared. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The selection of
one model over the other might be dictated by the nature of the application.
For example, if the cross-sectional units were countries and states, it may be
plausible to assume that the omitted effects are fixed in nature and not the
outcome of a random draw. However, if we are dealing with a sample of
individuals or firms drawn from a population, the assumption of a random
effects model has greater appeal. However, the choice of which model to choose
is ultimately dictated empirically. If it does not prove possible to
discriminate between the two models on the basis of the Hausman test, it may be
safest to use the fixed effects model, where the consequences of a correlation
between the fixed effects and the explanatory variables are less devastating
than is the case with the random effects model where the consequences of
failure result in inconsistent estimates. Of course, if the random effects are
found to be independent of the covariates, the random effects model is the most
appropriate because it provides a more <b><i><u>efficient</u></i></b> estimator than the
fixed effects estimator.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 10.0pt;">**This tutorial is culled from my lecture
note as given by Prof. Barry Reilly (Professor of Econometrics, University of
Sussex, UK).</span></i><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 10.0pt;"><o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">How
to Perform the Hausman Test in Stata<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">First</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Open
a log file, load data into Stata, use a do-file (to replicate your research)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Second</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Inform
Stata that you are using a panel with ‘<i>id</i>’
the cross-sectional indicator and <i>'year'</i>
the time period indicator to prepare for panel data analysis.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">xtset
id year</span></i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Third</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
Create year dummies (to capture time variations in the data) <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">tab
year, gen(yr)<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fourth</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Run
the fixed effects model and store the results<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">eststo
fixed: xtreg y x<sub>1</sub> x<sub>2</sub> x<sub>3</sub> x<sub>4</sub> yr2 –
yr..., fe i(c_id)<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fifth</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Run
the random effects model and store the results<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">eststo
random: xtreg y x<sub>1</sub> x<sub>2</sub> x<sub>3</sub> x<sub>4</sub> yr2 –
yr..., re i(c_id)<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Sixth</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: Run
the Hausman test<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">hausman
fixed random<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Seventh</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
Interpret results: </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">Reject the null
hypothesis if the prob-value is statistically significant at 5% level. It
implies that the individual effects (</span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">a<sub>i</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">)
correlate with the explanatory variables. Therefore use the fixed effect
estimator to run the analysis. Otherwise, use the random effects estimator.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">[Watch video tutorial on performing the
Hausman test in Stata]</span></b><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/FphcgTjfoC4/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/FphcgTjfoC4?feature=player_embedded" width="320"></iframe></div>
<div class="MsoNoSpacing" style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">If you still have
comments or questions regarding how to perform the Hausman test, kindly post
them in the comments section below…..</span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-57658075465816877212018-02-26T13:42:00.000+01:002018-02-27T07:40:33.791+01:00Time Series Analysis (Lecture 3): How to Perform Stationarity Test in Excel<h2 style="text-align: justify;">
What is Stationarity in Time Series Analysis?</h2>
<div style="text-align: justify;">
In econometrics, time series data are frequently used and they often pose distinct problems for econometricians. As it will be discussed with examples, most empirical work based on time series data assumes that the underlying series is stationary. Stationarity of a series (that is, a variable) implies that its mean, variance and covariance are constant over time. That is, these do not vary systematically over time. In order words, they are time invariant. However, if that is not the case, then the series is nonstationary. We will discuss some possible scenarios where two series, Y and X, are nonstationary and the error term, u, is also nonstationary. In that case, the error term will exhibit autocorrelation. Another likely scenario is where Y and X are nonstationary, but u is stationary. The implications of this will also be explored. In time series analysis, the words nonstationary, unit root or random walk model are used synonymously. In essence, of a series is considered to be nonstationary, it implies that such exhibit a unit root and exemplifies a random walk series.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Regressing two series that are nonstationary, likewise, yields a spurious (or nonsense) regression. That is, a regression whose outcome cannot be used for inferences or forecasting. In short, such results should not be taken seriously and must be discarded. A stationary series will tend to return to its mean (called mean reversion) and fluctuations around this mean (measured by its variance) will have a broadly constant breadth. But if a time series is not stationary in the sense just explained, it is called a nonstationary time series such will have a time-varying mean or a time-varying variance or both. In summary, a stationary time series is important because if such is nonstationary, its behaviour can be studied only for the time period under consideration. That is, each set of time series data will therefore be for a particular episode. As a result, it is not possible to generalise its relevance to other time periods. Therefore, for the purpose of forecasting, such (nonstationary) time series may be of little practical value</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<b>How to detect unit root in a series?</b></div>
<div style="text-align: justify;">
In a bivariate (2 variables) model or that involving multiple variables (called a multiple regression model), it is assumed that all the variables are stationary at level (that is, the order of integration of each of the variable is zero,<i> I</i>(0). It is important to state at this point, that the order of integration of a series in a regression model is determined by the outcome of a unit root test (or stationarity test). If the series is stationary at level after performing unit root test, then it is <i>I</i>(0), otherwise it is <i>I(d)</i> where d represents the number of times the series is differenced before it becomes stationary. But what if the assumption of stationarity at level of the series in a bivariate or multiple regression model is relaxed and we consequently allow for a unit root in each of the variables in the model, how can this be corrected? In general, this would require a different treatment from a conventional regression with stationary variables at <i>I</i>(0).</div>
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<br /></div>
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In particular, we focus on a class of linear combination of unit root processes known as cointegrated process. The generic representation for the order of integration of series is <i>I(d)</i> where d is the number of differencing to render the series stationary. Hence, a stationary series at level, <i>d</i> = 0 is a series with an <i>I</i>(0) process. Although, for any non-stationary series, ‘<i>d</i>’ can assume any value greater than zero, however, in applied research, only the unit root process of <i>I</i>(1) process is allowed, otherwise such series with higher order of integration (<i>d</i> > 1) should be excluded in the model as no meaningful policy implications or relevance can be drawn from such series. </div>
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<br /></div>
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Here is an example of a bivariate linear regression model:</div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;"> <i>Y<sub>t </sub></i>= </span><span style="font-family: "cambria math" , serif; font-size: 12pt;">𝛂₀</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + <i>bX<sub>t</sub></i> + <i>u<sub>t</sub></i>
[1]</span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">Assume <i>Y<sub>t</sub> </i>and<i> X<sub>t</sub></i>
are two random walk models that are <i>I</i>(1) processes and are independently
distributed as: <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<i>Y<sub>t </sub></i>= ρ<i>Y<sub>t-1</sub></i> + <i>v<sub>t</sub></i>,
-1
≤ ρ ≤ 1
[2]
<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<i>X<sub>t </sub></i>= ղ<i>X<sub>t-1</sub></i> + <i>e<sub>t</sub></i>,
-1 ≤ ղ ≤
1
[3]<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">and <i>v<sub>t</sub></i> and <i>e<sub>t</sub></i> have
zero mean, a constant variance and are orthogonal (these are <i>white
noise</i> error terms). <o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">We also assumed that <i>v<sub>t</sub></i> and <i>e<sub>t</sub></i> are
serially uncorrelated as well as mutually uncorrelated. As stated in [2] and
[3], both these time series are nonstationary; that is, they are <i>I</i>(1)
or exhibit stochastic trends. Suppose we regress <i>Y<sub>t</sub> </i>on<i> X<sub>t</sub></i>.
Since <i>Y<sub>t</sub> </i>on<i> X<sub>t</sub></i> are
uncorrelated <i>I</i>(1) processes, the <i>R</i><sup>2</sup> from
the regression of <i>Y </i>on<i> X</i> should tend to zero;
that is, there should not be any relationship between the two variables.
Equations [2] and [3] resemble the Markov first-order autoregressive model. If ρ
and ղ = 1, the equations become a random walk model without drift. If
ρ and ղ are in fact 1, then a unit root problem surfaces, that is, a
situation of nonstationarity; because we already know that in this case the
variance of <i>Y<sub>t</sub></i> is not stationary. The name unit
root is due to the fact that ρ = 1. Again, the terms nonstationary, random
walk, and unit root can be treated as synonymous. If, however, |ρ| ≤ 1, and
|ղ| ≤ 1, that is if their absolute values are less than one, then it can
be shown that both series <i>Y<sub>t</sub></i> and <i>X<sub>t</sub></i> are
stationary. In practice, then, it is important to find out if a time series
possesses a unit root.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">Given equations [2] and [3],
there should be no systematic relationship between <i>Y<sub>t</sub></i> and <i>X<sub>t</sub></i> as
they both drift away from equilibrium (i.e. they do not converge), and
therefore, we should expect that an ordinary least squares (OLS) estimate
of <i>b</i> should be close to zero, or insignificantly different
from zero, at least as the sample size increases. But this is not usually the
case. The fitted coefficients in this case may be statistically significant
even when there is no true relationship between the dependent variable and the
regressors. This is regarded as a spurious regression or correlation where, in
the case of our example, <i>b</i> takes any value randomly, and
its <i>t</i>-statistic indicates significance of the estimate.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , serif; font-size: 12pt;">But how can unit root
be detected?</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> There are some clues that tell you if a series is
nonstationary and if the regression of bivariate or multivariate relationships
are spurious. Some of these are:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; text-align: justify; text-indent: -18.0pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">1. Do a graphical plot of the series to visualise the
nature. Is it trending upwards or downwards? Does it exhibit a mean-reversion
or not? Or are there fluctuations around its mean?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; text-align: justify; text-indent: -18.0pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">2. Or carry out a regression analysis on two series
and observe the <i>R</i><sup>2</sup>. If it is above 0.9, it may suggest
that the variables are nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; text-align: justify; text-indent: -18.0pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">3. The rule-of-thumb: if the <i>R</i><sup>2</sup> obtained
from the regression is higher than the Durbin Watson (DW) statistic. The low DW
statistic evidences positive first order auto-correlation of the error terms.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">Using </span><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Gujarati
and Porter</span></a><span style="font-family: "times new roman" , serif; font-size: 12pt;"> Table 21.1 quarterly data from 1970q1 to 1991q4, examples of
nonstationary series and spurious regression can be seen from the <i>lnpce</i>
and <i>lnpdi</i> relationship. Since the series are measured in
billions of US dollars, the natural logarithms of the variables will be used in
analysing their essential features.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , serif; font-size: 12pt;">Nonstationary series:</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> the graphical plot of the two variables
shows an upward trend and none of the variables revert to their means. That is,
the data generating process of both series does not evolve around zero. That
clearly shows that the series are nonstationary.</span></div>
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<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhU0miGmENLsHK_xQWA04EOAs3rpp-YA6T-FKA8kebI-35ZiYEvXBq8q2MofOs3AxbA5o1L-JRUip4k20X2h44Wnma-9avmW74LR-qyIYqwJ6uRQlS2eomeUSthr8MUuvNKuAAXAsaHQVg/s1600/Excel+-+nonstationary+series.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Example of nonstationary series from cruncheconometrix.com.ng" border="0" data-original-height="281" data-original-width="442" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhU0miGmENLsHK_xQWA04EOAs3rpp-YA6T-FKA8kebI-35ZiYEvXBq8q2MofOs3AxbA5o1L-JRUip4k20X2h44Wnma-9avmW74LR-qyIYqwJ6uRQlS2eomeUSthr8MUuvNKuAAXAsaHQVg/s1600/Excel+-+nonstationary+series.PNG" title="Excel: Example of nonstationary series" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Example of a nonstationary series<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<br /></div>
<div class="MsoNormal">
<b><span style="background: lightgrey; font-size: 11pt;">Note:</span></b><span style="background: lightgrey; font-size: 11pt;"> To
generate the graph: Highlight the cells, go to <b>Insert >> Recommended Charts >> All Charts >> Line</b><o:p></o:p></span></div>
<div class="MsoNormal">
<span style="background: lightgrey; font-size: 11pt;"><b><br /></b></span></div>
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<b><span style="font-family: "times new roman" , serif; font-size: 12pt;">What is a spurious
regression? </span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;">Sometimes we expect to find
no relationship between two variables, yet a regression of one on the other
variable often shows a significant relationship. This situation exemplifies the
problem of spurious, or nonsense, regression. The regression of <i>lnpce</i> on <i>lnpdi</i> shows
how a spurious regression can arise if time series are not stationary. As
expected, because both variables are nonstationary, the result evidences that a
spurious regression has been undertaken.</span></div>
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<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgePEKmfBEiAnugVramhKyP62lRdDJ2xnMJMzZS7FVI6cNIB2u9sn5d61fbwQP4275jouMpfd3XFfvOR-wrpCf-jVcnEw95CTsNO0ldVoOYBvY2DWY8VwIzRu8QVaCYBaCs0B_54QjUt34/s1600/Excel+-+Spurious+regression.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Example of a spurious regression from cruncheconometrix.com.ng" border="0" data-original-height="395" data-original-width="611" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgePEKmfBEiAnugVramhKyP62lRdDJ2xnMJMzZS7FVI6cNIB2u9sn5d61fbwQP4275jouMpfd3XFfvOR-wrpCf-jVcnEw95CTsNO0ldVoOYBvY2DWY8VwIzRu8QVaCYBaCs0B_54QjUt34/s1600/Excel+-+Spurious+regression.png" title="Excel: Example of a spurious regression" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Example of a spurious regression<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<b style="text-align: center;"> </b><br />
<b style="text-align: center;"> [Watch video on how to compute the Durbin Watson <i>d</i> statistic]</b></div>
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/J_XTcBNGmu4/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/J_XTcBNGmu4?feature=player_embedded" width="320"></iframe></div>
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<b style="text-align: center;"><br /></b></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">As you can see, the
coefficient of <i>lnpdi</i> is highly statistically significant, and
the <i>R</i><sup>2</sup> value is statistically significantly
different from zero. From these results, you may be tempted to conclude that
there is a significant statistical relationship between both variables,
whereas <i>a priori</i> there may or may <i>not</i> be
none. This is simply the phenomenon of <b>spurious or nonsense regression</b>,
first discovered by Yule (1926). He showed that (spurious) correlation could
persist in nonstationary time series even if the sample is very large. That
there is something wrong in the preceding regression is suggested by the
extremely low Durbin–Watson value, which suggests very strong first-order
autocorrelation. According to Granger and Newbold, <i>R</i><sup>2</sup> >
DW is a good rule of thumb to suspect that the estimated regression is
spurious, as in the given example.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">Why is it important
to test for stationarity?</span></b><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">We usually consider a nonstationary
series for the following reasons:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , serif; font-size: 12pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , serif; font-size: 12pt;">To evaluate the behaviour of series over time. Is the series
trending upward or downward? This can be verified from performing a
stationarity test. In other words, the test can be used to evaluate the stability
or predictability of time series. If a series is nonstationary, that means the
series is unstable or unpredictable and therefore may not be valid for
inferences, prediction or forecasting.<o:p></o:p></span></div>
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<br /></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">2. To know how a series responds to shocks requires
carrying out a stationarity test. If such series is nonstationary, the impact
of shocks to the series are more likely to be permanent. Consequently, if a
series is stationary, impact of shocks will be temporary or brief.</span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">How to correct for
nonstationarity of a series?</span></b><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">What can be done with
nonstationarity in a time series knowing that performing OLS on such a model
yields spurious regression?<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">The Unit Root Test</span></b><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">We begin with equations [2]
and [3] which are unit root (stochastic) processes with white noise error
terms. If the parameters of the models are equal to 1, that is, in the case of
the unit root, both equations become random walk models without drift, which we
know is a nonstationary stochastic process. So, what can be done to correct
this? For instance, for equation [2], simply regress <i>Y<sub>t</sub></i> on
its (one-period) lagged value <i>Y<sub>t−1</sub></i> and find out if
the estimated ρ is statistically equal to 1? If it is, then <i>Y<sub>t</sub></i> is
nonstationary. Repeat same for the <i>X<sub>t</sub></i> series. This
is the general idea behind the unit root test of stationarity.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">For theoretical reasons,
equation [2] is manipulated as follows: Subtract <i>Y<sub>t−1</sub></i> from
both sides of [2] to obtain:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<i>Y<sub>t </sub></i>- <i>Y<sub>t-1 </sub></i>= ρ<i>Y<sub>t-1</sub></i> - <i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i>
[4]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
= (ρ - 1)<i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">and this can be stated
alternatively as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 72.0pt; text-align: justify; text-indent: 36.0pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">⃤ <i>Y<sub>t</sub></i> = δ<i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i>
[5]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">where δ = (ρ − 1)
and ⃤, as usual, is the first-difference operator. In practice, therefore,
instead of estimating [2], we estimate [5] and test the null hypothesis
that δ = 0. If δ = 0, then ρ = 1, that is we have a unit
root, meaning the time series under consideration is nonstationary.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Before we proceed to
estimate [5], it may be noted that if δ = 0, [5] will become:<o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">⃤ <i>Y<sub>t</sub></i> = <i>Y<sub>t-1</sub></i> - <i>Y<sub>t-1 </sub></i>= <i>v<sub>t</sub></i>
[6]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">(Remember to do the same
for <i>X<sub>t</sub></i> series)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Since <i>v</i><sub>t</sub> is a white
noise error term, it is stationary, which means that <b>the first
difference of a random walk time series is stationary</b>.</span></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHUTSG8t-YF9WK20CUekvOv2OPecOdNUc4F2Ly-aJIFa1YQ3klRHCCFKM-jj1CcBwHmqYOLPeasRym5Pz1aRWzfZqTCmpdYIJHMhyphenhyphenQKzihA7YXmkDfR89NatRr6u8fW80vPKfJOcUvVXA/s1600/Excel+-+Stationary+series.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Example of stationary series from cruncheconometrix.com.ng" border="0" data-original-height="280" data-original-width="484" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHUTSG8t-YF9WK20CUekvOv2OPecOdNUc4F2Ly-aJIFa1YQ3klRHCCFKM-jj1CcBwHmqYOLPeasRym5Pz1aRWzfZqTCmpdYIJHMhyphenhyphenQKzihA7YXmkDfR89NatRr6u8fW80vPKfJOcUvVXA/s1600/Excel+-+Stationary+series.PNG" title="Excel: Example of stationary series" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Example of stationary series<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Visual observation of the
differenced series shows that the three variables are stationary around the
mean. They all exhibit constant mean-reversions. That is, they fluctuate around
0. If we are to draw a trend line, such a line will be horizontal at 0.01.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Okay, having said all that.
Let us return to estimating equation [5]. This is quite simple, all that is
required is to take the first differences of <i>Y<sub>t</sub></i> and
regress on <i>Y<sub>t−1</sub></i> and see if the estimated slope
coefficient in this regression is statistically different from is zero or not.
If it is zero, we conclude that <i>Y<sub>t</sub></i> is
nonstationary. But if it is negative, we conclude that <i>Y<sub>t</sub></i> is
stationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , serif;">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , serif;"> Since
δ = (ρ − 1), for stationarity ρ must be less than one. For this to happen δ must
be negative!</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The only question is which
test do we use to find out if the estimated coefficient of <i>Y<sub>t−1</sub></i> in
[5] is zero or not? You might be tempted to say, why not use the usual <i>t</i> test?
Unfortunately, under the null hypothesis that δ = 0 (i.e., ρ = 1), the <i>t</i> value
of the estimated coefficient of <i>Y<sub>t−1</sub></i> does not follow
the <i>t</i> distribution even in large samples; that is, it does not
have an asymptotic normal distribution.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">What is the alternative?
Dickey and Fuller (DF) have shown that under the null hypothesis that δ = 0,
the estimated <i>t</i> value of the coefficient of <i>Y<sub>t−1</sub></i> in
[5] follows the <b><i>τ (tau)</i></b> statistic. These authors have
computed the critical values of the <i>tau statistic</i> on the basis
of Monte Carlo simulations.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , serif;">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , serif;"> Interestingly,
if the hypothesis that δ = 0 is rejected (i.e., the time series is stationary),
we can use the usual (Student’s) <i>t</i> test.</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The unit root test can be
computed under three (3) different null hypotheses. That is, under different
model specifications such as if the series is a:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">1. random walk (that is, model has no
constant, no trend)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">2. random walk with drift (that is, model
has a constant)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">3. random walk with drift and a trend (that
is, model has a constant and trend)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">In all cases, the null
hypothesis is that δ = 0; that is, there is a unit root and the alternative
hypothesis is that δ is less than zero; that is, the time series is stationary.
If the null hypothesis is rejected, it means that <i>Y<sub>t</sub></i> is
a stationary time series with zero mean in the case of [5], that <i>Y<sub>t</sub></i> is
stationary with a nonzero mean in the case of a random walk with drift model,
and that <i>Y<sub>t</sub></i> is stationary around a deterministic
trend in the case of random walk with drift around a trend.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">It is extremely important to
note that the critical values of the <i>tau test</i> to test the
hypothesis that δ = 0, are different for each of the preceding three
specifications of the DF test, which are now computed by all econometric
packages. In each case, if the computed absolute value of the <i>tau
statistic</i> (|τ|) exceeds the DF or MacKinnon critical <i>tau
values</i>, the null hypothesis of a unit root is rejected, in order words the
time series is stationary. On the other hand, if the computed |τ| does not
exceed the critical <i>tau</i> value, we fail to reject the null
hypothesis, in which case the time series is nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , serif;">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , serif;"> Students
often get confused in interpreting the outcome of a unit root test. For
instance, if the calculated <i>tau</i> statistic is -2.0872 and the DF <i>tau</i> statistic
is -3.672, you cannot reject the null hypothesis. Hence, the conclusion is that
the series is nonstationary. But if the calculated <i>tau</i> statistic
is -5.278 and the DF <i>tau</i> statistic is -3.482, you reject the
null hypothesis in favour of the alternative. Hence, the conclusion is that the
series is stationary.</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: lightgrey; font-family: "times new roman" , serif;">*Always use the appropriate critical τ values
for the indicated model specification.</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">How to Perform Unit
Root Test in Excel (see for <a href="http://cruncheconometrix.blogspot.com.ng/2018/02/time-series-analysis-lecture-3-how-to_21.html" target="_blank">Stata</a> and <a href="http://cruncheconometrix.blogspot.com.ng/2018/02/time-series-analysis-lecture-3-how-to.html" target="_blank">EViews</a>)</span></b><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: yellow; font-family: "times new roman" , serif; font-size: 12pt;">Example
dataset is from </span><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank"><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Gujarati and Porter</span></a><span style="background: yellow; font-family: "times new roman" , serif; font-size: 12pt;"> Table
21.1</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Several tests have been
developed in the literature to test for unit root. Prominent among these tests
are Augmented Dickey-Fuller, Phillips-Perron, Dickey-Fuller Generalised Least
Squares (DFGLS) and so on. But this tutorials limits testing to the use of ADF
and PP tests. Once the reader has good basic knowledge of these two techniques,
they can progress to conducting other stationarity test on their time series
variables.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">How to Perform the
Augmented Dickey-Fuller (ADF) Test</span></b><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">An important assumption of
the DF test is that the error terms are independently and identically
distributed. The ADF test adjusts the DF test to take care of possible serial
correlation in the error terms by adding the lagged difference terms of the
outcome (dependent) variable. For <i>Y<sub>t</sub></i> series, in
conducting the DF test, it is assumed that the error term <i>v<sub>t</sub></i> is
uncorrelated. But in case where it is correlated, Dickey and Fuller have
developed a test, known as the augmented Dickey–Fuller (ADF) test. This test is
conducted by “augmenting” the preceding three model specifications stated above
by adding the lagged values of the dependent variable.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">…so, let’s get started!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">First step: get the Data Analysis Add-in menu<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before you begin, ensure that the <b>DATA ANALYSIS</b> Add-in is in your tool
bar because without it, you cannot perform any regression analysis. To obtain
it follow this guide:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">File</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> >> <b>Options</b> >> <b>Add-ins</b> >> <b>Excel
Options</b> dialog box opens<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under <b>Active Application Add-ins</b>, choose <b>Analysis ToolPak</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In the <b>Manage</b> section, choose <b>Excel
Add-ins</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click <b>Go</b>, then <b>OK<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">If it is correctly done, you should see
this:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMVUq9I88BuQ061Um71xd4RlDWBhvk6NY6zqTDh4c1hQaNKUItwV_IjMQtbccp58OG8qXADMYQTpLyevLOXZ5etQqSekY8M70uSmKQxGV6G2q0pH7tuVEfe_olAGNPkw5OxjyJMKb1tdM/s1600/Excel+Add-in.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel Add-in Dialog Box from cruncheconometrix.com.ng" border="0" data-original-height="678" data-original-width="821" height="330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMVUq9I88BuQ061Um71xd4RlDWBhvk6NY6zqTDh4c1hQaNKUItwV_IjMQtbccp58OG8qXADMYQTpLyevLOXZ5etQqSekY8M70uSmKQxGV6G2q0pH7tuVEfe_olAGNPkw5OxjyJMKb1tdM/s400/Excel+Add-in.png" title="Excel Add-in Dialog Box" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel Add-in Dialog Box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">…and you have the <b>Data Analysis</b> menu to your extreme top-right corner under <b>Data</b> menu.<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXFpcxX8dYLs1va_16HZxT_xGLk0I_h08BR2Am4YmsCgEfEUvXqcWHho4kVMRE9uUbU2SRjYke-XJOURuXD61ONKZSoFKfmYKQO0-ztUC5tl3dhyEFB2hRuiU7iz9AldOqsFZ9YBIiiMA/s1600/Excel+-+Data+Analysis+Add-in.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel Add-in Icon from cruncheconometrix.com.ng" border="0" data-original-height="113" data-original-width="481" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXFpcxX8dYLs1va_16HZxT_xGLk0I_h08BR2Am4YmsCgEfEUvXqcWHho4kVMRE9uUbU2SRjYke-XJOURuXD61ONKZSoFKfmYKQO0-ztUC5tl3dhyEFB2hRuiU7iz9AldOqsFZ9YBIiiMA/s1600/Excel+-+Data+Analysis+Add-in.PNG" title="Excel Add-in Icon" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel Add-in Icon<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Second step: have your data ready <o:p></o:p></span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Using Gujarati and Porter Table 21.1 quarterly
data from 1970q1 to 1991q4. We are </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;"><u>only</u></b><span style="font-family: "times new roman" , serif; font-size: 12pt;">
considering the series of </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> in natural logarithms (because the
variable is initially measured in US$ billions).</span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Remember, that the ADF equation
is given as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">⃤<i>Y<sub>t</sub></i> = δ<i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i>
[5]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Hence, there is need to create 3
additional variables: the <b><i>difference of lnpce</i></b>, the <b><i>lag
of lnpce</i></b> and the <b><i>lagged difference of lnpce</i></b>. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , serif;">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , serif;"> The augmented Dickey–Fuller (ADF) test is conducted
by “augmenting” the model specifications by adding the lagged values of the
dependent variable.</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Here is the data in excel format:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwjEyqUMrgrCUWs58t5yOd0fltg3fzdFVTUj5HF4pQhEewkatlyh6GVE9B9eI-Um_zSbI0uBJ5zv5tUwt4xSGDyEYNsfm1Qzk2sss_7YBGXLFdHV16DtuW0Bswu_z7pL6V8_K5vUW8Bb8/s1600/Excel+-+ADF+Workfile+Data.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: lnpce Workfile from cruncheconometrix.com.ng" border="0" data-original-height="482" data-original-width="504" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwjEyqUMrgrCUWs58t5yOd0fltg3fzdFVTUj5HF4pQhEewkatlyh6GVE9B9eI-Um_zSbI0uBJ5zv5tUwt4xSGDyEYNsfm1Qzk2sss_7YBGXLFdHV16DtuW0Bswu_z7pL6V8_K5vUW8Bb8/s1600/Excel+-+ADF+Workfile+Data.png" title="Excel: lnpce Workfile" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: <i>lnpce</i> Workfile<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Third step: Run the regression in “level”</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Data</b> >> <b>Data Analysis</b>
(dialogue box opens) >> <b>Regression</b>
>> <b>OK >> </b>dialog box opens<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4nt1GMutR0AFTuii9TyEojZE7ftv7m2pBhqG4EnChfhZLlL9Bb6PGa1TqQ1yjbD-AglvuyQz6pdova_Lnf8RFSdeuoThLYdq7tlycMF_hGGQP2TApm9alGaq_0wiv3Y7CrS3sVWYlCL0/s1600/Excel+-+Regression+Dialogue+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Regression dialog box from cruncheconometrix.com.ng" border="0" data-original-height="365" data-original-width="418" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4nt1GMutR0AFTuii9TyEojZE7ftv7m2pBhqG4EnChfhZLlL9Bb6PGa1TqQ1yjbD-AglvuyQz6pdova_Lnf8RFSdeuoThLYdq7tlycMF_hGGQP2TApm9alGaq_0wiv3Y7CrS3sVWYlCL0/s1600/Excel+-+Regression+Dialogue+Box.png" title="Excel: Regression dialog box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Regression dialog box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Put
data range for <i>dlnpce</i> under <b>Input <i>Y</i>
Range</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Put
data range for <i>lnpce_1 </i>and <i>dlnpce_1</i> under <b>Input <i>X</i> Range</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Check
<b>label</b> box<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Check
<b>Confidence</b> <b>Level</b> box<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Check
<b>Output</b> <b>range</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click
<b>OK</b><o:p></o:p></span></div>
<div style="text-align: justify;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(You have simply told <b>Excel</b> to regress the dependent
variable, <i>dlnpce</i>, on the explanatory
variables, <i>lnpce_1 </i>and <i>dlnpce_1</i>), and the output is shown as:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF50vONmwEELQDCNoFDXilCWCaQ5w61NqdJ59Wu5WseKJ185joJVSxtxonBhW8zbxIUpxgY_-9N-bGEzr_sk32w0fPxWO6D_-N-26PtvbvFRzJukGh8Rie55FYxUzXCgKAalbIgcAt_z0/s1600/Excel+-+ADF+%2528intercept%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Augmented Dickey-Fuller Result for nonstationarity from cruncheconometrix.com.ng" border="0" data-original-height="391" data-original-width="522" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF50vONmwEELQDCNoFDXilCWCaQ5w61NqdJ59Wu5WseKJ185joJVSxtxonBhW8zbxIUpxgY_-9N-bGEzr_sk32w0fPxWO6D_-N-26PtvbvFRzJukGh8Rie55FYxUzXCgKAalbIgcAt_z0/s1600/Excel+-+ADF+%2528intercept%2529.png" title="Excel: Augmented Dickey-Fuller Result for nonstationarity" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Augmented Dickey-Fuller Result for nonstationarity<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPqnaE0I1E_6mauyugaGQeFwViQ0Rfva1BFpuBGNeEHQ5jMCqBlNRaKfA5Rzq2nll-CFnsnfkc58GzOQ8H0-5N3MC733vXUTIIhvrM21jEqROJ0t8TKu3F03S_SnRhXV51HUAiinK_ie4/s1600/Excel+-+ADF+%2528intercept%252C+Stata%2529.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Augmented Dickey-Fuller Critical Values from cruncheconometrix.com.ng" border="0" data-original-height="210" data-original-width="474" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPqnaE0I1E_6mauyugaGQeFwViQ0Rfva1BFpuBGNeEHQ5jMCqBlNRaKfA5Rzq2nll-CFnsnfkc58GzOQ8H0-5N3MC733vXUTIIhvrM21jEqROJ0t8TKu3F03S_SnRhXV51HUAiinK_ie4/s1600/Excel+-+ADF+%2528intercept%252C+Stata%2529.PNG" title="Excel: Augmented Dickey-Fuller Critical Values" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Augmented Dickey-Fuller Result Critical Values<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , serif;">Decision:</span></b><span style="background: lightgrey; font-family: "times new roman" , serif;"> The null hypothesis of a unit root cannot be rejected
against the one-sided alternative hypothesis if the computed absolute value of
the <i>tau statistic</i> is lower than the absolute value of the DF
or MacKinnon critical tau values and we conclude that the series is nonstationary;
otherwise (that is, if it is higher), then the series is stationary.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , serif;">Decision: </span></b><span style="background: lightgrey; font-family: "times new roman" , serif;">On the other hand, using the probability value, we
reject the null hypothesis of unit root if the computed probability value is
less than the chosen level of statistical significance.</span><span style="font-family: "times new roman" , serif;"> <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fourth step: Run the regression in “first difference”</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
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</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Having confirmed that <i>lnpce</i> is nonstationary, we need to run
the test again using its first difference. So, the next thing to do is to
generate the first difference of <i>dlnpce</i>
(that is, <i>D.dlnpce</i>) and estimate the
equation. The data for the first difference equation is shown here:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXzejtP6q4GolqmBvFs37sGGFQkpvAle3pOELqhKsvzR4LIm4fKXBTczFQZd1JtbrBwI32HnrQygTYsC3BS_c01Ii2KCQtWIOrZeuqn6PJo5cMDMn6wDOkySXEXnh56YCM_HDxmZ8zN2U/s1600/Excel+-+D.dlnpce+Workfile+Data.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Dlnpce Workfile from cruncheconometrix.com.ng" border="0" data-original-height="359" data-original-width="575" height="249" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXzejtP6q4GolqmBvFs37sGGFQkpvAle3pOELqhKsvzR4LIm4fKXBTczFQZd1JtbrBwI32HnrQygTYsC3BS_c01Ii2KCQtWIOrZeuqn6PJo5cMDMn6wDOkySXEXnh56YCM_HDxmZ8zN2U/s400/Excel+-+D.dlnpce+Workfile+Data.png" title="Excel: Dlnpce Workfile" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: <i>Dlnpce</i> Workfile<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
And the output of the regression is shown as:</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFbcfQ5QGnKK3a2gWZq9dcp9uwXyxg1XZU23Pv8mNlUfbLgGoSEQZgwznEE-0_1x84sXtmfzxMglh7hVk1o_h3pUZ1J8kIZnsVMw_NEb7OUASXR5Stc12LtBnICtKixoSrlikDTyy7BS4/s1600/Excel+-+ADF+%25281st+diff%252C+intercept%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel: Augmented Dickey-Fuller Result for Stationarity from cruncheconometrix.com.ng" border="0" data-original-height="429" data-original-width="551" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFbcfQ5QGnKK3a2gWZq9dcp9uwXyxg1XZU23Pv8mNlUfbLgGoSEQZgwznEE-0_1x84sXtmfzxMglh7hVk1o_h3pUZ1J8kIZnsVMw_NEb7OUASXR5Stc12LtBnICtKixoSrlikDTyy7BS4/s1600/Excel+-+ADF+%25281st+diff%252C+intercept%2529.png" title="Excel: Augmented Dickey-Fuller Result for Stationarity" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel: Augmented Dickey-Fuller Result for Stationarity<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">After unit root
testing, what next?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The outcome of unit root
testing matters for the empirical model to be estimated. The following scenarios
explain the implications of unit root testing for further analysis. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">Scenario 1:
When series under scrutiny are stationary in levels? </span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If <i>pce</i> and <i>pdi</i> are
stationary in levels, that is, they are <i>I</i>(0) series (integrated of
order zero). In this situation, performing a cointegration test is <b><i><u>not</u></i></b> necessary.
This is because any shock to the system in the short run quickly adjusts to the
long run. Consequently, only the long run model should be estimated. That
is, the model should be specified as: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <i>pce<sub>t </sub></i>=
</span><span style="color: black; font-family: "cambria math" , "serif"; font-size: 12.0pt;">𝛂₀</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> + <i>bpdi<sub>t</sub></i> + <i>u<sub>t</sub></i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In essence, the estimation
of short run model is not necessary if series are <i>I</i>(0). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">Scenario 2: When
series are stationary in first differences?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under this scenario, the series are assumed to be non-stationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">One special feature of these series is that they are of the same
order of integration.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under this scenario, the model in question is not entirely useless
although the variables are unpredictable. To verify further the relevance of
the model, there is need to test for cointegration. That is, can we
assume a long run relationship in the model despite the fact that the series
are drifting apart or trending either upward or downward?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If there is cointegration, that means the series in question are
related and therefore can be combined in a linear fashion. This implies that,
even if there are shocks in the short run, which may affect movement in the
individual series, they would converge with time (in the long run).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, there is no long run if series are not cointegrated. This
implies that, if there are shocks to the system, the model is not likely to
converge in the long run.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note that both long run and short run models must be estimated
when there is cointegration.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The estimation will require the use of vector autoregressive (VAR)
model analysis and VECM models.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If there is no cointegration, there is no long run and therefore,
only the short run model will be estimated. That is, run only VAR no VECM
analysis!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">There are however, two prominent cointegration tests for <i>I</i>(I)
series in the literature. They are Engle-Granger cointegration test and
Johansen cointegration test.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The Engle-Granger test is meant for single equation model while
Johansen is considered when dealing with multiple equations. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">Scenario 3: The
series are integrated of different order? </span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Should in case <i>lnpce</i> and <i>lnpdi</i> are
integrated of different orders, like the second scenario, cointegration test is
also required but the use of either Engle-Granger or Johansen cointegration are
no longer valid.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The appropriate cointegration test to apply is the Bounds test for
cointegration and the estimation technique is the autoregressive distributed
lag (ARDL) model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Similar to case 2, if series are not cointegrated based on Bounds
test, we are expected to estimate only the short run. That is run only the ARDL
model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, both the long run and short run models are valid if there
is cointegration. That is run both ARDL and ECM models.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In addition, there are
formal tests that can be carried out to see if despite the behaviour of the
series, there can still be a linear combination or long run relationship or
equilibrium among the series. The existence of the linear combination is what is
known as cointegration. Thus, the regression with <i>I</i>(1) series can
either be spurious or cointegrated. The basic single equation cointegration
tests are Johansen, Engle-Granger and Bounds cointegration tests. These will be
discussed in detail in subsequent tutorials.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In conclusion, I have
discussed what is meant by nonstationary series, how can a series with a unit
root be detected, and how can such series be made useful for empirical
research? You are encouraged to use your data or the sample </span><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">datasets</span></a><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> uploaded
to this bog to practise in order to get more hands-on knowledge.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , serif;"><b>[Watch video on how to perform stationarity test in Excel]</b></span></div>
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<br />Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-20702588374135655042018-02-23T10:20:00.000+01:002018-02-26T08:03:24.961+01:00Time Series Analysis (Lecture 3): How to Perform Stationarity Test in Stata<br />
<h2 style="text-align: center;">
<b style="text-align: justify;"><span style="font-family: "times new roman" , serif; font-size: 14pt; line-height: 107%;">What is Stationarity in Time Series Analysis?</span></b></h2>
<h2 style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 107%;">In econometrics, time series data are frequently used and they
often pose distinct problems for econometricians. As it will be discussed with
examples, most empirical work based on time series data assumes that the
underlying series is stationary. Stationarity of a series (that is, a variable)
implies that its mean, variance and covariance are constant over time. That is,
these do not vary systematically over time. In order words, they are <i>time</i> <i>invariant</i>.
However, if that is not the case, then the series is nonstationary. We will
discuss some possible scenarios where two series, <i>Y</i> and <i>X</i>,
are nonstationary and the error term, <i>u</i>, is also nonstationary. In
that case, the error term will exhibit autocorrelation. Another likely scenario
is where <i>Y</i> and <i>X</i> are nonstationary, but <i>u</i> is
stationary. The implications of this will also be explored. In time series
analysis, the words <i>nonstationary</i>, <i>unit root</i> or <i>random
walk</i> <i>model</i> are used synonymously. In essence, of a series
is considered to be nonstationary, it implies that such exhibit a unit root and
exemplifies a random walk series.<o:p></o:p></span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Regressing two series that
are nonstationary, likewise, yields a spurious (or nonsense) regression. That
is, a regression whose outcome cannot be used for inferences or forecasting. In
short, such results should not be taken seriously and must be discarded. A
stationary series will tend to return to its mean (called </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">mean
reversion</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">) and fluctuations around this mean (measured by its variance) will
have a broadly constant breadth. But if a time series is not stationary in the
sense just explained, it is called a nonstationary time series such will have a
time-varying mean or a time-varying variance or both. In summary, a stationary
time series is important because if such is nonstationary, its behaviour can be
studied only for the time period under consideration. That is, each set of time
series data will therefore be for a particular episode. As a result, it is not
possible to generalise its relevance to other time periods. Therefore, for the
purpose of forecasting, such (nonstationary) time series may be of little
practical value</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">How to detect unit
root in a series?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In a bivariate (2 variables)
model or that involving multiple variables (called a multiple regression
model), it is assumed that all the variables are stationary at level (that is,
the order of integration of each of the variable is zero, <i>I</i>(0). It
is important to state at this point, that the order of integration of a series
in a regression model is determined by the outcome of a unit root test (or
stationarity test). If the series is stationary at level after performing unit
root test, then it is <i>I</i>(0), otherwise it is <i>I</i>(<i>d</i>)
where <i>d</i> represents the number of times the series is
differenced before it becomes stationary. But what if the assumption of <i>stationarity
at level</i> of the series in a bivariate or multiple regression model is
relaxed and we consequently allow for a unit root in each of the variables in
the model, how can this be corrected? In general, this would require a
different treatment from a conventional regression with stationary variables
at <i>I</i>(0).<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In particular, we focus on a
class of linear combination of unit root processes known as cointegrated
process. The generic representation for the order of integration of series
is <i>I</i>(<i>d</i>) where <i>d</i> is the number of
differencing to render the series stationary. Hence, a stationary series at
level, <i>d</i> = 0 is a series with an <i>I</i>(0) process.
Although, for any non-stationary series, <i>‘d’</i> can assume any
value greater than zero, however, in applied research, only the unit root
process of <i>I</i>(1) process is allowed, otherwise such series with
higher order of integration (<i>d</i> > 1) should be excluded in the
model as no meaningful policy implications or relevance can be drawn from such
series. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Here is an example of a
bivariate linear regression model:<o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">
<i>Y<sub>t </sub></i>= </span><span style="color: black; font-family: "cambria math" , "serif"; font-size: 12.0pt;">𝛂₀</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> + <i>bX<sub>t</sub></i> + <i>u<sub>t</sub></i>
[1]<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Assume <i>Y<sub>t</sub> </i>and<i> X<sub>t</sub></i>
are two random walk models that are <i>I</i>(1) processes and are
independently distributed as: <o:p></o:p></span></div>
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<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">
<i>Y<sub>t </sub></i>= ρ<i>Y<sub>t-1</sub></i> + <i>v<sub>t</sub></i>,
-1
≤ ρ ≤ 1
[2]
<o:p></o:p></span></div>
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<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">
<i>X<sub>t </sub></i>= ղ<i>X<sub>t-1</sub></i> + <i>e<sub>t</sub></i>,
-1 ≤ ղ ≤
1
[3]<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">and <i>v<sub>t</sub></i> and <i>e<sub>t</sub></i> have
zero mean, a constant variance and are orthogonal (these are <i>white
noise</i> error terms). <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">We also assumed that <i>v<sub>t</sub></i> and <i>e<sub>t</sub></i> are
serially uncorrelated as well as mutually uncorrelated. As stated in [2] and
[3], both these time series are nonstationary; that is, they are <i>I</i>(1)
or exhibit stochastic trends. Suppose we regress <i>Y<sub>t</sub> </i>on<i> X<sub>t</sub></i>.
Since <i>Y<sub>t</sub> </i>on<i> X<sub>t</sub></i> are
uncorrelated <i>I</i>(1) processes, the <i>R</i><sup>2</sup> from
the regression of <i>Y </i>on<i> X</i> should tend to zero;
that is, there should not be any relationship between the two variables.
Equations [2] and [3] resemble the Markov first-order autoregressive model. If ρ
and ղ = 1, the equations become a random walk model without drift. If
ρ and ղ are in fact 1, then a unit root problem surfaces, that is, a
situation of nonstationarity; because we already know that in this case the
variance of <i>Y<sub>t</sub></i> is not stationary. The name unit
root is due to the fact that ρ = 1. Again, the terms nonstationary, random
walk, and unit root can be treated as synonymous. If, however, |ρ| ≤ 1, and
|ղ| ≤ 1, that is if their absolute values are less than one, then it can
be shown that both series <i>Y<sub>t</sub></i> and <i>X<sub>t</sub></i> are
stationary. In practice, then, it is important to find out if a time series
possesses a unit root.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Given equations [2] and [3],
there should be no systematic relationship between <i>Y<sub>t</sub></i> and <i>X<sub>t</sub></i> as
they both drift away from equilibrium (i.e. they do not converge), and
therefore, we should expect that an ordinary least squares (OLS) estimate
of <i>b</i> should be close to zero, or insignificantly different
from zero, at least as the sample size increases. But this is not usually the
case. The fitted coefficients in this case may be statistically significant
even when there is no true relationship between the dependent variable and the
regressors. This is regarded as a spurious regression or correlation where, in
the case of our example, <i>b</i> takes any value randomly, and
its <i>t</i>-statistic indicates significance of the estimate.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">But how can unit root
be detected?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> There are some clues that tell you if a series is
nonstationary and if the regression of bivariate or multivariate relationships
are spurious. Some of these are:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; text-align: justify; text-indent: -18.0pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">1. Do a graphical plot of the series to visualise
the nature. Is it trending upwards or downwards? Does it exhibit a
mean-reversion or not? Or are there fluctuations around its mean?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; text-align: justify; text-indent: -18.0pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">2. Or carry out a regression analysis on two series
and observe the <i>R</i><sup>2</sup>. If it is above 0.9, it may suggest
that the variables are nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; text-align: justify; text-indent: -18.0pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">3. The rule-of-thumb: if the <i>R</i><sup>2</sup> obtained
from the regression is higher than the Durbin Watson (DW) statistic. The low DW
statistic evidences positive first order auto-correlation of the error terms.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Using </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j?usp=sharing" target="_blank">Gujarati and Porter</a></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Table 21.1 quarterly data of 1970q1 to 1991q4, examples of
nonstationary series and spurious regression can be seen from the <i>pce</i>, <i>pdi</i> and <i>gdp</i> relationship.
Since the series are measured in billions of US dollars, the natural logarithms
of the variables will be used in analysing their essential features.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Nonstationary series:</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> the
graphical plot of the three variables shows an upward trend and none of the
variables revert to their means. That is, all three variables do not exhibit
mean reversions. That clearly tells us that the series are nonstationary.</span><b><span style="background: lightgrey; font-size: 11pt;"><o:p> </o:p></span></b></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfmU7ndJKvAfN365xa5sAcXZpYBKJP8MIiBqeVC3w_I3BKbulAY3CEfyAhYxXDckZ9E3EN2n-XXVs5pmN1ImPky0PS35b_iIRUFDVd0JbRKgJ11MXbZBtRPG-HLNGT4G-Wed9B4N1Trjo/s1600/Stata+-+Nonstationary+series.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: Plot of nonstationary series from cruncheconometrix.com.ng" border="0" data-original-height="415" data-original-width="493" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfmU7ndJKvAfN365xa5sAcXZpYBKJP8MIiBqeVC3w_I3BKbulAY3CEfyAhYxXDckZ9E3EN2n-XXVs5pmN1ImPky0PS35b_iIRUFDVd0JbRKgJ11MXbZBtRPG-HLNGT4G-Wed9B4N1Trjo/s1600/Stata+-+Nonstationary+series.png" title="Stata: Plot of nonstationary series" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8px;">Stata: Plot of nonstationary series<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span style="background: lightgrey; color: black; font-size: 11.0pt;"><o:p> </o:p></span></b><b><span style="background: lightgrey; font-size: 11pt;">Note:</span></b><span style="background: lightgrey; font-size: 11pt;"> Use
this syntax to generate the graph: <i>line
lnpce lnpdi lngdp qtrly, legend(size(medsmall))</i></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">What is a spurious
regression? </span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Sometimes we expect to find no relationship between two variables,
yet a regression of one on the other variable often shows a significant
relationship. This situation exemplifies the problem of spurious, or nonsense,
regression. The regression of <i>lnpce</i> on <i>lnpdi</i> shows
how spurious regressions can arise if time series are not stationary. As
expected, because both variables are nonstationary, the result evidences that a
spurious regression has been undertaken.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">But how do we know this? Take
a look at the <i>R</i><sup>2</sup> the value of <b>0.9944</b> is
higher than the Durbin Watson statistic of <b>0.57</b>. So, whenever
the <i>R</i><sup>2</sup> > DW, a spurious regression has occurred
because the variables are nonstationary.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i style="mso-bidi-font-style: normal;"><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Stata syntax: regress lnpce lnpdi</span></i><i style="mso-bidi-font-style: normal;"><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></i></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRM4xhjTKfhaLP33oxBtJ_U_AnOLjX5-Sf5KEMWsdkauA2pr4SnH0oYGDitlo3BtVkRALB8HcE7sM-H3RQXlmyCHBYiA8Mnj6LLMx1I3SIihBbAmZi_z65mvAvyYzdyZUCpku4cSwjjU4/s1600/Stata+-+Spurious+regression.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: Example of a spurious regression from cruncheconometrix.com.ng" border="0" data-original-height="327" data-original-width="578" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRM4xhjTKfhaLP33oxBtJ_U_AnOLjX5-Sf5KEMWsdkauA2pr4SnH0oYGDitlo3BtVkRALB8HcE7sM-H3RQXlmyCHBYiA8Mnj6LLMx1I3SIihBbAmZi_z65mvAvyYzdyZUCpku4cSwjjU4/s1600/Stata+-+Spurious+regression.png" title="Stata: Example of a spurious regression" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: Example of a spurious regression<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">As you can see, the coefficient
of </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">lnpdi</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is highly statistically significant, and the </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">R</i><sup style="font-family: "Times New Roman", serif;">2</sup><span style="font-family: "times new roman" , serif; font-size: 12pt;"> value
is statistically significantly different from zero. From these results, you may
be tempted to conclude that there is a significant statistical relationship
between both variables, whereas </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">a priori</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> there may or
may </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">not</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> be none. This is simply the phenomenon of </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">spurious
or nonsense regression</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">, first discovered by Yule (1926). He showed that
(spurious) correlation could persist in nonstationary time series even if the
sample is very large. That there is something wrong in the preceding regression
is suggested by the extremely low Durbin–Watson value, which suggests very
strong first-order autocorrelation. According to Granger and Newbold, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">R</i><sup style="font-family: "Times New Roman", serif;">2</sup><span style="font-family: "times new roman" , serif; font-size: 12pt;"> >
DW is a good rule of thumb to suspect that the estimated regression is
spurious, as in the given example.</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">Why is it important
to test for stationarity?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">We usually consider a
nonstationary series for the following reasons:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">To evaluate the behaviour of series over time. Is the series
trending upward or downward? This can be verified from performing a
stationarity test. In other words, the test can be used to evaluate the
stability or predictability of time series. If a series is nonstationary, that
means the series is unstable or unpredictable and therefore may not be valid
for inferences, prediction or forecasting.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">To know how a series responds to shocks requires carrying out a
stationarity test. If such series is nonstationary, the impact of shocks to the
series are more likely to be permanent. Consequently, if a series is
stationary, impact of shocks will be temporary or brief.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">How to correct for
nonstationarity?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">What can be done with
nonstationarity in a time series knowing that performing OLS on such a model
yields spurious regression?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">The Unit Root Test</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">We begin with equations [2]
and [3] which are unit root (stochastic) processes with white noise error
terms. If the parameters of the models are equal to 1, that is, in the case of
the unit root, both equations become random walk models without drift, which we
know is a nonstationary stochastic process. So, what can be done to correct
this? For instance, for equation [2], simply regress <i>Y<sub>t</sub></i> on
its (one-period) lagged value <i>Y<sub>t−1</sub></i> and find out if
the estimated ρ is statistically equal to 1? If it is, then <i>Y<sub>t</sub></i> is
nonstationary. Repeat same for the <i>X<sub>t</sub></i> series. This
is the general idea behind the unit root test of stationarity.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">For theoretical reasons,
equation [2] is manipulated as follows: Subtract <i>Y<sub>t−1</sub></i> from
both sides of [2] to obtain:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">
<i>Y<sub>t </sub></i>- <i>Y<sub>t-1 </sub></i>= ρ<i>Y<sub>t-1</sub></i> - <i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i>
[4]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">
= (ρ - 1)<i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">
<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">and this can be stated
alternatively as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 72.0pt; text-align: justify; text-indent: 36.0pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">⃤ <i>Y<sub>t</sub></i> = δ<i>Y<sub>t-1 </sub></i>+ <i>v<sub>t</sub></i>
[5]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">where δ = (ρ − 1)
and ⃤, as usual, is the first-difference operator. In practice, therefore,
instead of estimating [2], we estimate [5] and test the null hypothesis
that δ = 0. If δ = 0, then ρ = 1, that is we have a unit
root, meaning the time series under consideration is nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before we proceed to
estimate [5], it may be noted that if δ = 0, [5] will become:<o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">⃤ <i>Y<sub>t</sub></i> = <i>Y<sub>t-1</sub></i> - <i>Y<sub>t-1 </sub></i>= <i>v<sub>t</sub></i>
[6]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">(Remember to do the same
for <i>X<sub>t</sub></i> series)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since <i>v</i><sub>t</sub> is
a white noise error term, it is stationary, which means that <b>the first
difference of a random walk time series is stationary</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjq6WNqtp7m09AodUmyW3P4Y37GKoxi-UpGr6ybdPCipKTKa2JWWJedWyAt4KgrYtEIN5iopf3r1L5YZcmdVvtzwjht597Do7gFDoS4pa6w5GTWrT0qYbAXuWP_jjFsis1h5xb48CGSVfo/s1600/Stata+-+Stationary+series.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: Example of a stationary series from cruncheconometrix.com.ng" border="0" data-original-height="369" data-original-width="516" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjq6WNqtp7m09AodUmyW3P4Y37GKoxi-UpGr6ybdPCipKTKa2JWWJedWyAt4KgrYtEIN5iopf3r1L5YZcmdVvtzwjht597Do7gFDoS4pa6w5GTWrT0qYbAXuWP_jjFsis1h5xb48CGSVfo/s1600/Stata+-+Stationary+series.png" title="Stata: Example of a stationary series" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: Example of a stationary series<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Visual observation of the
differenced series shows that the three variables are stationary around the mean.
They all exhibit constant mean-reversions. That is, they fluctuate around 0. If
we are to draw a trend line, such a line will be horizontal at 0.01.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Okay, having said all that.
Let us return to estimating equation [5]. This is quite simple, all that is
required is to take the first differences of <i>Y<sub>t</sub></i> and
regress on <i>Y<sub>t−1</sub></i> and see if the estimated slope
coefficient in this regression is statistically different from is zero or not.
If it is zero, we conclude that <i>Y<sub>t</sub></i> is
nonstationary. But if it is negative, we conclude that <i>Y<sub>t</sub></i> is
stationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";"> Since
δ = (ρ − 1), for stationarity ρ must be less than one. For this to happen δ
must be negative!</span><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The only question is which
test do we use to find out if the estimated coefficient of <i>Y<sub>t−1</sub></i> in
[5] is zero or not? You might be tempted to say, why not use the usual <i>t</i> test?
Unfortunately, under the null hypothesis that δ = 0 (i.e., ρ = 1), the <i>t</i> value
of the estimated coefficient of <i>Y<sub>t−1</sub></i> does not follow
the <i>t</i> distribution even in large samples; that is, it does not
have an asymptotic normal distribution.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">What is the alternative?
Dickey and Fuller (DF) have shown that under the null hypothesis that δ = 0,
the estimated <i>t</i> value of the coefficient of <i>Y<sub>t−1</sub></i> in
[5] follows the <b><i>τ (tau)</i></b> statistic. These authors have
computed the critical values of the <i>tau statistic</i> on the basis
of Monte Carlo simulations.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";"> Interestingly,
if the hypothesis that δ = 0 is rejected (i.e., the time series is stationary),
we can use the usual (Student’s) <i>t</i> test.</span><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The unit root test can be
computed under three (3) different null hypotheses. That is, under different
model specifications such as if the series is a:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">1. random walk (that is, model has no
constant, no trend)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">2. random walk with drift (that is, model
has a constant)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">3. random walk with drift and a trend (that
is, model has a constant and trend)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In all cases, the null
hypothesis is that δ = 0; that is, there is a unit root and the alternative
hypothesis is that δ is less than zero; that is, the time series is stationary.
If the null hypothesis is rejected, it means that <i>Y<sub>t</sub></i> is
a stationary time series with zero mean in the case of [5], that <i>Y<sub>t</sub></i> is
stationary with a nonzero mean in the case of a random walk with drift model,
and that <i>Y<sub>t</sub></i> is stationary around a deterministic
trend in the case of random walk with drift around a trend.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">It is extremely important to
note that the critical values of the <i>tau test</i> to test the
hypothesis that δ = 0, are different for each of the preceding three
specifications of the DF test, which are now computed by all econometric
packages. In each case, if the computed absolute value of the <i>tau
statistic</i> (|τ|) exceeds the DF or MacKinnon critical <i>tau
values</i>, the null hypothesis of a unit root is rejected, in order words the
time series is stationary. On the other hand, if the computed |τ| does not
exceed the critical <i>tau</i> value, we fail to reject the null
hypothesis, in which case the time series is nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";"> Students
often get confused in interpreting the outcome of a unit root test. For
instance, if the calculated <i>tau</i> statistic is -2.0872 and the
MacKinnon <i>tau</i> statistic is -3.672, you cannot reject the null
hypothesis. Hence, the conclusion is that the series is nonstationary. But if
the calculated <i>tau</i> statistic is -5.278 and the MacKinnon <i>tau</i> statistic
is -3.482, you reject the null hypothesis in favour of the alternative. Hence,
the conclusion is that the series is stationary.</span><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">*Always use the appropriate critical τ values
for the indicated model specification.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">How to Perform Unit
Root Test in Stata (see here for <a href="https://cruncheconometrix.blogspot.com.ng/2018/02/time-series-analysis-lecture-3-how-to.html" target="_blank">EViews</a>)</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: yellow; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Example
dataset is from </span><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j?usp=sharing" target="_blank">Gujarati and Porter</a></span><span style="background: yellow; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Table
21.1</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Several tests have
been developed in the literature to test for unit root. Prominent among these
tests are Augmented Dickey-Fuller, Phillips-Perron, Dickey-Fuller Generalised
Least Squares (DF-GLS) and so on. But this tutorials limits testing to the use
of ADF and PP tests. Once the reader has good basic knowledge of these two
techniques, they can progress to conducting other stationarity test on their
time series variables.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">How to Perform the
Augmented Dickey-Fuller (ADF) Test</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">An important assumption of
the DF test is that the error terms are independently and identically
distributed. The ADF test adjusts the DF test to take care of possible serial
correlation in the error terms by adding the lagged difference terms of the
outcome (dependent) variable. For <i>Y<sub>t</sub></i> series, in
conducting the DF test, it is assumed that the error term <i>v<sub>t</sub></i> is
uncorrelated. But in case where it is correlated, Dickey and Fuller have
developed a test, known as the augmented Dickey–Fuller (ADF) test. This test is
conducted by “augmenting” the preceding three model specifications stated above
by adding the lagged values of the dependent variable.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">As mentioned earlier,
approaches will be limited to using the ADF and PP tests. Either of these tests
can be used and when both are used, the reader can compare the outcomes to see
if there are similarities or differences in the results.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Using Gujarati and Porter Table 21.1 quarterly data on <i style="mso-bidi-font-style: normal;">pce, pdi </i>and<i style="mso-bidi-font-style: normal;"> gdp</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Load data into Stata<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">We are considering the pair of <i>lnpce</i> and <i>lnpdi</i> in
natural logarithms (because variables are measured in US$ billions)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">4.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="background: white; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Inform Stata that you are about to perform a time series
analysis by typing this code into the <b style="mso-bidi-font-weight: normal;">Command</b>
box: <i style="mso-bidi-font-style: normal;">tsset qtrly</i></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 39.0pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 39.0pt; text-align: justify;">
<span style="background: white; font-family: "times new roman" , "serif"; font-size: 12.0pt;">and
you will obtain this:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQGW2AzRXZxu5tcar0Iruo8-vqJ117sH82KJXq9DVPYWzfQNTWVLepRA3uZWOvxNjmf0hmKCSUFrp9jYn6g2btMMbQrcCCL1I-B8nNjyIGMftmePjxhW9PqEcW55oOdOldTOVgbjqjsW4/s1600/Stata+-+tsset+command.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: tsset command from cruncheconometrix.com.ng" border="0" data-original-height="51" data-original-width="357" height="45" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQGW2AzRXZxu5tcar0Iruo8-vqJ117sH82KJXq9DVPYWzfQNTWVLepRA3uZWOvxNjmf0hmKCSUFrp9jYn6g2btMMbQrcCCL1I-B8nNjyIGMftmePjxhW9PqEcW55oOdOldTOVgbjqjsW4/s320/Stata+-+tsset+command.png" title="Stata: tsset command" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: tsset command<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="margin-left: 39.0pt; text-align: justify;">
<span style="background: white; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_11"
o:spid="_x0000_i1034" type="#_x0000_t75" style='width:267.75pt;height:38.25pt;
visibility:visible;mso-wrap-style:square'>
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o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="background: white; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: white; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br />Stata now recognises
that you are about conducting a time series analysis using quarterly data from
1<sup>st</sup> quarter of 1970 to the 4<sup>th</sup> quarter of 1991. If you
don’t issue this command, Stata will not run your analysis.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: lightgrey; font-family: "times new roman" , "serif";">Note: if you
are using a yearly data, type the syntax <i style="mso-bidi-font-style: normal;">tsset
year</i> and if it is a monthly data type <i style="mso-bidi-font-style: normal;">tsset
month</i>.</span><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The Augmented Dickey-Fuller
(ADF) Test</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Unit root test for <i>lnpce</i>:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go
to <b>Statistics</b> >> <b>Time series</b> >> <b style="mso-bidi-font-weight: normal;">Tests >> Augmented Dickey-Fuller unit
root test </b>>> dialog box opens <b><o:p></o:p></b></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTHdc6WBc54dSpgKCGEFSzZuM0QyCERaNr5IpL7oanwIpbGNri3ltXcgYfQ4j8X4dHgjZdtA8E-_5TzNPzZgO1M_v72bX1EbCnuHOBVgWwd9rMyXmHkM9YlBC3repM7zZIEDovtohZbw4/s1600/Stata+-+ADF+dialog+box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: Augmented Dickey-Fuller dialog box from cruncheconometrix.com.ng" border="0" data-original-height="327" data-original-width="429" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTHdc6WBc54dSpgKCGEFSzZuM0QyCERaNr5IpL7oanwIpbGNri3ltXcgYfQ4j8X4dHgjZdtA8E-_5TzNPzZgO1M_v72bX1EbCnuHOBVgWwd9rMyXmHkM9YlBC3repM7zZIEDovtohZbw4/s1600/Stata+-+ADF+dialog+box.png" title="Stata: Augmented Dickey-Fuller dialog box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: Augmented Dickey-Fuller Dialog Box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div align="center" class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1033"
type="#_x0000_t75" style='width:321.75pt;height:245.25pt;visibility:visible;
mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image008.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span></b><b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: left;">
<span style="font-family: "symbol"; font-size: 12pt; text-align: justify; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify; text-indent: -14.2pt;">Under <b>Variable</b>,
select <i>lnpce</i></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under <b style="mso-bidi-font-weight: normal;">Options</b>, the
choice of model is very important since the distribution statistic under the
null hypothesis differs across these three cases. Therefore, specify whether to
<b style="mso-bidi-font-weight: normal;">“suppress constant term”, “include trend
term”, </b>or<b style="mso-bidi-font-weight: normal;"> “include drift term”</b>. Thus,
our demonstration will involve these options.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If the <b style="mso-bidi-font-weight: normal;">“display regression
table”</b> box is checked, Stata reports the test statistic together with the
estimated test regression. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Depending on the structure of your data, include the number of <b style="mso-bidi-font-weight: normal;">“lagged differences”</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Decision:</span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";"> The null hypothesis of a unit root is rejected
against the one-sided alternative hypothesis if the computed absolute value of
the <i>tau statistic</i> exceeds the DF or MacKinnon critical tau
values and we conclude that the series is stationary; otherwise (that is, if it
is lower), then the series is non-stationary.</span><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Decision: </span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Another way of stating this is that in failing to
reject the null hypothesis of a unit root, the computed τ value should be <b><i><u>more</u></i></b> negative
than the critical τ value. Since in general δ is expected to be negative, the
estimated τ statistic will have a negative sign. Therefore, a large negative τ
value is generally an indication of stationarity.</span><span style="color: black; font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">Decision: </span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif";">On the other hand, using the probability value, we
reject the null hypothesis of unit root if the computed probability value is
less than the chosen level of statistical significance.</span><span style="color: black; font-family: "times new roman" , "serif";"> <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Having specified the <b style="mso-bidi-font-weight: normal;">“suppress constant term”</b> and
checked the <b style="mso-bidi-font-weight: normal;">“display regression table”</b>
box, the ADF result is given as: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDXQFmhsjBnIDKKb-8hezgsT-7Yhgt4DXv7K7Jb2Pu0V63883gPnOQ4NuygVwwq-2XRA34HijRNjSqFclqX29OJpFc-LU3B5R7eNd2m7gHu_qh6m4QY1y54JbM-rFu84t2b_UfBOOyPR0/s1600/Stata+-+ADF+%2528suppress+constant%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: ADF test, level, "suppress constant" option from cruncheconometrix.com.ng" border="0" data-original-height="214" data-original-width="592" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDXQFmhsjBnIDKKb-8hezgsT-7Yhgt4DXv7K7Jb2Pu0V63883gPnOQ4NuygVwwq-2XRA34HijRNjSqFclqX29OJpFc-LU3B5R7eNd2m7gHu_qh6m4QY1y54JbM-rFu84t2b_UfBOOyPR0/s1600/Stata+-+ADF+%2528suppress+constant%2529.png" title="Stata: ADF test, level, "suppress constant" option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: ADF test, level, "suppress constant" option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;">Following similar
procedures, the select </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-indent: -14.2pt;">“include trend
term”</b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;"> for the ADF unit root test yields:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5CZszM-n7qTC4C6Z0DA3RuAOgOKINbO8NCQkpoIBMh0bc5K29WkqFsXIBcb-0CBgqVtW_ArfchE1LX6GSIJMC0JSjsCj2t6rlEc836M6FPfLOV5rEiwJZL_HDVNFeJIKWWnmi0D_YhA4/s1600/Stata+-+ADF+%2528with+trend%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: ADF Test, level, "include trend term" option from cruncheconometrix.com.ng" border="0" data-original-height="279" data-original-width="593" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5CZszM-n7qTC4C6Z0DA3RuAOgOKINbO8NCQkpoIBMh0bc5K29WkqFsXIBcb-0CBgqVtW_ArfchE1LX6GSIJMC0JSjsCj2t6rlEc836M6FPfLOV5rEiwJZL_HDVNFeJIKWWnmi0D_YhA4/s1600/Stata+-+ADF+%2528with+trend%2529.png" title="Stata: ADF Test, level, "include trend term" option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: ADF test, level, "include trend term" option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The result for the “</span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">include drift term</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">” option for the ADF
unit root test is shown below:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2LMLgljvImH1Lqni8BLNoCEQiniUi57Irx-ffSZUBINRbN_LTUj49ZbwgrhY_9DysJ6jmSd90zB8VspVv4qWSuDUGwvdXvQmBB2Fm_bpRYO9y6vp28Wq0FAMi24dBJAmoTduB_p6EPKs/s1600/Stata+-+ADF+%2528with+drift%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: ADF test, level, "include drift term" option from cruncheconometrix.com.ng" border="0" data-original-height="282" data-original-width="595" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2LMLgljvImH1Lqni8BLNoCEQiniUi57Irx-ffSZUBINRbN_LTUj49ZbwgrhY_9DysJ6jmSd90zB8VspVv4qWSuDUGwvdXvQmBB2Fm_bpRYO9y6vp28Wq0FAMi24dBJAmoTduB_p6EPKs/s1600/Stata+-+ADF+%2528with+drift%2529.png" title="Stata: ADF test, level, "include drift term" option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: ADF test, level, "include drift term" option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
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<div class="MsoNoSpacing" style="text-align: left;">
<span style="background-color: #cccccc;"><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">Note:</b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> the null hypothesis for the three ADF specifications </span><u style="font-family: "Times New Roman", serif; font-size: 12pt; font-weight: bold; text-align: justify;"><i>cannot</i> </u><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">be rejected at the 5% level, confirming that </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"><i>lnpce</i></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> is nonstationary which is a confirmation of the graphical plot. Notice that the interpolated Dickey-Fuller </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">tau </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">statistic differ across all specifications. </span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: lightgrey; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: lightgrey; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Do same for the <i>lnpdi</i> series.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Having confirmed that <i style="mso-bidi-font-style: normal;">lnpce</i> is nonstationary, we need to run
the tests again using its first difference. So, the next thing to do is to
generate the first difference of <i style="mso-bidi-font-style: normal;">lnpce</i>
and run the test across the three specifications.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">To generate the difference
variable, the syntax is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i style="mso-bidi-font-style: normal;"><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">generate dlnpce=d.lnpce<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">1<sup>st</sup> difference with <b>“suppress constant” </b>option
result:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3BGd4Da0MV-fRGDMt_96qfYR12rJS6mAeKuttvZsFYzfjN5e3DTdIJAOkRsyOu5-LWJWQ2kIpWGpNJ55A8gxqVerb_axTUm70PqnhjjyZZ_QH_EidHVO861PoLuAv3BsF6FoFMLYOkXo/s1600/Stata+-+ADF+%25281st+diff%252C+none%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: ADF test, 1st difference, "suppress constant" option from cruncheconometrix.com.ng" border="0" data-original-height="222" data-original-width="584" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3BGd4Da0MV-fRGDMt_96qfYR12rJS6mAeKuttvZsFYzfjN5e3DTdIJAOkRsyOu5-LWJWQ2kIpWGpNJ55A8gxqVerb_axTUm70PqnhjjyZZ_QH_EidHVO861PoLuAv3BsF6FoFMLYOkXo/s1600/Stata+-+ADF+%25281st+diff%252C+none%2529.png" title="Stata: ADF test, 1st difference, "suppress constant" option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: ADF test, 1st difference, "suppress constant" option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">1</span><sup style="font-family: "Times New Roman", serif;">st</sup><span style="font-family: "times new roman" , serif; font-size: 12pt;"> difference
with </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">“include trend term” </b><span style="font-family: "times new roman" , serif; font-size: 12pt;">option result: </span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOK0UHGcYMaFL0yIYNBONVJccfvQbGAatveJ91IeCAtiQMBMXyjz_8Ftes1v-w0oMU69w_LWdmk5UrMJOEtI-yYx5T0hJwmU7OAPS_GM3BvJfGLv1xcFpbTtlOlv2swtm0xqE86I-YBcQ/s1600/Stata+-+ADF+%25281st+diff%252C+with+trend%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: ADF test, 1st difference, "include trend term" option from cruncheconometrix.com.ng" border="0" data-original-height="276" data-original-width="595" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOK0UHGcYMaFL0yIYNBONVJccfvQbGAatveJ91IeCAtiQMBMXyjz_8Ftes1v-w0oMU69w_LWdmk5UrMJOEtI-yYx5T0hJwmU7OAPS_GM3BvJfGLv1xcFpbTtlOlv2swtm0xqE86I-YBcQ/s1600/Stata+-+ADF+%25281st+diff%252C+with+trend%2529.png" title="Stata: ADF test, 1st difference, "include trend term" option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: ADF test, 1st difference, "include trend term" option<br />
Source: CrunchEconometrix </td></tr>
</tbody></table>
<div class="MsoListParagraph">
<span style="font-family: "symbol"; font-size: 12pt; text-align: justify; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify; text-indent: -14.2pt;">1<sup>st</sup> difference with <b>“include drift term”</b> option result:</span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEighoYGs-XjrK1JLxo8dowav1egCbS3YlhjfCNcyf-XmPuWfoNY04BzBwFqSK_292FwiPyBvQATLmi73Uef2pwr8VVJ6vziq9ZjtsXZ93-kCgWdTiyCkf-BXmdp8YBBIMh6lb9yDGcnL2k/s1600/Stata+-+ADF+%25281st+diff%252C+with+drift%2529.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: ADF test, 1st difference, "include drift term" option from cruncheconometrix.com.ng" border="0" data-original-height="278" data-original-width="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEighoYGs-XjrK1JLxo8dowav1egCbS3YlhjfCNcyf-XmPuWfoNY04BzBwFqSK_292FwiPyBvQATLmi73Uef2pwr8VVJ6vziq9ZjtsXZ93-kCgWdTiyCkf-BXmdp8YBBIMh6lb9yDGcnL2k/s1600/Stata+-+ADF+%25281st+diff%252C+with+drift%2529.PNG" title="Stata: ADF test, 1st difference, "include drift term" option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: ADF test, 1st difference, "include drift term" option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoListParagraph">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">Having done the
first-difference analysis, and the trend term is not statistically significant,
we conclude that the null hypotheses of a unit root is rejected and that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">lnpce</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> series is difference-stationary.
That is, </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;"><i>lnpce</i> is stationary at 1<sup>st</sup> difference with
a constant </b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">hence, carrying out a </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">“2<sup>nd</sup> difference”</b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> test
is unnecessary.</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify; text-indent: -14.2pt;">
<span style="background: lightgrey; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Again, do same for the <i style="mso-bidi-font-style: normal;">d<span style="mso-bidi-font-style: italic;">lnpdi</span></i> series.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">How to Perform the
Phillips-Perron (PP) Test</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Phillips and Perron use
nonparametric statistical methods to take care of the serial correlation in the
error terms <b><i><u>without</u></i></b> adding lagged difference
terms. Procedures for testing for unit root using the PP test differs a bit
from that of ADF.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Statistics</b> >> <b>Time series</b> >>
<b style="mso-bidi-font-weight: normal;">Tests >> Phillips-Perron unit root
test </b>>> dialog box opens<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4h-kfZ-0opKT7KyiE-mCqqfgMMgaK4Xeo1EdAkusltsKZ5eqJJCjYR6k78LjsJWSZzpgwjuIEKMpkc7Npmm59ok2nOUqxwfG0ny4yrn-BDSl4msK3ke4WiZWw5g8_er2nPYL1v3-rQrU/s1600/Stata+-+PP+dialog+box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata: Phillips-Perron Dialog Box from cruncheconometrix.com.ng" border="0" data-original-height="327" data-original-width="389" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4h-kfZ-0opKT7KyiE-mCqqfgMMgaK4Xeo1EdAkusltsKZ5eqJJCjYR6k78LjsJWSZzpgwjuIEKMpkc7Npmm59ok2nOUqxwfG0ny4yrn-BDSl4msK3ke4WiZWw5g8_er2nPYL1v3-rQrU/s1600/Stata+-+PP+dialog+box.png" title="Stata: Phillips-Perron Dialog Box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata: Phillips-Perron Dialog Box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</o:p></span></div>
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<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_1319"
o:spid="_x0000_i1025" type="#_x0000_t75" style='width:291.75pt;height:245.25pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image019.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fill the details of the
variables in the <b style="mso-bidi-font-weight: normal;">“Variable”</b> box and
indicate which specification to run, then click <b style="mso-bidi-font-weight: normal;">OK</b>. Or, you can use the following codes for the different
specifications:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">For the <b style="mso-bidi-font-weight: normal;">“level”</b> specification, the syntax is:<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">(For <b style="mso-bidi-font-weight: normal;">“Suppress constant term in regression”</b>)<o:p></o:p></span></div>
<div class="MsoNormal">
<i style="mso-bidi-font-style: normal;"><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">pperron lnpce, noconstant regress<o:p></o:p></span></i></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">(For <b style="mso-bidi-font-weight: normal;">“include trend term in regression”</b>)<i style="mso-bidi-font-style: normal;"><o:p></o:p></i></span></div>
<div class="MsoNormal">
<i style="mso-bidi-font-style: normal;"><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">pperron lnpce, trend regress</span></i><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">For the <b style="mso-bidi-font-weight: normal;">“1<sup>st</sup> difference”</b> specification, the syntax is:<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">(For <b style="mso-bidi-font-weight: normal;">“Suppress constant term in regression”</b>)<o:p></o:p></span></div>
<div class="MsoNormal">
<i style="mso-bidi-font-style: normal;"><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">pperron dlnpce, noconstant regress</span></i><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"> <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">(For <b style="mso-bidi-font-weight: normal;">“include trend term in regression”</b>)<i style="mso-bidi-font-style: normal;"><o:p></o:p></i></span></div>
<div class="MsoNormal">
<i style="mso-bidi-font-style: normal;"><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;">pperron dlnpce, trend regress</span></i><span style="color: black; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"> <o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In comparing the results from both procedures, the same conclusion is arrived at. That is, <i style="mso-bidi-font-style: normal;">lnpce</i> is difference-stationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note:</span></b><span style="background: lightgrey; color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> The asymptotic distribution of the PP test is the
same as the ADF test statistic.</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">After unit root
testing, what next?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The outcome of unit root
testing matters for the empirical model to be estimated. The following
scenarios explain the implications of unit root testing for further
analysis. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">Scenario 1:
When series under scrutiny are stationary in levels? </span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If <i>pce</i> and <i>pdi</i> are
stationary in levels, that is, they are <i>I</i>(0) series (integrated of
order zero). In this situation, performing a cointegration test is <b><i><u>not</u></i></b> necessary.
This is because any shock to the system in the short run quickly adjusts to the
long run. Consequently, only the long run model should be estimated. That
is, the model should be specified as: <o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <i>pce<sub>t </sub></i>=
</span><span style="color: black; font-family: "cambria math" , "serif"; font-size: 12.0pt;">𝛂₀</span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;"> + <i>bpdi<sub>t</sub></i> + <i>u<sub>t</sub></i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In essence, the estimation
of short run model is not necessary if series are <i>I</i>(0). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;">Scenario 2: When
series are stationary in first differences?</span></b><span style="color: black; font-family: "times new roman" , "serif"; font-size: 14.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under this scenario, the series are assumed to be non-stationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">One special feature of these series is that they are of the same
order of integration.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under this scenario, the model in question is not entirely useless
although the variables are unpredictable. To verify further the relevance of
the model, there is need to test for cointegration. That is, can we
assume a long run relationship in the model despite the fact that the series
are drifting apart or trending either upward or downward?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If there is cointegration, that means the series in question are
related and therefore can be combined in a linear fashion. This implies that,
even if there are shocks in the short run, which may affect movement in the
individual series, they would converge with time (in the long run).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, there is no long run if series are not cointegrated. This
implies that, if there are shocks to the system, the model is not likely to
converge in the long run.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note that both long run and short run models must be estimated
when there is cointegration.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The estimation will require the use of vector autoregressive (VAR)
model analysis and VECM models.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">If there is no cointegration, there is no long run and therefore,
only the short run model will be estimated. That is, run only VAR no VECM
analysis!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">There are however, two prominent cointegration tests for <i>I</i>(I)
series in the literature. They are Engle-Granger cointegration test and
Johansen cointegration test.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The Engle-Granger test is meant for single equation model while
Johansen is considered when dealing with multiple equations. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 14pt;">Scenario 3: The
series are integrated of different orders? </span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Should in case <i>lnpce</i> and <i>lnpdi</i> are
integrated of different orders, like the second scenario, cointegration test is
also required but the use of either Engle-Granger or Johansen cointegration are
no longer valid.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">The appropriate cointegration test to apply is the Bounds test for
cointegration and the estimation technique is the autoregressive distributed
lag (ARDL) model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Similar to case 2, if series are not cointegrated based on Bounds
test, we are expected to estimate only the short run. That is run only the ARDL
model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="color: black; font-family: "symbol"; font-size: 12.0pt;"><span style="mso-list: Ignore;">·<span style="font: 7.0pt "Times New Roman";"> </span></span></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, both the long run and short run models are valid if there
is cointegration. That is run both ARDL and ECM models.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12.0pt;">In addition, there are
formal tests that can be carried out to see if despite the behaviour of the
series, there can still be a linear combination or long run relationship or
equilibrium among the series. The existence of the linear combination is what
is known as cointegration. Thus, the regression with <i>I</i>(1) series
can either be spurious or cointegrated. The basic single equation cointegration
tests are Johansen, Engle-Granger and Bounds cointegration tests. These will be
discussed in detail in subsequent tutorials.<o:p></o:p></span></div>
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<br />
<div style="text-align: center;">
<b>[Watch video clip on performing the ADF unit root test in Stata]</b></div>
<div style="text-align: center;">
<b><br /></b></div>
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/m7rl-5t2OeA/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/m7rl-5t2OeA?feature=player_embedded" width="320"></iframe></div>
<div style="text-align: center;">
<b><br /></b></div>
<span style="color: black; font-family: "times new roman" , "serif"; font-size: 12pt;">In conclusion, I have
discussed what is meant by nonstationary series, how can a series with a unit
root be detected, and how can such series be made useful for empirical
research? You are encouraged to use your data or the sample </span><span style="font-family: "times new roman" , "serif"; font-size: 12pt;"><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j?usp=sharing" target="_blank">datasets</a></span><span style="color: black; font-family: "times new roman" , "serif"; font-size: 12pt;"> uploaded
to this bog to practise in order to get more hands-on knowledge.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div style="margin-bottom: .0001pt; margin: 0cm;">
<span style="color: black;">Please
post your comments below….<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<br />Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-59729716683950840422018-02-23T10:15:00.000+01:002018-02-26T08:03:51.692+01:00Time Series Analysis (Lecture 3): How to Perform Stationarity Test in EViews<h1 align="center" style="text-align: center;">
How to Perform Unit Root Test in EViews</h1>
<h2 style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">What
is Stationarity in Time Series Analysis?</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In econometrics, time series data are
frequently used and they often pose distinct problems for econometricians. As
it will be discussed with examples, most empirical work based on time series
data assumes that the underlying series is stationary. Stationarity of a series
(that is, a variable) implies that its mean, variance and covariance are
constant over time. That is, these do not vary systematically over time. In order
words, they are <i>time</i> <i>invariant</i>. However, if that is not the
case, then the series is nonstationary. We will discuss some possible scenarios
where two series, <i>Y</i> and <i>X</i>, are nonstationary and the error term,
<i>u</i>, is also nonstationary. In that
case, the error term will exhibit autocorrelation. Another likely scenario is
where <i>Y</i> and <i>X</i> are nonstationary, but <i>u</i>
is stationary. The implications of this will also be explored. In time series
analysis, the words <i>nonstationary</i>, <i>unit root</i> or <i>random walk</i> <i>model</i> are
used synonymously. In essence, of a series is considered to be nonstationary,
it implies that such exhibit a unit root and exemplifies a random walk series.</span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Regressing two series that are
nonstationary, likewise, yields a spurious (or nonsense) regression. That is, a
regression whose outcome cannot be used for inferences or forecasting. In short,
such results should not be taken seriously and must be discarded. A stationary
series will tend to return to its mean (called </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">mean reversion</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">) and fluctuations around this mean (measured by its variance)
will have a broadly constant breadth. But if a time series is not stationary in
the sense just explained, it is called a nonstationary time series such will
have a time-varying mean or a time-varying variance or both. In summary, a
stationary time series is important because if such is nonstationary, its
behaviour can be studied only for the time period under consideration. That is,
each set of time series data will therefore be for a particular episode. As a
result, it is not possible to generalise its relevance to other time periods.
Therefore, for the purpose of forecasting, such (nonstationary) time series may
be of little practical value</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">How
to detect unit root in a series?<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In a bivariate (2 variables) model or
that involving multiple variables (called a multiple regression model), it is
assumed that all the variables are stationary at level (that is, the order of
integration of each of the variable is zero, <i>I</i>(0). It is important to state at this point, that the order of
integration of a series in a regression model is determined by the outcome of a
unit root test (or stationarity test). If the series is stationary at level
after performing unit root test, then it is <i>I</i>(0),
otherwise it is <i>I</i>(<i>d</i>) where <i>d</i> represents the number of times the series is differenced before
it becomes stationary. But what if the assumption of <i>stationarity at level</i> of the series in a bivariate or multiple
regression model is relaxed and we consequently allow for a unit root in each
of the variables in the model, how can this be corrected? In general, this
would require a different treatment from a conventional regression with
stationary variables at <i>I</i>(0). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In particular, we focus on a class of
linear combination of unit root processes known as cointegrated process. The
generic representation for the order of integration of series is <i>I</i>(<i>d</i>)
where <i>d</i> is the number of differencing
to render the series stationary. Hence, a stationary series at level, <i>d</i> = 0 is a series with an <i>I</i>(0) process. Although, for any
non-stationary series, <i>‘d’</i> can assume
any value greater than zero, however, in applied research, only the unit root
process of <i>I</i>(1) process is allowed,
otherwise such series with higher order of integration (<i>d</i> > 1) should be excluded in the model as no meaningful policy
implications or relevance can be drawn from such series. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Here is an example of a bivariate linear
regression model: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">= 𝛂₀</span><span style="font-family: "calibri" , sans-serif; font-size: 11pt; line-height: 107%; position: relative; top: 3pt;"></span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + <i>b</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;">X<sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + <i>u</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> [1]</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Assume <i>Y<sub>t</sub> </i>and<i> X<sub>t</sub></i> are two random walk models that are <i>I</i>(1) processes and are independently
distributed as: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">= </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ρ</span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t-1</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + </span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> <i>v</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">, -1 ≤ </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ρ </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">≤
1 [2] </span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">X<sub>t </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">= </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ղ</span><i style="font-family: "Times New Roman", serif; font-size: 16px;">X<sub>t-1</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + </span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> <i>e</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">, -1 ≤ </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ղ</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> ≤
1 [3]</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">and <i>v<sub>t</sub></i>
and <i>e<sub>t</sub></i> have zero mean, a
constant variance and are orthogonal (these are <i>white noise</i> error terms). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We also assumed that <i>v<sub>t</sub></i> and <i>e<sub>t</sub></i> are serially uncorrelated as well as mutually
uncorrelated. As stated in [2] and [3], both these time series are
nonstationary; that is, they are <i>I</i>(1)
or exhibit stochastic trends. Suppose we regress <i>Y<sub>t</sub> </i>on<i> X<sub>t</sub></i>. Since <i>Y<sub>t</sub> </i>on<i> X<sub>t</sub></i> are uncorrelated <i>I</i>(1) processes, the <i>R</i><sup>2</sup> from the regression of <i>Y </i>on<i> X</i> should tend to
zero; that is, there should not be any relationship between the two variables. Equations
[2] and [3] resemble the Markov first-order autoregressive model. If ρ and </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ղ</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> =
1, the equations become a random walk model without drift. If ρ and </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ղ</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> are in fact 1, then a unit root problem
surfaces, that is, a situation of nonstationarity; because we already know that
in this case the variance of <i>Y<sub>t</sub></i>
is not stationary. The name unit root is due to the fact that ρ = 1. Again, the
terms nonstationary, random walk, and unit root can be treated as synonymous.
If, however, |ρ| ≤ 1, and |</span><span style="font-family: "calibri" , sans-serif; font-size: 11pt; line-height: 107%; position: relative; top: 3pt;"><span style="font-family: "times new roman" , serif; font-size: 16px;">ղ</span></span><span style="font-family: "times new roman" , serif; font-size: 12pt;">| ≤ 1, that is if their absolute values
are less than one, then it can be shown that both series </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">Y<sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> and </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">X<sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">
are stationary. In practice, then, it is important to find out if a time series
possesses a unit root.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Given equations [2] and [3], there
should be no systematic relationship between <i>Y<sub>t</sub></i> and <i>X<sub>t</sub></i>
as they both drift away from equilibrium (i.e. they do not converge), and
therefore, we should expect that an ordinary least squares (OLS) estimate of <i>b</i> should be close to zero, or
insignificantly different from zero, at least as the sample size increases. But
this is not usually the case. The fitted coefficients in this case may be
statistically significant even when there is no true relationship between the
dependent variable and the regressors. This is regarded as a spurious
regression or correlation where, in the case of our example, <i>b</i> takes any value randomly, and its <i>t</i>-statistic indicates significance of
the estimate. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">But
how can unit root be detected?</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> There are some clues that tell you if a
series is nonstationary and if the regression of bivariate or multivariate
relationships are spurious. Some of these are:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Do
a graphical plot of the series to visualise the nature. Is it trending upwards
or downwards? Does it exhibit a mean-reversion or not? Or are there fluctuations
around its mean?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Or
carry out a regression analysis on two series and observe the <i>R</i><sup>2</sup>. If it is above 0.9, it may suggest that the
variables are nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; mso-list: l2 level1 lfo1; text-align: justify; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The rule-of-thumb: if the <i>R</i><sup>2</sup> obtained
from the regression is higher than the Durbin Watson (DW) statistic. The low DW
statistic evidences positive first order auto-correlation of the error terms. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Using <a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table 21.1
quarterly data of 1970q1 to 1991q4, examples of nonstationary series and
spurious regression can be seen from the <i>pce</i>,
<i>pdi</i> and <i>gdp</i> relationship. Since the series are measured in billions of US
dollars, the natural logarithms of the variables will be used in analysing
their essential features.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Nonstationary
series:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">
the graphical plot of the three variables shows an upward trend and none of the
variables revert to their means. That is, all three variables do not
exhibit mean reversions. That clearly tells us that the series are
nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEho_G4awNtOovXgaVEFHD19OymumNnsKPYHs1muhdxlP1ZCAi4S0jzcbugJPGCluBHDwRL2vFlEfhlF5PPHsfKdLQrcgJIXyRwWAo-HLwN15RUbBy16EwhoHyJZ7kQcrHSK0AV69gDougc/s1600/EViews+-+nonstationary+series.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Example of nonstationary series from cruncheconometrix.com.ng" border="0" data-original-height="470" data-original-width="508" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEho_G4awNtOovXgaVEFHD19OymumNnsKPYHs1muhdxlP1ZCAi4S0jzcbugJPGCluBHDwRL2vFlEfhlF5PPHsfKdLQrcgJIXyRwWAo-HLwN15RUbBy16EwhoHyJZ7kQcrHSK0AV69gDougc/s1600/EViews+-+nonstationary+series.png" title="EViews - Example of nonstationary series" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Example of nonstationary series<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">What
is a spurious regression? </span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;">Sometimes we expect to find no relationship between
two variables, yet a regression of one on the other variable often shows a
significant relationship. This situation exemplifies the problem of spurious, or
nonsense, regression. The regression of <i>lnpce</i>
on <i>lnpdi</i> shows how spurious
regressions can arise if time series are not stationary. As expected, because
both variables are nonstationary, the result evidences that a spurious
regression has been undertaken.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">But how do we know this? Take a look at
the <i>R</i><sup>2</sup> the value of <b>0.9944</b> is higher than the Durbin Watson
statistic of <b>0.57</b>. So, whenever the <i>R</i><sup>2</sup> > DW, a spurious
regression has occurred because the variables are nonstationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcI1UqPLbRIdo0ZLP-QL0HGZgy1X-YTQDu3fdV9yevuSJsFRU3ubZmsbfE7uAgXc5gPpJwooYrKWZ6Lv4phaC4N7k9sCqW7erlPtRdyPY6yB29-yBo3draz9Aj3HEcRRdnGGKaS_PA_aY/s1600/EViews+-+Spurious+regression.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Example of a spurious regression from cruncheconometrix.com.ng" border="0" data-original-height="311" data-original-width="441" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcI1UqPLbRIdo0ZLP-QL0HGZgy1X-YTQDu3fdV9yevuSJsFRU3ubZmsbfE7uAgXc5gPpJwooYrKWZ6Lv4phaC4N7k9sCqW7erlPtRdyPY6yB29-yBo3draz9Aj3HEcRRdnGGKaS_PA_aY/s1600/EViews+-+Spurious+regression.png" title="EViews - Example of a spurious regression" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Example of a spurious regression<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">As you can see, the coefficient of </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">lnpdi</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is highly statistically
significant, and the </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">R</i><sup style="font-family: "Times New Roman", serif;">2</sup><span style="font-family: "times new roman" , serif; font-size: 12pt;">
value is statistically significantly different from zero. From these results,
you may be tempted to conclude that there is a significant statistical
relationship between both variables, whereas </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">a priori</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> there may or may </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">not</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">
be none. This is simply the phenomenon of <b>spurious or nonsense regression</b>, first
discovered by Yule (1926). He showed that (spurious) correlation could persist
in nonstationary time series even if the sample is very large. That there is
something wrong in the preceding regression is suggested by the extremely low
Durbin–Watson value, which suggests very strong first-order autocorrelation.
According to Granger and Newbold, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">R</i><sup style="font-family: "Times New Roman", serif;">2</sup><span style="font-family: "times new roman" , serif; font-size: 12pt;">
> DW is a good rule of thumb to suspect that the estimated regression is
spurious, as in the given example.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Why
is it important to test for stationarity? <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We usually consider a nonstationary
series for the following reasons: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; mso-list: l7 level1 lfo2; text-align: justify; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">To
evaluate the behaviour of series over time. Is the series trending upward or
downward? This can be verified from performing a stationarity test. In other
words, the test can be used to evaluate the stability or predictability of time
series. If a series is nonstationary, that means the series is unstable or
unpredictable and therefore may not be valid for inferences, prediction or
forecasting. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; mso-list: l7 level1 lfo2; text-align: justify; text-indent: -18.0pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="margin-left: 18.0pt; mso-list: l7 level1 lfo2; text-align: justify; text-indent: -18.0pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">To
know how a series responds to shocks requires carrying out a stationarity test.
If such series is nonstationary, the impact of shocks to the series are more
likely to be permanent. Consequently, if a series is stationary, impact of
shocks will be temporary or brief. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">How
to correct for nonstationarity?<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">What can be done with nonstationarity in
a time series knowing that performing OLS on such a model yields spurious
regression? <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The
Unit Root Test<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We begin with equations [2] and [3]
which are unit root (stochastic) processes with white noise error terms. If the
parameters of the models are equal to 1, that is, in the case of the unit root,
both equations become random walk models without drift, which we know is a
nonstationary stochastic process. So, what can be done to correct this? For
instance, for equation [2], simply regress <i>Y<sub>t</sub></i>
on its (one-period) lagged value <i>Y<sub>t−1</sub></i>
and find out if the estimated ρ is statistically equal to 1? If it is, then <i>Y<sub>t</sub></i> is nonstationary. Repeat
same for the <i>X<sub>t</sub></i> series. This
is the general idea behind the unit root test of stationarity. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">For theoretical reasons, equation [2] is
manipulated as follows: Subtract <i>Y<sub>t−1</sub></i>
from both sides of [2] to obtain:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">- </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t-1 </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">= </span><span style="font-family: "times new roman" , serif; font-size: 16px;">ρ</span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t-1</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">- </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t-1 </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">+ </span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> <i>v</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: center;"> [4]</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: center;"> = (</span><span style="font-family: "times new roman" , serif; font-size: 16px;">ρ </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: center;">- 1)</span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t-1 </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">+ </span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> <i>v</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><!--[if gte msEquation 12]><m:oMath><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>=(</m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>⍴-</m:r><m:r>1)</m:r><m:r>Y</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r><m:r>-</m:r><m:r>1</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>+</m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>v</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:95.25pt;height:14.25pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image016.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">and this can be stated alternatively as:</span></div>
<div class="MsoNoSpacing" style="margin-left: 72.0pt; text-align: justify; text-indent: 36.0pt;">
<!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>∆</m:r><m:r>Y</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>= </m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>δY</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r><m:r>-</m:r><m:r>1</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>+</m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>v</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:87.75pt;height:14.25pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image018.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">⃤ </span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: 36pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: center;">= </span><span style="font-family: "times new roman" , serif; font-size: 16px; text-indent: 0px;">δ</span><i style="font-family: "Times New Roman", serif; font-size: 16px;">Y<sub>t-1 </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">+ </span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> <i>v</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: 36pt;"> [5]</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">where </span><span style="font-family: "times new roman" , serif; font-size: 16px;">δ</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = (ρ − 1) and ⃤</span><!--[if gte msEquation 12]><m:oMath><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>∆</m:r></span></i></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:7.5pt;height:14.25pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image020.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">, as usual, is the first-difference
operator. In practice, therefore, instead of estimating [2], we estimate [5]
and test the null hypothesis that </span><span style="font-family: "times new roman" , serif; font-size: 16px;">δ</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = 0. If </span><span style="font-family: "times new roman" , serif; font-size: 16px;">δ</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = 0, then ρ = 1, that is we have a
unit root, meaning the time series under consideration is nonstationary. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before we proceed to estimate [5], it
may be noted that if </span><span style="font-family: "times new roman" , serif; font-size: 16px;">δ</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = 0, [5] will become:<o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>∆</m:r><m:r>Y</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>=</m:r></span></i><m:d><m:dPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>Y</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>-</m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>Y</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r><m:r>-</m:r><m:r>1</m:r></span></i></m:sub></m:sSub></m:e></m:d><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>=</m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>v</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:114pt;height:14.25pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image022.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify; text-indent: 48px;">⃤ </span><i style="font-family: "Times New Roman", serif; font-size: 16px; text-align: justify; text-indent: 48px;">Y<sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify; text-indent: 36pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: 48px;">= </span><i style="font-family: "Times New Roman", serif; font-size: 16px; text-align: justify;">Y<sub>t-1</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">- </span><i style="font-family: "Times New Roman", serif; font-size: 16px; text-align: justify;">Y<sub>t-1</sub></i><i style="font-family: "Times New Roman", serif; font-size: 16px; text-align: justify; text-indent: 48px;"><sub> </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify; text-indent: 48px;">= </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify; text-indent: 48px;"> <i>v</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px; text-align: justify; text-indent: 48px;"><sub>t</sub></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> [6]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(Remember to do the same for <i>X<sub>t</sub></i> series)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since <i>v</i><sub>t</sub> is a white noise error term, it is stationary, which
means that <b>the first difference of a
random walk time series is stationary</b>. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHg-yCz3TD4nDAkcvVIqzq3nw1_cB-M1Tjz0cTL3uiYR3r9vBLd0FMZXadEyXxWIp3_4ClyqhTKTKF13D-gXB8E5Li77JvWoojd4S2ef7aF1zxoKa3or6PTyd7w1OToT-eYYGDguD2Wio/s1600/EViews+-+Stationary+series.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Example of stationary series from cruncheconometrix.com.ng" border="0" data-original-height="470" data-original-width="517" height="363" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHg-yCz3TD4nDAkcvVIqzq3nw1_cB-M1Tjz0cTL3uiYR3r9vBLd0FMZXadEyXxWIp3_4ClyqhTKTKF13D-gXB8E5Li77JvWoojd4S2ef7aF1zxoKa3or6PTyd7w1OToT-eYYGDguD2Wio/s400/EViews+-+Stationary+series.png" title="EViews - Example of stationary series (in first difference)" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Example of stationary series (in first difference)<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</span></div>
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<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Visual observation of the differenced
series shows that the three variables are stationary around the mean. They all
exhibit constant mean-reversions. That is, they fluctuate around 0. If we are
to draw a trend line, such a line will be horizontal at 0.01.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Okay, having said all that. Let us
return to estimating equation [5]. This is quite simple, all that is required
is to take the first differences of <i>Y<sub>t</sub></i>
and regress on <i>Y<sub>t−1</sub></i> and
see if the estimated slope coefficient in this regression is statistically
different from is zero or not. If it is zero, we conclude that <i>Y<sub>t</sub></i> is nonstationary. But if
it is negative, we conclude that <i>Y<sub>t</sub></i>
is stationary. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , "serif";"> Since δ = (ρ − 1), for
stationarity ρ must be less than one. For this to happen δ must be negative!</span><span style="font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The only question is which test do we
use to find out if the estimated coefficient of <i>Y<sub>t−1</sub></i> in [5] is zero or not? You might be tempted to say,
why not use the usual <i>t</i> test?
Unfortunately, under the null hypothesis that δ = 0 (i.e., ρ = 1), the <i>t</i> value of the estimated coefficient of <i>Y<sub>t−1</sub></i> does not follow the <i>t</i> distribution even in large samples;
that is, it does not have an asymptotic normal distribution. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">What is the alternative? Dickey and
Fuller (DF) have shown that under the null hypothesis that δ = 0, the estimated
<i>t</i> value of the coefficient of <i>Y<sub>t−1</sub></i> in [5] follows the <b><i>τ
(tau)</i></b> statistic. These authors have computed the critical values of the
<i>tau statistic</i> on the basis of Monte
Carlo simulations.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , "serif";"> Interestingly, if the hypothesis
that δ = 0 is rejected (i.e., the time series is stationary), we can use the
usual (Student’s) <i>t</i> test.</span><span style="font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The unit root test can be computed under
three (3) different null hypotheses. That is, under different model
specifications such as if the series is a:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">random
walk (that is, model has no constant, no trend)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">random
walk with drift (that is, model has a constant)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l3 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">random
walk with drift and a trend (that is, model has a constant and trend)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In all cases, the null hypothesis is
that δ = 0; that is, there is a unit root and the alternative hypothesis is
that δ is less than zero; that is, the time series is stationary. If the null
hypothesis is rejected, it means that <i>Y<sub>t</sub></i>
is a stationary time series with zero mean in the case of [5], that <i>Y<sub>t</sub></i> is stationary with a
nonzero mean in the case of a random walk with drift model, and that <i>Y<sub>t</sub></i> is stationary around a
deterministic trend in the case of random walk with drift around a trend.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">It is extremely important to note that
the critical values of the <i>tau test</i>
to test the hypothesis that δ = 0, are different for each of the preceding
three specifications of the DF test, which are now computed by all econometric
packages. In each case, if the computed absolute value of the <i>tau statistic</i> (|τ|) exceeds the DF or MacKinnon
critical <i>tau values</i>, the null
hypothesis of a unit root is rejected, in order words the time series is
stationary. On the other hand, if the computed |τ| does not exceed the critical
<i>tau</i> value, we fail to reject the null
hypothesis, in which case the time series is nonstationary. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , "serif";"> Students often get confused in
interpreting the outcome of a unit root test. For instance, if the calculated <i>tau</i> statistic is -2.0872 and the
MacKinnon <i>tau</i> statistic is -3.672,
you cannot reject the null hypothesis. Hence, the conclusion is that the series
is nonstationary. But if the calculated <i>tau</i>
statistic is -5.278 and the MacKinnon <i>tau</i>
statistic is -3.482, you reject the null hypothesis in favour of the
alternative. Hence, the conclusion is that the series is stationary.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: lightgrey; font-family: "times new roman" , "serif";">*Always use the appropriate
critical τ values for the indicated model specification.</span><span style="font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">How
to Perform Unit Root Test in EViews (See here for <a href="https://cruncheconometrix.blogspot.com.ng/2018/02/time-series-analysis-lecture-3-how-to_21.html" target="_blank">Stata</a>)<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Example
dataset is from <a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> T21.1</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Several tests have been developed in the
literature to test for unit root. Prominent among these tests are Augmented Dickey-Fuller,
Phillips-Perron, Dickey-Fuller Generalised Least Squares (DFGLS) and so on. But
this tutorials limits testing to the use of ADF and PP tests. Once the reader
has good basic knowledge of these two techniques, they can progress to
conducting other stationarity test on their time series variables. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">How
to Perform the Augmented Dickey-Fuller (ADF) Test</span></b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">An important assumption of the DF test
is that the error terms are independently and identically distributed. The ADF
test adjusts the DF test to take care of possible serial correlation in the
error terms by adding the lagged difference terms of the outcome (dependent)
variable. For <i>Y<sub>t</sub></i> series,
in conducting the DF test, it is assumed that the error term <i>v<sub>t</sub></i> is uncorrelated. But in
case where it is correlated, Dickey and Fuller have developed a test, known as
the augmented Dickey–Fuller (ADF) test. This test is conducted by “augmenting”
the preceding three model specifications stated above by adding the lagged
values of the dependent variable.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">As mentioned earlier, approaches will be
limited to using the ADF and PP tests. Either of these tests can be used and
when both are used, the reader can compare the outcomes to see if there are
similarities or differences in the results.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l6 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Load
workfile into EViews. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l6 level1 lfo4; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We
are considering only <i>lnpce</i> and <i>lnpdi</i> in natural logarithms. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The
Augmented Dickey-Fuller (ADF) Test<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Unit root test for <i>lnpce</i>:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Double
click the <i>lnpce</i> series to open it.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go
to <b>View</b> >> <b>Unit root test</b> >> dialog box
opens >> Under <b>Test Type</b>, select
<b>Augmented Dickey Test</b></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "symbol"; font-size: 12pt; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;">Decide
whether to test for a unit root in the <b>level</b>,
<b>1st difference</b>, or <b>2nd difference</b> of the series. Ideally, always
start with the <b>level</b> and if we fail
to reject the test in levels then continue with testing for the first difference.
Hence, we first click on <b>'Level'</b> in
the dialog box to see what happens in the levels of the series and then
continue, if appropriate, with the first and second differences. </span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Also,
the choice of model is very important since the distribution statistic under
the null hypothesis differs across these three cases. Therefore, specify whether
to include an <b>intercept</b>, <b>trend and intercept</b>, or <b>none</b> in the regression. It is more
appropriate to consider the three possible test regressions when dealing with stationarity
test. Thus, our demonstration will involve these three options: <b>“none”</b>, <b>“constant”</b>, <b>“constant</b> <b>and</b> <b>trend”</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo5; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We
have to also specify the number of lagged dependent variables to be included in
the model in order to correct for the presence of serial correlation. Thus, the
number of lags to be included in the model would be determined either
automatically or manually. I prefer to allow AIC automatically decide the lag
length. Due to the fact that I have a quarterly data, AIC automatically chose
11 lags which I modified to 8.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Consequently, EViews reports the test
statistic together with the estimated test regression. The null hypothesis of a
unit root is rejected against the one-sided alternative hypothesis if the
computed absolute value of the <i>tau
statistic</i> exceeds the DF or MacKinnon critical tau values and we conclude
that the series is stationary; otherwise (that is, if it is lower), then the
series is non-stationary. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Another way of stating this is that in
failing to reject the null hypothesis of a unit root, the computed τ value
should be <b><i><u>more</u></i></b> negative than the critical τ value. Since in
general δ is expected to be negative, the estimated τ statistic will have a
negative sign. Therefore, a large negative τ value is generally an indication
of stationarity. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">On the other hand, using the probability
value, we reject the null hypothesis of unit root if the computed probability
value is less than the chosen level of statistical significance. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Having
specified the <b>“none”</b> option where
both the intercept and trend are excluded in the test regression, the unit root
test dialog box is shown thus:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg50xEE8VNUe6uqMQT1dkGiGyZimTNqLlM6Gi2DLsY75fIsLsP4vX4cQbrwDuTIysXvDDAsWJOIaJkRO_9KXpe5ghMYnev9ykh3-XhImxwVa20DFnpKnwFahrfp6AwOIzhkAreqvSv2QBo/s1600/EViews+-+URT+dialog+box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Unit root test dialog box from cruncheconometrix.com.ng" border="0" data-original-height="322" data-original-width="394" height="326" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg50xEE8VNUe6uqMQT1dkGiGyZimTNqLlM6Gi2DLsY75fIsLsP4vX4cQbrwDuTIysXvDDAsWJOIaJkRO_9KXpe5ghMYnev9ykh3-XhImxwVa20DFnpKnwFahrfp6AwOIzhkAreqvSv2QBo/s400/EViews+-+URT+dialog+box.png" title="EViews - Unit root test dialog box" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8px;">EViews - Unit root test dialog box<br />
Source: CrunchEconometrix<br />
<br /></td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "symbol"; font-size: 12pt; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;">The
ADF unit root test results for the selected regression option, “<b>none</b>” appears as follows:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbfd01cSpss4TZhp74bIezh_9HxpI36-RMmyO5uSJ_7QMO6lBom33OToY_3xKKIVrF2hzCrp5BgGylkaJopV6KNV07KO4t9HjcqaMfi36hR-75yJHE5wljqQJ_unOEsRSxP9mnEOddRV4/s1600/EViews+-+ADF+%2528none%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Augmented Dickey-Fuller test from cruncheconometrix.com.ng" border="0" data-original-height="539" data-original-width="604" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbfd01cSpss4TZhp74bIezh_9HxpI36-RMmyO5uSJ_7QMO6lBom33OToY_3xKKIVrF2hzCrp5BgGylkaJopV6KNV07KO4t9HjcqaMfi36hR-75yJHE5wljqQJ_unOEsRSxP9mnEOddRV4/s1600/EViews+-+ADF+%2528none%2529.png" title="EViews - Augmented Dickey-Fuller test ("none") option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Augmented Dickey-Fuller test ("none") option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">Following similar procedures, the select
“</span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">intercept</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">” for the ADF unit root
test yields:<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVd_Xeu3JaS3tihhwgWXjC0JeLe-fOZWqibjn5jY911TQnxeOl5rK0qsLmwGWPvP6Nc6ZDB_Jus5ggrzrMxtlP0ouowiNegaB4e6qi8Gyg-h0S9-ffiMafCdLprw4x8CNAJCQ9_Px9LpA/s1600/EViews+-+ADF+%2528intercept%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Augmented Dickey-Fuller test from cruncheconometrix.com.ng" border="0" data-original-height="512" data-original-width="529" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVd_Xeu3JaS3tihhwgWXjC0JeLe-fOZWqibjn5jY911TQnxeOl5rK0qsLmwGWPvP6Nc6ZDB_Jus5ggrzrMxtlP0ouowiNegaB4e6qi8Gyg-h0S9-ffiMafCdLprw4x8CNAJCQ9_Px9LpA/s1600/EViews+-+ADF+%2528intercept%2529.png" title="EViews - Augmented Dickey-Fuller test ("intercept") option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Augmented Dickey-Fuller test ("intercept") option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<!--[if gte vml 1]><v:shape
id="Picture_x0020_28" o:spid="_x0000_i1029" type="#_x0000_t75" style='width:396.75pt;
height:384pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image030.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">The result for the “</span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">trend and intercept</b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">” options for the ADF unit root test is shown
below:</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSVuH-hG9T2IoNuX9IdIjwtbpZtuuzxNMDFg1EIiZqkh2a3DFp4DFmmGuZj1RgbQove6fVeNs0xGO_AcFzEycSknplWbwHxY2XOs24OYeDFQjASJsk1ghvHPhlukUAoS7M67DZGTLMrDA/s1600/EViews+-+ADF+%2528trend+and+intercept%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Augmented Dickey-Fuller test from cruncheconometrix.com.ng" border="0" data-original-height="499" data-original-width="526" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSVuH-hG9T2IoNuX9IdIjwtbpZtuuzxNMDFg1EIiZqkh2a3DFp4DFmmGuZj1RgbQove6fVeNs0xGO_AcFzEycSknplWbwHxY2XOs24OYeDFQjASJsk1ghvHPhlukUAoS7M67DZGTLMrDA/s1600/EViews+-+ADF+%2528trend+and+intercept%2529.png" title="EViews - Augmented Dickey-Fuller test ("trend and intercept") option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Augmented Dickey-Fuller test ("trend and intercept") option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="background: lightgrey; font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="background: lightgrey; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Do same for the <i>lnpdi</i> series.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The three ADF specifications all confirm
that <i>lnpce</i> is nonstationary with a
trend which is also a confirmation of the graphical plot. The next thing to do
is to run the specifications with the <b>“1<sup>st</sup>
difference”</b> option, and if the series is still nonstationary, the <b>“2<sup>nd</sup> difference”</b> option is
conducted.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1<sup>st</sup>
difference with <b>“none” </b>option
result:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikcmGGIkdNaQSpByYSc1bWFY0bIzPpAzdjzGpT80oR-OzuLh4HFbF6gpl0I-Nubos3ahfD67n8PXPb0zFfam8tTarjhr0Tb0TwfI5gLuoTtCWmu96jNZoqopBiazcIuFC2umNAI0wPKwI/s1600/EViews+-+ADF+%25281st+diff%252C+none%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Augmented Dickey-Fuller test 1st difference from cruncheconometrix.com.ng" border="0" data-original-height="516" data-original-width="519" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikcmGGIkdNaQSpByYSc1bWFY0bIzPpAzdjzGpT80oR-OzuLh4HFbF6gpl0I-Nubos3ahfD67n8PXPb0zFfam8tTarjhr0Tb0TwfI5gLuoTtCWmu96jNZoqopBiazcIuFC2umNAI0wPKwI/s1600/EViews+-+ADF+%25281st+diff%252C+none%2529.png" title="EViews - Augmented Dickey-Fuller test 1st difference ("none") option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Augmented Dickey-Fuller test 1st difference ("none") option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1<sup>st</sup>
difference with <b>“intercept” </b>option
result:</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> </span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDFW99z9AM1dOxgIme_X-AdD0keF0B47ZYzGYSCRKW9LZlIB2qJJ-xVwmTmHj-tzrTZUjsb0vUz3XXTzdVhwvJVqepObIENSEsjVnHyiLSM1sW6sidhi9bjW5GTxdEozKos0HiGy-zGAM/s1600/EViews+-+ADF+%25281st+diff%252C+intercept%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Augmented Dickey-Fuller test 1st difference from cruncheconometrix.com.ng" border="0" data-original-height="470" data-original-width="487" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDFW99z9AM1dOxgIme_X-AdD0keF0B47ZYzGYSCRKW9LZlIB2qJJ-xVwmTmHj-tzrTZUjsb0vUz3XXTzdVhwvJVqepObIENSEsjVnHyiLSM1sW6sidhi9bjW5GTxdEozKos0HiGy-zGAM/s1600/EViews+-+ADF+%25281st+diff%252C+intercept%2529.png" title="EViews - Augmented Dickey-Fuller test 1st difference ("intercept") option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Augmented Dickey-Fuller test 1st difference ("intercept") option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "symbol"; font-size: 12pt; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;">1<sup>st</sup>
difference with <b>“trend and intercept” </b>option
result:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnRd8GyWP4tftK9bcxpiCUD9Szj_vNxs8OGsy1ZoC1DGg7BphdT1MtsyZDcteEO3Tjx40wpMVeyYZWN-VUFyAVS-Lp9Yh-aFof0VUzD7fA2Pzq0tJnQJkzAEXyA5pTK_fMwIzhtQSMoaY/s1600/EViews+-+ADF+%25281st+diff%252C+trend+and+intercept%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Augmented Dickey-Fuller test 1st difference from cruncheconometrix.com.ng" border="0" data-original-height="524" data-original-width="514" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnRd8GyWP4tftK9bcxpiCUD9Szj_vNxs8OGsy1ZoC1DGg7BphdT1MtsyZDcteEO3Tjx40wpMVeyYZWN-VUFyAVS-Lp9Yh-aFof0VUzD7fA2Pzq0tJnQJkzAEXyA5pTK_fMwIzhtQSMoaY/s1600/EViews+-+ADF+%25281st+diff%252C+trend+and+intercept%2529.png" title="EViews - Augmented Dickey-Fuller test 1st difference ("trend and intercept") option" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Augmented Dickey-Fuller test 1st difference ("trend and intercept") option<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The three ADF specifications all confirm
that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">lnpce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is stationary at 1</span><sup style="font-family: "Times New Roman", serif;">st</sup><span style="font-family: "times new roman" , serif; font-size: 12pt;">
difference but at varying significance levels. Given that I am willing to
reject the null hypothesis at the 5% level, then the conclusion is that </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;"><i>lnpce</i>
is stationary at 1<sup>st</sup> difference with a constant</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> because it is
only at that specification that the null hypothesis of a unit root is rejected.
Hence, carrying out a </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">“2<sup>nd</sup>
difference”</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> test is unnecessary.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="background: lightgrey; font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="background: lightgrey; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Again, do same for the <i>lnpdi</i> series.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">How
to Perform the Phillips-Perron (PP) Test</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Phillips and Perron use nonparametric statistical methods to take
care of the serial correlation in the error terms <b><i><u>without</u></i></b> adding
lagged difference terms. </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">Procedures for testing for unit root using PP test are similar to that of ADF earlier discussed except for the </span><b style="font-family: "times new roman", serif; font-size: 12pt;">Test Type</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> options.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: lightgrey; font-family: "times new roman" , "serif";">Note:</span></b><span style="background: lightgrey; font-family: "times new roman" , "serif";"> The asymptotic distribution of
the PP test is the same as the ADF test statistic.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">After
unit root testing, what next? <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The outcome of unit root testing matters
for the empirical model to be estimated. The following scenarios explain the
implications of unit root testing for further analysis. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Scenario
1: When series under scrutiny are
stationary in levels? <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If <i>pce</i>
and <i>pdi</i> are stationary in levels,
that is, they are <i>I</i>(0) series
(integrated of order zero). In this
situation, performing a cointegration test is <b><i><u>not</u></i></b> necessary.
This is because any shock to the system in the short run quickly adjusts to the
long run. Consequently, only the long run model should be estimated. That is, the model should be specified as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
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style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>pce</m:r></span></i></m:e><m:sub><i
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style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>a</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>0</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>+ </m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>bpdi</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>+</m:r></span></i><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>u</m:r></span></i></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>t</m:r></span></i></m:sub></m:sSub></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%;"><!--[if gte vml 1]><v:shape
id="_x0000_i1025" type="#_x0000_t75" style='width:120.75pt;height:14.25pt'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image035.png"
o:title="" chromakey="white"/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <i>pce</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t </sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">= 𝛂₀</span><span style="font-family: "calibri" , sans-serif; font-size: 11pt; line-height: 15.6933px; position: relative; top: 3pt;"></span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + <i>b</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;">pdi<sub>t</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> + <i>u</i></span><i style="font-family: "Times New Roman", serif; font-size: 16px;"><sub>t</sub></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In essence, the estimation of short run
model is not necessary if series are <i>I</i>(0). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Scenario
2: When series are stationary in first differences?<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under
this scenario, the series are assumed to be non-stationary. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">One
special feature of these series is that they are of the same order of
integration. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under
this scenario, the model in question is not entirely useless although the
variables are unpredictable. To verify further the relevance of the model,
there is need to test for cointegration.
That is, can we assume a long run relationship in the model despite the
fact that the series are drifting apart or trending either upward or downward? <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If
there is cointegration, that means the series in question are related and therefore
can be combined in a linear fashion. This implies that, even if there are
shocks in the short run, which may affect movement in the individual series,
they would converge with time (in the long run). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">However,
there is no long run if series are not cointegrated. This implies that, if
there are shocks to the system, the model is not likely to converge in the long
run. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note
that both long run and short run models must be estimated when there is
cointegration. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
estimation will require the use of vector autoregressive (VAR) model analysis
and VECM models.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If
there is no cointegration, there is no long run and therefore, only the short
run model will be estimated. That is, run only VAR no VECM analysis!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">There
are however, two prominent cointegration tests for <i>I</i>(I) series in the literature. They are Engle-Granger cointegration
test and Johansen cointegration test. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo6; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
Engle-Granger test is meant for single equation model while Johansen is
considered when dealing with multiple equations. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Scenario
3: The series are integrated of different order? <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l5 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Should
in case <i>lnpce</i> and <i>lnpdi</i> are integrated of different
orders, like the second scenario, cointegration test is also required but the
use of either Engle-Granger or Johansen cointegration are no longer valid. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l5 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The
appropriate cointegration test to apply is the Bounds test for cointegration
and the estimation technique is the autoregressive distributed lag (ARDL) model.
<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l5 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Similar
to case 2, if series are not cointegrated based on Bounds test, we are expected
to estimate only the short run. That is run only the ARDL model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l5 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">However,
both the long run and short run models are valid if there is cointegration.
That is run both ARDL and ECM models.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In addition, there are formal tests that
can be carried out to see if despite the behaviour of the series, there can
still be a linear combination or long run relationship or equilibrium among the
series. The existence of the linear combination is what is known as
cointegration. Thus, the regression with <i>I</i>(1)
series can either be spurious or cointegrated. The basic single equation
cointegration tests are Johansen, Engle-Granger and Bounds cointegration tests.
These will be discussed in detail in subsequent tutorials.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br />
<div style="text-align: center;">
<b>[Watch video clip on performing ADF stationarity test in EViews]</b></div>
<div style="text-align: center;">
<b><br /></b></div>
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/ovpHuz6YMLc/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/ovpHuz6YMLc?feature=player_embedded" width="320"></iframe></div>
<div style="text-align: center;">
<b><br /></b></div>
<div style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">In conclusion, I have discussed what is
meant by nonstationary series, how can a series with a unit root be detected, and how can such series be made useful for empirical research?
You are encouraged to use your data or the sample </span><a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" style="font-family: "times new roman", serif; font-size: 12pt; text-align: justify;" target="_blank">datasets</a><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> uploaded to this bog
to practise in order to get more hands-on knowledge.</span></div>
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 107%;">Please post your
comments below….</span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-70629365219320296282018-02-14T08:00:00.000+01:002018-02-14T15:02:35.506+01:00Time Series Analysis (Lecture 2): Choosing Optimal Lags in Stata<br />
<h2 style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">General Overview on Lag Selection</span></b></h2>
<h2 style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Since
this blog is tailored for beginners in econometrics, I will not be engaging an
advanced discussion on the topic but an introductory approach by which a
beginner can understand the essence of using lags in a model and the pitfalls
that may occur if lags are excessively used. Interested readers who require advanced
information on selecting optimal lags can consult appropriate econometric textbooks.
Having said that, i</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">n economics the dependence of a variable <b style="mso-bidi-font-weight: normal;"><i>Y</i></b><i> </i>(outcome variable or
regressand) on another variable(s) <b style="mso-bidi-font-weight: normal;"><i>X</i></b><i>
</i>(the predictor variable or regressor) is rarely instantaneous. Very often, <b style="mso-bidi-font-weight: normal;"><i>Y</i></b><i> </i>responds to <b style="mso-bidi-font-weight: normal;"><i>X</i></b><i> </i>with a lapse of time.
Such a lapse of time is called a <i>lag</i>. <span style="background: white;">Therefore,
in time series analysis, some level of care must be exercised when including
lags in a model.<o:p></o:p></span></span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">So
how many lags should be used in a model?</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> T<span style="mso-bidi-font-weight: bold;">here is no hard-and-fast-rule on the choice of
lag length. It is </span>basically an empirical issue. As noted in Damodar Gujarati
<i style="mso-bidi-font-style: normal;">Basic Econometrics, </i>there is no <i style="mso-bidi-font-style: normal;">a priori</i> guide as to what the maximum
length of the lag should be. The researcher must bear in mind that, as one
estimates successive lags, there are fewer degrees of freedom left, making
statistical inference somewhat unstable. Economists are usually not that lucky
to have a long series of data so that they can go on estimating numerous lags.
More importantly, in economic time series data, successive values (lags) tend
to be highly correlated increasing the likelihood of multicollinearity in the
model. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Also, from
Jeffery Wooldridge’s <i style="mso-bidi-font-style: normal;">Introductory
Econometrics: A Modern Approach</i> with annual data, the number of lags is
typically small, 1 or 2 lags in order not to lose degrees of freedom. With
quarterly data, 1 to 8 lags is appropriate, and for monthly data, 6, 12 or 24
lags can be used given sufficient data points. Again, in the words of Damodar Gujarati
<i style="mso-bidi-font-style: normal;">Basic Econometrics</i> “the sequential
search for the lag length opens the researcher to the charge of <span style="mso-bidi-font-weight: bold;">data mining”<b>. </b>He further stated that </span>the
nominal and true level of significance to test statistical hypotheses becomes
an important issue in such sequential searches”. For instance, if the lag
length, <i>k</i>, is incorrectly specified, the researcher will have to contend
with the problem of misspecification errors. In addition, because of the lags
involved, distributed and or autoregressive models raise the topic of causality
in economic variables. <span style="background: white;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Hence,
b</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">efore
you estimate a time series equation, it is necessary to decide on the maximum
lag length. Like I mentioned earlier, this is purely an empirical question.
Suppose there are 40 observations in all, by including too many lagged values,
your model consumes degrees of freedom, not to mention introducing the
likelihood of multicollinearity occurring. As noted in my previous tutorial on
multicollinearity, it leads to imprecise estimation; that is, the standard
errors tend to be inflated in relation to the estimated coefficients. As a
result, based on the routinely computed <i>t </i>ratios, we may tend to declare
(erroneously), that a lagged coefficient(s) is statistically insignificant. In
the same vein, including too few lags will lead to specification errors. The
easiest way out of this quagmire, is to decide using a criterion like the
Akaike or Schwarz and choose that model that gives the lowest values of these
criteria. Most econometric packages easily compute these optimal lag length but
note some trial and error is inevitable. <o:p></o:p></span></div>
<h1 style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="background: white; color: red; font-size: 13.0pt; line-height: 107%;">Choosing
Optimal Lags in Stata<o:p></o:p></span></b></h1>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman", serif; font-size: 13pt;">For this tutorial, I will extract data from </span><b style="font-family: "times new roman", serif; font-size: 13pt;"><a href="https://drive.google.com/drive/u/0/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table 21.1</b><span style="font-family: "times new roman", serif; font-size: 13pt;"> dataset. It is a quarterly data on United States from 1970 to 1991, which is 88 observations. The variables are </span><i style="font-family: "times new roman", serif; font-size: 13pt;">gdp</i><span style="font-family: "times new roman", serif; font-size: 13pt;"> (gross domestic product), </span><i style="font-family: "times new roman", serif; font-size: 13pt;">pdi</i><span style="font-family: "times new roman", serif; font-size: 13pt;"> (personal disposable income) and </span><i style="font-family: "times new roman", serif; font-size: 13pt;">pce</i><span style="font-family: "times new roman", serif; font-size: 13pt;"> (personal consumption expenditure).</span><br />
<div class="MsoNoSpacing" style="-webkit-text-stroke-width: 0px; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: justify; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="margin: 0px;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13pt;"><br /></span></div>
</div>
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step 1: Load data into Stata<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB5IR37qFNcrfuKloiUIhicTQW5HdEtBhBddfJMnJ6-k-rOh5D151T_67Gx73vowvDub5Ca1-0uQYiACNkub222i3qLZASg_uTVFHVi0kU7EDZsEHKY7huY9TYzemYbg0HYBppmktEhr4/s1600/Stata+-+Workfile.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata Dataset from cruncheconometrix.com.ng" border="0" data-original-height="520" data-original-width="677" height="305" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB5IR37qFNcrfuKloiUIhicTQW5HdEtBhBddfJMnJ6-k-rOh5D151T_67Gx73vowvDub5Ca1-0uQYiACNkub222i3qLZASg_uTVFHVi0kU7EDZsEHKY7huY9TYzemYbg0HYBppmktEhr4/s400/Stata+-+Workfile.png" title="Stata Dataset" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata Dataset<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><!--[if gte vml 1]><v:shapetype
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</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p> </o:p></span><b><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step 2: Prepare Stata for Analysis</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Inform
Stata that you are about to perform a time series analysis by typing this code
into the <b style="mso-bidi-font-weight: normal;">Command</b> box: <i style="mso-bidi-font-style: normal;">tsset qtrly<o:p></o:p></i></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">and
you will obtain this response:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIICEfcOe1NacWzl10I8Ckwic7JiSzM_Qkhq7a32ca4pIc6Zuh-EZMcfRtCCVlhpXGLYC_HFYFvA2W88J-73MD9LPBFafAdTfZ-k2K7EDRhC0Pghus5z2ATUh8E4SJ-PFFv-ISr7yFI2w/s1600/Stata+-+tsset+command.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata - tsset Command from cruncheconometrix.com.ng" border="0" data-original-height="51" data-original-width="357" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIICEfcOe1NacWzl10I8Ckwic7JiSzM_Qkhq7a32ca4pIc6Zuh-EZMcfRtCCVlhpXGLYC_HFYFvA2W88J-73MD9LPBFafAdTfZ-k2K7EDRhC0Pghus5z2ATUh8E4SJ-PFFv-ISr7yFI2w/s1600/Stata+-+tsset+command.png" title="Stata - tsset Command" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata - tsset Command<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background-color: white; font-family: "times new roman" , serif; font-size: 13pt;">Stata
now recognises that you are about conducting a time series analysis using quarterly
data from 1</span><sup style="font-family: "Times New Roman", serif;">st</sup><span style="background-color: white; font-family: "times new roman" , serif; font-size: 13pt;"> quarter of 1970 to the 4</span><sup style="font-family: "Times New Roman", serif;">th</sup><span style="background-color: white; font-family: "times new roman" , serif; font-size: 13pt;"> quarter of 1991.
If you don’t issue this command, Stata will not run your analysis.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step 3: Obtain Model Lag Length<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Type
this code into the <b style="mso-bidi-font-weight: normal;">Command</b> box: <i style="mso-bidi-font-style: normal;">varsoc gdp pce pdi<o:p></o:p></i></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">…and the Stata output for the
model (shown below) indicates that lag 2 is the optimal lag and that AIC is the
best criterion for the model given it has the lowest value, <b style="mso-bidi-font-weight: normal;"><span style="color: red;">26.8144</span></b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgka2pJUX079ESStWl_4mZ-rKBGobUq7c3igq57wfq29f14JpeUakyvySxxlYhdG611LgRjMSFVZc3DwI1HwIOS0nw0DPgHlPkbIDhmC61QYe1mMf2F2lHTLxjhsZocabTQ7tH72NQcr2Q/s1600/Stata+-+Model+lag.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata - Optimal Lags for the Model from cruncheconometrix.com.ng" border="0" data-original-height="260" data-original-width="582" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgka2pJUX079ESStWl_4mZ-rKBGobUq7c3igq57wfq29f14JpeUakyvySxxlYhdG611LgRjMSFVZc3DwI1HwIOS0nw0DPgHlPkbIDhmC61QYe1mMf2F2lHTLxjhsZocabTQ7tH72NQcr2Q/s1600/Stata+-+Model+lag.png" title="Stata - Optimal Lags for the Model" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata - Optimal Lags for the Model<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-size: 13.0pt; mso-ansi-language: EN-GB; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape
id="Picture_x0020_18" o:spid="_x0000_i1028" type="#_x0000_t75" style='width:436.5pt;
height:195pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image004.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step 4: Obtain Variables Lag Length</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Optimal
lags can be obtained for the respective variables and the rule-of-thumb remains
the same. We select that lag identified by the criterion which gives the lowest
value. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">So,
for <i style="mso-bidi-font-style: normal;">gdp</i>, type this code into the <b style="mso-bidi-font-weight: normal;">Command</b> box: <i style="mso-bidi-font-style: normal;">varsoc gdp</i> <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">…and
the Stata output indicates that the optimal lag length for <i style="mso-bidi-font-style: normal;">gdp</i> is 2.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-size: 13.0pt; mso-ansi-language: EN-GB; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape
id="Picture_x0020_24" o:spid="_x0000_i1027" type="#_x0000_t75" style='width:420.75pt;
height:195pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh07lYVOZ6RGbq5TNEvV2sMYC-VaChJgyGP0cEXSt_8kil3Slx4X-QEFs9KyyHfjepYI9bf9bhdjfL2jfHwKAWyC03udbYvB7S2iGjI5cd532aSY38p7dBtEoSD8t8Ke2iTXXzFxiEHxzQ/s1600/Stata+-+Lag+for+gdp.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata - Optimal Lags for gdp from cruncheconometrix.com.ng" border="0" data-original-height="249" data-original-width="561" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh07lYVOZ6RGbq5TNEvV2sMYC-VaChJgyGP0cEXSt_8kil3Slx4X-QEFs9KyyHfjepYI9bf9bhdjfL2jfHwKAWyC03udbYvB7S2iGjI5cd532aSY38p7dBtEoSD8t8Ke2iTXXzFxiEHxzQ/s640/Stata+-+Lag+for+gdp.png" title="Stata - Optimal Lags for gdp" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata - Optimal Lags for <i>gdp</i><br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p> </o:p></span><span style="background: white; font-family: "times new roman" , serif; font-size: 13pt;">To obtain optimal lag for <i>pce</i>, type: </span><i><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">varsoc pce</span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">…Stata
output indicates that the optimal lag length for <i style="mso-bidi-font-style: normal;">pce</i> is 4.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV8Lz16j_otQDxVAFuimkHldm3TJkvCZne1vJCRt-cbFXFMJJkRmz8TsnkEvNlqjCWBFNDvDP8AhplVptQtwj1JBMw7dboD4D8L9X1BUTMmMr6K1Y_o8H_cAGg36RDQNKbwgsi7d963BY/s1600/EViews+-+Lag+for+pce.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata - Optimal Lags for pce from cruncheconometrix.com.ng" border="0" data-original-height="245" data-original-width="563" height="278" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgV8Lz16j_otQDxVAFuimkHldm3TJkvCZne1vJCRt-cbFXFMJJkRmz8TsnkEvNlqjCWBFNDvDP8AhplVptQtwj1JBMw7dboD4D8L9X1BUTMmMr6K1Y_o8H_cAGg36RDQNKbwgsi7d963BY/s640/EViews+-+Lag+for+pce.png" title="Stata - Optimal Lags for pce" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata - Optimal Lags for <i>pce</i><br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p> </o:p></span><i><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">varsoc pdi</span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">…Stata
output indicates that the optimal lag length for <i style="mso-bidi-font-style: normal;">pdi</i> is 1.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-size: 13.0pt; mso-ansi-language: EN-GB; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape
id="Picture_x0020_31" o:spid="_x0000_i1025" type="#_x0000_t75" style='width:418.5pt;
height:183.75pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGnRgWnVLekdy6m54lFf5jNzi7qBKmPa2_AU9mtYvsaPszP5KXeuG3ZdFVz4dgZGQ6bl3oSVyKUMoPJO5WOExKem8C5L8pVKkCRZPqjV0BmYOtgl2bmfTLZe17EaHF8NGsuihxo2-1qjI/s1600/Stata+-+Lag+for+pdi.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata - Optimal Lags for pdi from cruncheconometrix.com.ng" border="0" data-original-height="245" data-original-width="558" height="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGnRgWnVLekdy6m54lFf5jNzi7qBKmPa2_AU9mtYvsaPszP5KXeuG3ZdFVz4dgZGQ6bl3oSVyKUMoPJO5WOExKem8C5L8pVKkCRZPqjV0BmYOtgl2bmfTLZe17EaHF8NGsuihxo2-1qjI/s640/Stata+-+Lag+for+pdi.png" title="Stata - Optimal Lags for pdi" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata - Optimal Lags for <i>pdi</i><br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Caveat</span></i></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">:
There are also cases where the <span style="background: white;">used lag length
is that which is most selected by the criterion named after
the econometricians who developed them, like HQ, SIC, AIC and LR,
etc. Some researchers prefer Schwartz criterion when the variables are
more than 4 and use the AIC when the variables are less than 4. As, mentioned
in the introductory part of this tutorial, the decision on the choice of lag is
purely an empirical issue. Generally, we choose the lag length for which the
values of most of these lag length criteria are minimized, indicated by asterisks
in the EViews output.</span><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">[Watch video tutorial
on lag selection using Stata]<o:p></o:p></span></b><br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/7llGl3yy3kE/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/7llGl3yy3kE?feature=player_embedded" width="320"></iframe></div>
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Having
gone through this tutorial, it will be easy for you to determine the optimal
lag for your model regardless of the analytical package used. The basics are the same. Remember that “Lag
length criteria” indicate a definite way of selecting the optimal lag
after estimating the initial VAR model (in EViews). Also VAR and ARDL models
are susceptible to arbitrary use of lags as this may erode the degrees of
freedom, weaken the significance of the coefficients, may induce
auto-correlation and weaken the strength of diagnostic tests.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Try these steps on
your models and if there are further and comments, do post them below…..</span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-72691346936651295892018-02-12T10:30:00.000+01:002018-02-12T10:00:52.888+01:00Excel: How to Interpret Regression Output<h2 style="text-align: center;">
<b><span style="color: windowtext; font-family: "times new roman" , serif;">How to Interpret Regression Output in Excel</span></b></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The dissertation buzz is on and students
are doing everything possible to meet up with the deadline. This is the current
atmosphere in tertiary institutions, at least for those with undisrupted
academic calendar </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">. The students
are in different stages of their <i>project</i>,
as it is commonly called. Some are yet to wrap up their chapter one which gives
the “study background” and the framing of research hypotheses, objectives and
questions. Some have moved on to chapter two reviewing relevant literature
related to their scope of study. Others have gone further in developing both
the theoretical and empirical frameworks for chapter three, but not without the
usual teething lags…but they’ll get around it, somehow </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">. A handful have even done better
by progressing to chapter four attempting to analyse their data.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since, chapters one to three are
relative to each students’ scope of research, however, a regression output is
common to all (although actual outcomes differ). It is based on this that I decided
to do a tutorial in explaining the basic features in a regression output. Likewise,
this write-up is in response to requests received from readers on (1) what some
specific figures in a regression output are and (2) how to interpret their
results. Let me state here that regardless of the analytical software whether
Stata, EViews, SPSS, R, Python, Excel etc. what you obtain in a regression
output is common to all analytical packages (except where slight variations
occur).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">For instance, in undertaking an ordinary
least squares (OLS) estimation using any of these applications, the regression
output will churn out the ANOVA (analysis of variance) table, <i>F</i>-statistic, <i>R</i>-squared, prob-values, coefficient, standard error, <i>t</i>-statistic, degrees of freedom, 95%
confidence interval and so on. These are the features of a regression output.
However, the issue is: what do these mean and how can they be interpreted and
related to a research.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Hence, the essence of this tutorial is
to teach students the relevance of these features and how to interpret their
results. I will be using <b>Excel</b>
analytical package to explain a regression output, but you can practise along
using any analytical package of your choice. (See tutorial for <a href="http://cruncheconometrix.blogspot.com.ng/2018/01/how-to-interpret-regression-output-in.html" target="_blank">Stata</a> and <a href="http://cruncheconometrix.blogspot.com.ng/2018/02/how-to-interpret-regression-output-in_3.html" target="_blank">EViews</a> users).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">An Example: Use <a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table7_12.dta
or Table7_12.xlsx dataset</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note: I will not be discussing
stationarity or cointegration analysis in this tutorial (that will come later on). Since the issue on how to understand the features of a regression output and interpret results, I will just be doing a <span style="background: yellow; mso-highlight: yellow;">simple linear regression</span>
analysis (a bi-variate analysis) with only one explanatory variable. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The dataset is on the United States from
1960 to 2009 (50 years data). The outcome variable is consumption expenditure (<i>pce</i>) and the explanatory variable is
income (<i>income</i>).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">First step: get the Data Analysis Add-in menu<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before you begin, ensure that the <b>DATA ANALYSIS</b> Add-in is in your tool
bar because without it, you cannot perform any regression analysis. To obtain it
follow this guide:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">File</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> >> <b>Options</b> >> <b>Add-ins</b> >> <b>Excel
Options</b> dialog box opens<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under <b>Active Application Add-ins</b>, choose <b>Analysis ToolPak</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In the <b>Manage</b> section, choose <b>Excel
Add-ins</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click <b>Go</b>, then <b>OK</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">If it is correctly done, you should see
this:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVsNEjrQ5EeZnflTJ3MRMnMjYViHsNOr9pU3UKfl8qubogh3ztVpFN35TSxpiIM1tslaSNjXDwvigYPAMirArJOLSjFLfajNI6iBuDhWSbzP9Fws5BVGpZzsA64F8W3XPtM1yK-ZbcpMI/s1600/Excel+Add-in.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel Add-in Dialog Box from cruncheconometrix.com.ng" border="0" data-original-height="678" data-original-width="821" height="528" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVsNEjrQ5EeZnflTJ3MRMnMjYViHsNOr9pU3UKfl8qubogh3ztVpFN35TSxpiIM1tslaSNjXDwvigYPAMirArJOLSjFLfajNI6iBuDhWSbzP9Fws5BVGpZzsA64F8W3XPtM1yK-ZbcpMI/s640/Excel+Add-in.png" title="Excel Add-in Dialog Box" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel Add-in Dialog Box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">And you will have the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Data Analysis</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> menu to your extreme top-right corner under </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Data</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> menu:</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgi7KR4WSpDQlt-8E_cGVIz7_7H4QfzwzF-ByzvJRbG88pWVWnzbDp_MXB3urRBgp-tLUtqgeG77akTPhmSBtelcX2rUSn-9mzTQ-UEAPi2GR-gpF8obgqGT2yp-7Vn5A1G0Lp0yUODvTE/s1600/Data+Analysis+menu.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Excel Data Analysis Menu from cruncheconometrix.com.ng" border="0" data-original-height="363" data-original-width="1366" height="170" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgi7KR4WSpDQlt-8E_cGVIz7_7H4QfzwzF-ByzvJRbG88pWVWnzbDp_MXB3urRBgp-tLUtqgeG77akTPhmSBtelcX2rUSn-9mzTQ-UEAPi2GR-gpF8obgqGT2yp-7Vn5A1G0Lp0yUODvTE/s640/Data+Analysis+menu.png" title="Excel Data Analysis Menu" width="640" /></a></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_8" o:spid="_x0000_i1032" style="height: 97.5pt; mso-wrap-style: square; visibility: visible; width: 366pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image002.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Second step: have your data ready <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Here is the data in excel format:<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqT91-z0Qf7dK_HfpW8IVSUc96OlGgDB_VVMBm9ANXNVev6ucTDtar0eQP3SJaDEvjLRaJpC-86mxlOcT6og2GvzDkPQRsc6N3zoSgT6wFDs5rXOKtnlGN0wgkO2gK5_pQ96i58ZG5x9E/s1600/Data+in+excel+format.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Data in excel format from cruncheconometrix.com.ng" border="0" data-original-height="650" data-original-width="665" height="624" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqT91-z0Qf7dK_HfpW8IVSUc96OlGgDB_VVMBm9ANXNVev6ucTDtar0eQP3SJaDEvjLRaJpC-86mxlOcT6og2GvzDkPQRsc6N3zoSgT6wFDs5rXOKtnlGN0wgkO2gK5_pQ96i58ZG5x9E/s640/Data+in+excel+format.png" title="Data in excel format" width="640" /></a></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_2" o:spid="_x0000_i1031" style="height: 190.5pt; mso-wrap-style: square; visibility: visible; width: 405.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Third step: Visualise the relationship between the variables<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before analysing the data, it is good to
always graph the dependent and key explanatory variable (using a scatter plot)
in order to observe the pattern between them. This gives you what to expect in
your actual analysis. Here’s the procedure:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Highlight
the 2 columns that contain the variables<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go
to <b>Insert</b> >> <b>Charts</b> >> <b>Scatter</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><b><br /></b></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT9HobaV3rmYn71IF6CKGkmdJJFA0U1k43Q0xv8cMZw6gMouLurpTpN77OG9JhfldLiZ00UHEfnwajW1_XGM9lSmdfThINywDMvzX0MYzm0DwdBx9mCbFUGETqc69UZERyozv8va2eFBE/s1600/Excel+-+scatter+plot.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel - Scatter plot of pce and income from cruncheconometrix.com.ng" border="0" data-original-height="287" data-original-width="485" height="378" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT9HobaV3rmYn71IF6CKGkmdJJFA0U1k43Q0xv8cMZw6gMouLurpTpN77OG9JhfldLiZ00UHEfnwajW1_XGM9lSmdfThINywDMvzX0MYzm0DwdBx9mCbFUGETqc69UZERyozv8va2eFBE/s640/Excel+-+scatter+plot.PNG" title="Excel - Scatter plot of pce and income" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel - Scatter plot of pce and income<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><b><br /></b></span></div>
<div align="center" class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_10" o:spid="_x0000_i1030" style="height: 181.5pt; mso-wrap-style: square; visibility: visible; width: 306.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image004.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The graph indicates a positive
relationship between the two variables. This seems plausible because the higher
your income, the higher will be your consumption, except you are very frugal</span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The graph can be formatted by adding a
trend line (see video on how to do this). In Excel, adding a trendline also
gives you the linear prediction:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpGaUr4z_hzLmgeEm4R0JBdjLZlzNdRJfTfhGFfIecdlkU4_z55qdXfYNtHEOXLJUJnyja_sdHq5SzaWE9k9G-MdDbmGvzAyIiCoVi0bGG85p6K_3UU38fmsMZp9jhXK0v72yOYutfRQA/s1600/Excel+-+scatter+plot+with+prediction.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Excel - Scatter plot with trendline from cruncheconometrix.com.ng" border="0" data-original-height="290" data-original-width="490" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpGaUr4z_hzLmgeEm4R0JBdjLZlzNdRJfTfhGFfIecdlkU4_z55qdXfYNtHEOXLJUJnyja_sdHq5SzaWE9k9G-MdDbmGvzAyIiCoVi0bGG85p6K_3UU38fmsMZp9jhXK0v72yOYutfRQA/s640/Excel+-+scatter+plot+with+prediction.PNG" title="Excel - Scatter plot with trendline" width="640" /></a></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_11" o:spid="_x0000_i1029" style="height: 183.75pt; mso-wrap-style: square; visibility: visible; width: 311.25pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">As can be seen from the second graph, we
have the linear prediction for <i>pce</i>
and the <i>R</i><sup>2</sup>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fourth step: The scientific investigation<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Now we want to scientifically
investigate the relationship between <i>pce</i>
and <i>income</i>. To do this, go to <b>Data</b> >> <b>Data Analysis</b> (dialogue box opens) >> <b>Regression</b> >> <b>OK. </b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5yGJBjfUcpsLjOgCJjC4CzI56YJ_rDU6hjXnE4omsg7W5wcoYtznc5MrFCvwz44vHOmjHiD8V-LN5fKIjtnVV3ANjXiie2TrExLXlrfZVPeqMbwEvbwrc0BQDsd-QnKKvWQqLG4nachE/s1600/Excel+-+Data+Analysis+Dialogue+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel - Data Analysis Dialogue Box from cruncheconometrix.com.ng" border="0" data-original-height="193" data-original-width="395" height="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5yGJBjfUcpsLjOgCJjC4CzI56YJ_rDU6hjXnE4omsg7W5wcoYtznc5MrFCvwz44vHOmjHiD8V-LN5fKIjtnVV3ANjXiie2TrExLXlrfZVPeqMbwEvbwrc0BQDsd-QnKKvWQqLG4nachE/s640/Excel+-+Data+Analysis+Dialogue+Box.png" title="Excel - Data Analysis Dialogue Box" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel - Data Analysis Dialogue Box<br />
Source: Crunch Econometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Once you click </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">OK</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">, the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Regression</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">
dialogue box opens:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWSCVqt7Bp614xIk84d5BEU6yI87mfXD-KakzZDFqxnixatbLX1s-tTHyRm4VpLtei_KZFTHfYofB9JwpfkvyYXcbcs2uLKhMwXuToCp4szr7NRe1WTH6Peaj5JeFQJVzXhxttRzKBe80/s1600/Excel+-+Regression+Dialogue+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel - Regression Dialogue Box from cruncheconometrix.com.ng" border="0" data-original-height="365" data-original-width="418" height="558" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWSCVqt7Bp614xIk84d5BEU6yI87mfXD-KakzZDFqxnixatbLX1s-tTHyRm4VpLtei_KZFTHfYofB9JwpfkvyYXcbcs2uLKhMwXuToCp4szr7NRe1WTH6Peaj5JeFQJVzXhxttRzKBe80/s640/Excel+-+Regression+Dialogue+Box.png" title="Excel - Regression Dialogue Box" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel - Regression Dialogue Box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "symbol"; font-size: 12pt; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.2pt;">Put
data range for <i>pce</i> under <b>Input <i>Y</i>
Range</b></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Put
data range for <i>income</i> under <b>Input <i>X</i>
Range</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Check
<b>label</b> box<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Check
<b>Confidence</b> <b>Level</b> box<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Check
<b>Output</b> <b>range</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo3; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click
<b>OK</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(You have simply told <b>Excel</b> to regress the dependent
variable, <i>pce</i>, on the explanatory
variable, <i>income</i>), and the output is
shown as:<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcmEDN60MAB17PGtNhIpNb1RITKDlMqebAw1mTr8q3Na7KtQk_TOeMI1fhP0gVGRbjJem5kjP2v2P3VdShGi1bU20DHYaHYOwmUocO8e0HaN-r9XQe6i9gAfJn2RMqv-QKSYGey4KCv_Q/s1600/Excel+-+Regression+Output.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="Excel - Regression Output from cruncheconometrix.com.ng" border="0" data-original-height="387" data-original-width="793" height="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcmEDN60MAB17PGtNhIpNb1RITKDlMqebAw1mTr8q3Na7KtQk_TOeMI1fhP0gVGRbjJem5kjP2v2P3VdShGi1bU20DHYaHYOwmUocO8e0HaN-r9XQe6i9gAfJn2RMqv-QKSYGey4KCv_Q/s640/Excel+-+Regression+Output.PNG" title="Excel - Regression Output" width="640" /></a></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_14" o:spid="_x0000_i1026" style="height: 178.5pt; mso-wrap-style: square; visibility: visible; width: 366.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image008.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fifth step: The features of a regression output<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The Excel output gives the <b>Regression Statistics</b> and the <b>ANOVA</b> table. So what do these figures
mean? I will explain each feature in turns.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under
“Regression Statistics”:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">R</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: gives the
variation in <i>pce</i> that is explained by
<i>income</i>. The higher the <i>R</i><sup>2</sup>, the better the model and
the more predictive power the variables have. Although, an <i>R</i><sup>2</sup> that equals 1 will elicit some suspicion. The R is
actually the correlation coefficient between the 2 variables. This implies that
</span><!--[if gte msEquation 12]><m:oMath><m:rad><m:radPr><m:degHide m:val="on"/><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>R</m:r></span></i></m:e><m:sup><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>2</m:r></span></i></m:sup></m:sSup></m:e></m:rad></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><v:shape id="_x0000_i1025" style="height: 16.5pt; width: 21pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image009.png">
</v:imagedata></v:shape></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtlvNcORyRiqj4ufaARkPElKSG3O32Piluhu3FFlhl9HHGlwAAU2bgnXYMScunigx6XQwoq7Vn3xdKLkGaPBefYLvJklyDVG3iKrCccTa76xC4KYNFhTgPiqn-_6zE8jrI5CcXn2H4lXM/s1600/R2.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="21" data-original-width="37" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtlvNcORyRiqj4ufaARkPElKSG3O32Piluhu3FFlhl9HHGlwAAU2bgnXYMScunigx6XQwoq7Vn3xdKLkGaPBefYLvJklyDVG3iKrCccTa76xC4KYNFhTgPiqn-_6zE8jrI5CcXn2H4lXM/s1600/R2.PNG" /></a></div>
= the correlation coefficient.<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span><br />
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></i></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Adjusted R</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;">: this is the <i>R</i><sup>2</sup> adjusted as you increase your explanatory variables.
It reduces as more explanatory variables are added.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Standard
Error</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
this is the standard error of the regression<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Observations</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: the data span
is from 1960 to 2009 = 50 years<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under
“ANOVA” (analysis of variance):<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Source</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: there are two
sources of variation on the dependent variable, <i>pce</i>. Those explained by the regression (i.e, the <b><span style="color: blue;">Model</span></b>)
and those due to randomness (<b><span style="color: blue;">Residuals</span></b>)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">df</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is <span style="color: blue;">degree of freedom</span> calculated as k-1 (for the model) and
n-k (for the residuals)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">SS</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: implies <span style="color: blue;">sum of squared residuals</span> for the Regression
(explained variation in <i>pce</i>) and
Residuals (unexplained variation in <i>pce</i>).
After doing the regression analysis, all the points on <i>pce</i> do not fall on the predicted line. Those points outside the
line are known as <b>residuals</b>. Those
that can be explained by the regression are known as <b>Explained Sum of Squares</b> (ESS) while those that are due to random
forces, which are outside the model are known as <b>Residual Sum of Squares</b> (RSS).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpGaUr4z_hzLmgeEm4R0JBdjLZlzNdRJfTfhGFfIecdlkU4_z55qdXfYNtHEOXLJUJnyja_sdHq5SzaWE9k9G-MdDbmGvzAyIiCoVi0bGG85p6K_3UU38fmsMZp9jhXK0v72yOYutfRQA/s1600/Excel+-+scatter+plot+with+prediction.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Excel - predicted Value of pce from cruncheconometrix.com.ng" border="0" data-original-height="290" data-original-width="490" height="378" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpGaUr4z_hzLmgeEm4R0JBdjLZlzNdRJfTfhGFfIecdlkU4_z55qdXfYNtHEOXLJUJnyja_sdHq5SzaWE9k9G-MdDbmGvzAyIiCoVi0bGG85p6K_3UU38fmsMZp9jhXK0v72yOYutfRQA/s640/Excel+-+scatter+plot+with+prediction.PNG" title="Excel - Predicted Value of pce" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Excel - Predicted Value of pce<br />
Source: CrunchEconomterix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">As observed from the graph, all the
points do not fall on the predicted line. Some lie above, while some are
beneath the line. These are all the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;"><span style="color: blue;">residuals</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> (in order words, the remnants obtained
after the regression analysis). If the predicted line falls above a point, it
means that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is over-predicted
(that is, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce – pce<sub>hat</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is
negative) and if it is beneath a point, it implies that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is under-predicted (that is, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce – pce<sub>hat</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is positive). The sum and mean of the residuals
equals zero.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">MS</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: implies <span style="color: blue;">mean sum of squared residuals </span>obtained by dividing <b><i>SS</i></b> by <i><b>df</b></i> i.e. <b><i>SS/df</i></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">F</span></i></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: captures whether the
explanatory variable, <i>income</i> is
significant in explaining the outcome variable, <i>pce</i>. The higher the <i>F</i>-stat, the better for the model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Significance
<i>F</i></span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the probability value
that indicates the statistical significance of the <i>F</i> ratio. A <i>significance</i>-value
that is less than 0.05 is often preferred.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Coefficient</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the
slope coefficient. The estimate for <i>income</i>.
The sign of the coefficient also tells you the direction of the relationship. A
positive (negative) sign implies a positive (negative) relationship.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Intercept</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the
hypothetical outcome on <i>pce</i> if <i>income</i> is zero. It is also the intercept
for the model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Standard
error</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
this is the standard deviation for the coefficient. That is, since you are not
so sure about the exact value for <i>income</i>,
there will be some variation in the prediction for the coefficient. Therefore,
the standard error shows how much deviation occurs from predicting the slope
coefficient estimate.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">t</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-stat</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this measures
the number of standard errors that the coefficient is from zero. It is obtained
by: <b><i>coeff/std. error</i></b></span><!--[if gte msEquation 12]><m:oMath><m:f><m:fPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>coeff</m:r></span></i></m:num><m:den><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>std</m:r><m:r>.<span
style='mso-spacerun:yes'> </span></m:r><m:r>error</m:r></span></i></m:den></m:f></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 7.0pt;"><v:shape id="_x0000_i1025" style="height: 21pt; width: 42.75pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png"></v:imagedata></v:shape></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">.
A <i>t</i>-stat above 2 is sufficient
evidence against the null hypothesis</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">P-value</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: there are
several interpretations for this. (1) it is smallest evidence required to
reject the null hypothesis, (2) it is the probability that one would have
obtained the slope coefficient value from the data if the actual slope
coefficient is zero, (3) the p-value looks up the <i>t</i>-stat table using the degree of freedom (df) to show the number of
standard errors the coefficient is from zero, (4) tells whether the
relationship is significant or not.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, if the <i>p</i>-value is 0.3, then it means that you are only 70% (that is,
(100-30)% ) confident that the slope coefficient is non-zero. This is not good
enough. This is because a very low <i>p</i>-value
gives a higher level of confidence in rejecting the null hypothesis. Hence, a <i>p</i>-value of 0.02, implies that you are
98% (that is, (100 - 2)% ) confident that the slope coefficient is non-zero
which is more re-assuring! </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Lower
and Upper 95%</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
these are the confidence intervals. If the coefficient is significant, this
interval will contain that slope coefficient but it will not, if otherwise.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Assignment:</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Use <a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table7_12.dta or
Table7_12.xlsx dataset.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(1)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> With <i>pce</i>
as the dependent variable and <i>gdpi</i> as
the explanatory variable, plot the graph of <i>pce</i>
and <i>gdpi</i>, what do you observe?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(2)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Run your regression. Can you interpret the
table and the features?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(3)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Plot the predicted line. What are your
observations?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">I have taken you through the basic
features of a regression output using <b>Excel data analysis software</b> on ordinary
least squares (OLS) model in a simple linear regression. So, you now have the
basic idea of what the <i>F</i>-stat, <i>t</i>-stat, df, SS, MS, prob>F, p>|t|,
confidence interval, <i>R</i><sup>2</sup>,
coefficient, standard error stand for. <o:p></o:p></span><br />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span>
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"></span><br />
<div class="MsoNoSpacing" style="text-align: justify;">
<div style="text-align: center;">
<b>[Watch video on "How to interpret regression output in Excel"]</b><br />
<div>
<br /></div>
</div>
</div>
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Hjn4OLH6pnw/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Hjn4OLH6pnw?feature=player_embedded" width="320"></iframe></div>
<div style="text-align: center;">
<br /></div>
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Practice the assignment and if you still
have further questions, kindly post them below....<o:p></o:p></span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-1821831498522335892018-02-12T08:00:00.000+01:002018-02-12T08:00:05.730+01:00Time Series Analysis (Lecture 2): Choosing Optimal Lags in EViews<h2 style="text-align: justify;">
<b><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">General Overview on Lag Selection</span></b></h2>
<h2 style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Since
this blog is tailored for beginners in econometrics, I will not be engaging an
advanced discussion on the topic but an introductory approach by which a
beginner can understand the essence of using lags in a model and the pitfalls
that may occur if lags are excessively used. Interested readers who require more advanced
information on lag selection can consult appropriate econometric textbooks. Having said that, i</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">n
economics the dependence of a variable <b><i>Y</i></b><i>
</i>(outcome variable or regressand) on another variable(s) <b><i>X</i></b><i> </i>(the predictor variable
or regressor) is rarely instantaneous. Very often, <b><i>Y</i></b><i> </i>responds to <b><i>X</i></b><i>
</i>with a lapse of time. Such a lapse of time is called a <i>lag</i>. <span style="background: white;">Therefore, in time series analysis, some level of care
must be exercised when including lags in a model.<o:p></o:p></span></span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">So
how many lags should be used in a model?</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> There is no hard-and-fast-rule on the choice of
lag length. It is basically an empirical issue. As noted in Damodar Gujarati
<i>Basic Econometrics, </i>there is no <i>a priori</i> guide as to what the maximum
length of the lag should be. The researcher must bear in mind that, as one
estimates successive lags, there are fewer degrees of freedom left, making
statistical inference somewhat unstable. Economists are usually not that lucky
to have a long series of data so that they can go on estimating numerous lags.
More importantly, in economic time series data, successive values (lags) tend
to be highly correlated increasing the likelihood of <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/multicollinearity.html" target="_blank">multicollinearity</a> in the
model. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Also, from
Jeffery Wooldridge’s <i>Introductory
Econometrics: A Modern Approach</i> with annual data, the number of lags is
typically small, 1 or 2 lags in order not to lose degrees of freedom. With
quarterly data, 1 to 8 lags is appropriate, and for monthly data, 6, 12 or 24
lags can be used given sufficient data points. Again, in the words of Damodar Gujarati
<i>Basic Econometrics</i> “the sequential
search for the lag length opens the researcher to the charge of data mining”<b>. </b>He further stated that the
nominal and true level of significance to test statistical hypotheses becomes
an important issue in such sequential searches”. For instance, if the lag
length, <i>k</i>, is incorrectly specified, the researcher will have to contend
with the problem of misspecification errors. In addition, because of the lags
involved, distributed and or autoregressive models raise the topic of causality
in economic variables. <span style="background: white;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Hence,
b</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">efore
you estimate a time series equation, it is necessary to decide on the maximum
lag length. Like I mentioned earlier, this is purely an empirical question.
Suppose there are 40 observations in all, by including too many lagged values,
your model consumes degrees of freedom, not to mention introducing the
likelihood of multicollinearity occurring. As noted in my previous tutorial on
multicollinearity, it leads to imprecise estimation; that is, the standard
errors tend to be inflated in relation to the estimated coefficients. As a
result, based on the routinely computed <i>t </i>ratios, we may tend to declare
(erroneously), that a lagged coefficient(s) is statistically insignificant. In
the same vein, including too few lags will lead to specification errors. The
easiest way out of this quagmire, is to decide using a criterion like the
Akaike or Schwarz and choose that model that gives the lowest values of these
criteria. Most econometric packages easily compute these optimal lag length but
note some trial and error is inevitable. <o:p></o:p></span></div>
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<br /></div>
<h1 style="text-align: justify;">
<b><span style="background: white; color: red; font-size: 13.0pt; line-height: 107%;">Choosing Optimal
Lags in EViews<o:p></o:p></span></b></h1>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">For instance, if
there are limited observations in a vector autoregressive (VAR) estimation, it
is often advised to use the Akaike Selection Criterion (AIC) in selecting the
lag length that "prefers" the more parsimonious models. However, the
information criterion with the smallest criterion value evidences the most
ideal lag length to employ. Most researchers prefer using the Akaike
information criterion (AIC) but my valuable advice is always to select that
criterion with the smallest value, because that ensures the model will be
stable. Let us begin by showing how you can select the optimal lag order for
your model and variables using the EViews analytical package. </span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background-color: yellow; font-family: "times new roman" , serif; font-size: 13pt;">Please note that
in EViews, the procedure is simply to <i>run
an initial VAR on the variables at level with the default settings and
obtain the results</i>. I will go through the steps in detail.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">For this
tutorial, I will extract data from <b><a href="https://drive.google.com/drive/u/0/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table 21.1</b> dataset. It is a quarterly data on United States from 1970 to
1991, which is 88 observations. The variables are <i>gdp</i> (gross domestic product), <i>pdi</i>
(personal disposable income) and <i>pce</i>
(personal consumption expenditure). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="color: blue; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step 1: Load Data into EViews<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">To import the
Excel file into EViews, go to: <b>File</b>
>> <b>Import</b> >> <b>Import from file</b> >> <b>Next</b> >> <b>Finish</b>. If it is correctly done, you obtain:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjealRw2zlgQr26NEPvTK6h23IyxJmfbYs92Wsd3u_P60lLLvTUs_HBWTlnZD5cde4H_CtBH2_bzlQLxfsoUs0c7qRkHUO_tGIolLhhImSeHNcQyVUBMfmNWWrgG553PEtRo0HhZ6123JM/s1600/EViews+Workfile.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Workfile from cruncheconometrix.com.ng" border="0" data-original-height="454" data-original-width="505" height="574" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjealRw2zlgQr26NEPvTK6h23IyxJmfbYs92Wsd3u_P60lLLvTUs_HBWTlnZD5cde4H_CtBH2_bzlQLxfsoUs0c7qRkHUO_tGIolLhhImSeHNcQyVUBMfmNWWrgG553PEtRo0HhZ6123JM/s640/EViews+Workfile.png" title="EViews Workfile" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews Workfile<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">From the EViews
interface, the three variables </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">gdp</i><span style="font-family: "times new roman" , serif; font-size: 13pt;">, </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 13pt;"> and </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">pdi</i><span style="font-family: "times new roman" , serif; font-size: 13pt;"> are individually shown. Double-clicking on each variable shows
them in separate sheets, like is:</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWQgY9g3skqMUSvK0YrHqdxcxcrZ6dOpvG0uOPolf3TaCeimdYiNOXQZ3ZbARFJu2BB_x6v7owGUbnzPnH2_3YVc78_xK0awIu5aIe7Ob5mBmb-hRyLgzwD-SYz7Hqnh5tj7B0-1BiOQI/s1600/EViews+-+Variables+Separate.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Creating Group Data from cruncheconometrix.com.ng" border="0" data-original-height="462" data-original-width="765" height="386" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWQgY9g3skqMUSvK0YrHqdxcxcrZ6dOpvG0uOPolf3TaCeimdYiNOXQZ3ZbARFJu2BB_x6v7owGUbnzPnH2_3YVc78_xK0awIu5aIe7Ob5mBmb-hRyLgzwD-SYz7Hqnh5tj7B0-1BiOQI/s640/EViews+-+Variables+Separate.png" title="EViews Creating Group Data" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews Creating Group Data<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_3"
o:spid="_x0000_i1034" type="#_x0000_t75" style='width:387.75pt;height:272.25pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image003.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br />Step
2: Create Group Data</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">But because I
need to obtain the optimal lag for the model, it becomes necessary to open this
data as a GROUP by putting all three variables in a worksheet. To do that: </span><span style="font-family: "times new roman" , serif; font-size: 13pt;">Press down the
</span><b style="font-family: "Times New Roman", serif; font-size: 13pt;">Cntrl key</b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> >> click on </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><i>gdp</i>, <i>pce</i> </b><span style="font-family: "times new roman" , serif; font-size: 13pt;">and</span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"> <i>pdi</i></b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> >> Right click on any part of the screen >> </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;">Open</b><span style="font-family: "times new roman" , serif; font-size: 13pt;">
>> </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;">as Group:</b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><b><br /></b></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEii-KpwJpEgIqKWdMrVOE9LeNOj6SXlnd4zhdK21HBg7gvqmlx4wotJf3OTwPUkBAywBFC_Ss7yI-RuajeQZOPLihySI6Py5rH0pKNVTZIQdqqeqylAhPW-rY3LbGGFAgqEgbdU3suA83E/s1600/EViews+-+Open+as+Group.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews - Open as Group Data from cruncheconometrix.com.ng" border="0" data-original-height="456" data-original-width="496" height="586" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEii-KpwJpEgIqKWdMrVOE9LeNOj6SXlnd4zhdK21HBg7gvqmlx4wotJf3OTwPUkBAywBFC_Ss7yI-RuajeQZOPLihySI6Py5rH0pKNVTZIQdqqeqylAhPW-rY3LbGGFAgqEgbdU3suA83E/s640/EViews+-+Open+as+Group.png" title="EViews - Open as Group Data" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Open as Group Data<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><b><br /></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">When you click <b>"as Group"</b>, you should have this:</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9_NpXXQHDcid0NtjesiVJaUUDQ4CpzDQwQT0zMV_DQ7TiMRIfG_LdQqpi5Ygr5n6DoQSeqMlPZyqNyKI8MNKSNnV8iVgAm4M0UkeDCDYv9E1H5Jz4CUBjWzPR9HCh-diTGnvgWQkq7z4/s1600/EViews+-+Group+Variable.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Group Data from cruncheconometrix" border="0" data-original-height="449" data-original-width="634" height="452" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9_NpXXQHDcid0NtjesiVJaUUDQ4CpzDQwQT0zMV_DQ7TiMRIfG_LdQqpi5Ygr5n6DoQSeqMlPZyqNyKI8MNKSNnV8iVgAm4M0UkeDCDYv9E1H5Jz4CUBjWzPR9HCh-diTGnvgWQkq7z4/s640/EViews+-+Group+Variable.png" title="EViews Group Data" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="font-size: 12.8px;">EViews Group Data<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step
3: Run Unrestricted VAR model</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Now that our
variables are grouped, next is to run an <b><i>unrestricted VAR model</i></b> with <b><i>the
level</i></b> of the variables and taking different lags before deciding which
model is the best. Remember, I am using quarterly data which allows me to use
up to 8 lags. But if yours is a yearly data you can use 2 lags at the most in
order not to lose too many degrees of freedom or if monthly data, up to 24
lags. The <b>unrestricted VAR</b> is chosen
only on the assumption that the three variables are not co-integrated. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Note: if the variables are
cointegrated, you should run the vector error correction model</span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">To run the <i>unrestricted VAR model</i>, go to: <b>Quick</b> >> <b>Estimate VAR</b> >> Dialog box opens:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKXJrFRVsSsDDrmXIIY7CuuWZpG_yAds6l5m-U_u1UnZ0zVSRmZhzNS5Af8Uf794uyCBa9zugzbTkNAeCDgi2qvbIjK5UBtHnqnX89Ere3nfKXlrmzH347oVKtPKXSw21bOJ8x-BOWwZs/s1600/EViews+-+VAR+Estimation+DB.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews VAR Specification from cruncheconometrix.com.ng" border="0" data-original-height="431" data-original-width="563" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKXJrFRVsSsDDrmXIIY7CuuWZpG_yAds6l5m-U_u1UnZ0zVSRmZhzNS5Af8Uf794uyCBa9zugzbTkNAeCDgi2qvbIjK5UBtHnqnX89Ere3nfKXlrmzH347oVKtPKXSw21bOJ8x-BOWwZs/s1600/EViews+-+VAR+Estimation+DB.png" title="EViews VAR Specification" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews VAR Specification<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><!--[if gte vml 1]><v:shape id="Picture_x0020_6"
o:spid="_x0000_i1031" type="#_x0000_t75" style='width:276.75pt;height:224.25pt;
visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image009.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Type in all the
variables names in the <b>Endogenous
variables box </b>(note under VAR, there is no exogenous variable, all
variables are endogenous). Since between 1 to 8 lags can be used because I am
using a quarterly data, I begin with 4 lags before deciding which model is the
best.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Click <b>OK</b>….here is the output (to save space
only relevant part shown):<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1NWxY-i1rzIGDdjwDl701L2q5KwBytTyKTI6JYCjF_siAyWjS623R-MugBnqztzr4hkncevJIjEKRFDjL6l8x-K-vsMeqOP4uvvTupE7t0PwkCuLCQ9Zk5l88pmIPxjvU6Zv_eF6SV8g/s1600/EViews+-+Lag+4+Output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Regression Output from cruncheconometrix.com.ng" border="0" data-original-height="523" data-original-width="495" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1NWxY-i1rzIGDdjwDl701L2q5KwBytTyKTI6JYCjF_siAyWjS623R-MugBnqztzr4hkncevJIjEKRFDjL6l8x-K-vsMeqOP4uvvTupE7t0PwkCuLCQ9Zk5l88pmIPxjvU6Zv_eF6SV8g/s1600/EViews+-+Lag+4+Output.png" title="EViews Regression Output" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews Regression Output<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">The EViews
output reports among others, the AIC and Schwarz criterion. You will also observe
that the output returned 2 sets of results, those identified by </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: red;">red bracket</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;">
are for the respective endogenous variables with each column representing the
result for </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">gdp, pce </i><span style="font-family: "times new roman" , serif; font-size: 13pt;">and</span><i style="font-family: "Times New Roman", serif; font-size: 13pt;"> pdi</i><span style="font-family: "times new roman" , serif; font-size: 13pt;"> in that order. But the results we
are most interested in are those identified by the </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: blue;">blue bracket</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;">. These are the
estimates for the VAR system. However, at this moment, we are only interested
in the criterion. Hence, between the AIC and Schwartz, the former’s criterion
of </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;">26.85144</b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> is lower than that of
Schwartz at </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;">27.98004</b><span style="font-family: "times new roman" , serif; font-size: 13pt;">. Therefore, we
conclude based on this output that the lag selection must be based on the AIC.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Step
4: Choose Optimal Lag length for the Model<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">However, we
cannot be running the unrestricted VAR model using different lag lengths before
deciding on the best model to adopt, there is a simplified way of obtaining the
optimal lag structure at once given a variety of information criteria. To do
that, click on <b>View</b> >> <b>Lag</b> <b>Structure</b> >> <b>Lag</b> <b>Length</b> <b>Criteria</b> >> the <b>Lag
Specification</b> dialog box opens:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXAE0RjpiF2ErtgR6DgP0OYQtxv3E3j0BMBlJ5S5K_u50kaP0o7fqFa90Z5sLBWZGZObqH-CaD6RnPl5WeceV5gtzMGYqPypL_Q1md0HXRskcBb0J0GA6c-k8DCCrL8OOe8-CbPxYP8tc/s1600/EViews+-+Lag+Specification.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews LagSpecification Dialogue Box from cruncheconometrix.com.ng" border="0" data-original-height="151" data-original-width="185" height="326" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXAE0RjpiF2ErtgR6DgP0OYQtxv3E3j0BMBlJ5S5K_u50kaP0o7fqFa90Z5sLBWZGZObqH-CaD6RnPl5WeceV5gtzMGYqPypL_Q1md0HXRskcBb0J0GA6c-k8DCCrL8OOe8-CbPxYP8tc/s400/EViews+-+Lag+Specification.png" title="EViews Lag Specification Dialogue Box" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews Lag Specification Dialogue Box<br />
Source:CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background-color: yellow; font-family: "times new roman" , serif; font-size: 13pt;">Note: I put in 8 lags because I am at liberty to use up
to 8 lags due to the nature of my data (quarterly). So, if yours is a yearly
data, you may put in 2.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Click <b>OK</b> to obtain the various information
criterion from lag 0 to 8 shown below:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh584cYOl2Kx2QvCA4g_0pxTf2BGXVplY3GjKlPpmAoXBfJIwi6J3NH5swAqMxwOyzHxwrzByUZBuMP3g-fM8LMsWo-QY3AFzkIOzuIOvBmupctUPRLz85_i-jntca1zVVUq9hcqRp4Bj8/s1600/EViews+-+Lag+Criterion+0+to+8.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Lag Structure for the Model from cruncheconometrix.com.ng" border="0" data-original-height="484" data-original-width="493" height="627" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh584cYOl2Kx2QvCA4g_0pxTf2BGXVplY3GjKlPpmAoXBfJIwi6J3NH5swAqMxwOyzHxwrzByUZBuMP3g-fM8LMsWo-QY3AFzkIOzuIOvBmupctUPRLz85_i-jntca1zVVUq9hcqRp4Bj8/s640/EViews+-+Lag+Criterion+0+to+8.png" title="EViews Lag Structure for the Model" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews Model Lag Structure<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">From the output,
the selected lag order is indicated by an </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;">asterisk
sign (*)</b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> which is distributed between lags 1 and 2, but mostly on lag order
2. The rule-of-thumb is to select the criterion with the lowest value which
again is the AIC at </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: #009900;">26.90693</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> this is because the lower the value,
the better the model. We can conclude that the optimal lag length for the model
is 2 and the best criterion to adopt for the model is AIC.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The same
procedure can be adopted in obtaining the respective lags for each variable.
For instance to obtain for <i>gdp</i>:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Double click on <i>gdp</i> >> <b>Quick</b>
>> Run the unrestricted VAR >> <b>OK</b>
>> Obtain the output<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Click <b>View</b> >> <b>Lag</b> <b>Structure</b> >> <b>Lag Length Criteria</b> >> <b>Lag
Specification</b> dialog box opens >> <b>OK</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">…and you obtain
this:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-size: 13.0pt; mso-ansi-language: EN-GB; mso-fareast-language: EN-GB; mso-no-proof: yes;"><!--[if gte vml 1]><v:shape
id="Picture_x0020_1" o:spid="_x0000_i1027" type="#_x0000_t75" style='width:351.75pt;
height:369.75pt;visibility:visible;mso-wrap-style:square'>
<v:imagedata src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image017.png"
o:title=""/>
</v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtfnKVejYrxZd-GcH2nlaHkTMPblxCv3leL3K6zZLFeobTqC95zKK7onOgdl_5nvH71B-o5jU7Fh2QE-c8w9Aiw0h9F7z4tVaWcPtsJXp7ELqkJ5Ya6c7Tol4yceP71Put72Lo6cbIdnI/s1600/EViews+-+Lag+for+gdp.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Lag Structure for gdp from cruncheconometrix.com.ng" border="0" data-original-height="493" data-original-width="493" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtfnKVejYrxZd-GcH2nlaHkTMPblxCv3leL3K6zZLFeobTqC95zKK7onOgdl_5nvH71B-o5jU7Fh2QE-c8w9Aiw0h9F7z4tVaWcPtsJXp7ELqkJ5Ya6c7Tol4yceP71Put72Lo6cbIdnI/s640/EViews+-+Lag+for+gdp.png" title="EViews Lag Structure for gdp" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Lag Structure for <i>gdp</i><br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">From the output,
the best criterion that fits the <i>gdp</i>
model is the AIC with the lowest figure of <b><span style="color: red;">9.937278</span></b> meaning that the optimal lag length for <i>gdp</i> is 2.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">Doing the same
procedure for </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 13pt;">, here is the
result:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeiSpH-lvT1EQXs8VBtC0IhkMSvTZ3t29v_Isi45cKXDXefNT8wY1r2cDZKcdwXkudQ5qjmb1EMSB10wOco5tGjXq-cLjh_HMsgspfHarShVeamwUEsOTVfrDcnQczQtP0c72r9u-NQ1E/s1600/EViews+-+Lag+for+pce.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Lag Structure for pce from cruncheconometrix.com.ng" border="0" data-original-height="487" data-original-width="492" height="632" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeiSpH-lvT1EQXs8VBtC0IhkMSvTZ3t29v_Isi45cKXDXefNT8wY1r2cDZKcdwXkudQ5qjmb1EMSB10wOco5tGjXq-cLjh_HMsgspfHarShVeamwUEsOTVfrDcnQczQtP0c72r9u-NQ1E/s640/EViews+-+Lag+for+pce.png" title="EViews Lag Structure for pce" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Lag Structure for <i>pce</i><br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">From the output,
the optimal lag length for </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 13pt;"> model
is 4 given the AIC value at </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: blue;">8.698617</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> which the lowest among the criterion, hence
it is the best criterion for the </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 13pt;">
model. For </span><i style="font-family: "Times New Roman", serif; font-size: 13pt;">pdi</i><span style="font-family: "times new roman" , serif; font-size: 13pt;">, the optimal lag
length is 1 given the AIC value at </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: magenta;">9.602079</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;"> shown below:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhE_uuF7Vf87oKKB5ejeRlhv63kEsEtQh45XsY9KJpPVSA7M0vh8xmD6nPyY0pIp-sjrwrGgjKRvl_XtrjxyFEqEMYHwLXD_Y7BlsXHLe9rBu1QGDUwuQCw5Xz_xo3xeNAa4IXdNUhkfUg/s1600/EViews+-+Lag+for+pdi.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews Lag Structure for pdi from cruncheconometrix.com.ng" border="0" data-original-height="489" data-original-width="485" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhE_uuF7Vf87oKKB5ejeRlhv63kEsEtQh45XsY9KJpPVSA7M0vh8xmD6nPyY0pIp-sjrwrGgjKRvl_XtrjxyFEqEMYHwLXD_Y7BlsXHLe9rBu1QGDUwuQCw5Xz_xo3xeNAa4IXdNUhkfUg/s640/EViews+-+Lag+for+pdi.png" title="EViews Lag Structure for pdi" width="634" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews - Lag Structure for <i>pdi</i><br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Caveat</span></i></b><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">: There
are also cases where the <span style="background: white;">used lag length is that
which is most selected by the criterion named after the econometricians
who developed them, like HQ, SIC, AIC and LR, etc. Some researchers prefer
Schwartz criterion when the variables are more than 4 and use the AIC when the
variables are less than 4. As, mentioned in the introductory part of this
tutorial, the decision on the choice of lag is purely an empirical issue. Generally,
we choose the lag length for which the values of most of these lag length
criteria are minimised, indicated by asterisks in the EViews output.</span></span><br />
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><span style="background: white;"><br /></span></span>
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><span style="background: white;"></span></span><br />
<div class="MsoNoSpacing" style="-webkit-text-stroke-width: 0px; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: justify; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
</div>
<br />
<div align="center" class="MsoNoSpacing" style="-webkit-text-stroke-width: 0px; color: black; font-family: "Times New Roman"; font-size: medium; font-style: normal; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: center; text-decoration-color: initial; text-decoration-style: initial; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;">
<div style="margin: 0px;">
<b><span lang="EN-US" style="background: white; font-family: "times new roman" , "serif"; font-size: 13pt;">[Watch video tutorial on optimal lag selection using EViews]<o:p></o:p></span></b></div>
</div>
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<div class="separator" style="clear: both; text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/jtb_4fqxBZE/0.jpg" src="https://www.youtube.com/embed/jtb_4fqxBZE?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Having
gone through this tutorial, it will be easy to understand and know how to determine the optimal
lags for a model regardless of the analytical package used. Remember that the “Lag
length criteria” indicates a definite way of selecting the optimal lags after estimating the initial VAR model. Also VAR and ARDL models
are susceptible to arbitrary use of lags as this may erode the degrees of
freedom, weaken the significance of the coefficients, may induce
auto-correlation and weaken the strength of diagnostic tests.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt; line-height: 107%;">Try the outlined steps
on your models and if there are further and comments, do post them below…..</span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-36462832783651189472018-02-07T08:19:00.001+01:002018-02-08T11:57:43.830+01:00Panel Data Analysis (Lecture 1): Sourcing Data, Theoretical Framework and Model Specification<h2>
</h2>
<h2 style="text-align: justify;">
<b><span lang="EN-US" style="background: yellow; color: red; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Caution: This tutorial is
only a guide and should not be adopted in its entirety. Endeavour to consult your
tutor and other resource materials for proper guidance!</span></b></h2>
<h2>
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">Introduction<br /><div style="text-align: justify;">
<span style="font-size: 13pt; font-weight: normal;">The dissertation
fervor is heating up with the usual twists and turns. In view of these and in
response to readers’ requests, I will be starting a series of lectures on how to run time
series and panel data analyses. These will be in parts and supported with short
video tutorials posted to YouTube (so ensure to hook up to get the hands-on
training). In order not to leave anyone out, these practical lectures will be
carried out using three (3) analytical packages that is common among final-year
students – Stata, EViews and Excel. Also, real country-level and longitudinal data will be used
(but subject to my modifications to prevent unethical conduct from readers).
Lastly, only quantitative research will be addressed.</span></div>
</span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">For </span><span style="color: red; font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;"><a href="http://cruncheconometrix.blogspot.com.ng/2018/02/time-series-analysis-lecture-1-sourcing.html" target="_blank">time series analysis</a></span><span style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">, the lectures will only cover: data sourcing, model
specification, lag selection, unit root testing, cointegration test, vector
autoregressive model (VAR), autoregressive distributed lag model (ARDL), vector
error correction mechanism (VECM), Granger causality tests, CUSUMSQ test and
other post-estimation tests. While for </span><span style="color: red; font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">panel data analysis</span><span style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">, the lectures will only cover:
setting up a panel data in Stata and EViews, data sourcing, model
specification, Hausman test, fixed effects (FE) model, random effects (RE)
model and generalised methods of moments (GMM).</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">So, in order to
get prompt tutorials, the moment I click the “post” button, I will encourage
you to subscribe for these blog posts. Use the <span style="background: yellow; mso-highlight: yellow;">“Follow by Email”</span>
menu on my blog <i><a href="https://cruncheconometrix.blogspot.com.ng/">https://cruncheconometrix.blogspot.com.ng</a></i>,
activate the link once you receive the notification in your email (check your
spam box too) and you are good to go! Likewise, follow that up by subscribing
to my YouTube videos for those short hands-on video clips. Click on this link<i> <a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos?view_as=subscriber" target="_blank">CrunchEconometrix</a></i><a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos?view_as=subscriber" target="_blank"> YouTube videos</a> and subscribe!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Data
Sourcing<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">“I can’t get
data!!!”, “what’s a proxy?”, “I have data but not for all the groups”,“ how do I
go about modeling my theoretical framework?”, “how do I construct my empirical
model?”, “in fact, I’m confused!”…so many questions and believe me the
chattering seems endless. First, I always tell students to relax! Secondly, I
tell them that the moment the research area has been identified, and the topic
streamlined, the next thing to do is to go on data-search. Okay, think about this:
of what use is an empirical research if there is no data (or you have
insufficient data to test your hypothesis)? So before, you proceed to writing
chapter 1 (that is, the study background), make certain that you have the data
handy.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Primary
Data Sources</b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Regardless of
the field of study or research discipline, primary data gathering requires the
use of questionnaires, interviews, focus group discussions etc. It may require
one of these or a combination of 2 or 3 data-gathering methods. So, if you are using
primary data, ensure to get out these materials and distribute to the
respondents in order to harvest responses within the shortest time frame. Getting
a good number of responses is a precursor to having a quality research and unbiased
results. However, these structured tutorials will not be extended to analysing
primary data….my sincere apologies!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Secondary
Data Sources<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Since, research
is not limited to those in the field of <a href="http://cruncheconometrix.blogspot.com.ng/2018_01_21_archive.html" target="_blank">economics</a>, it is important that
researchers identify those databases hosting the relevant data required for their
work. As an economist, I will indicate some databases/sources where students
can go source for their data. </span><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Here are some which can be
accessed (for macro and micro datasets):</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Coal Information<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA CO2 Emissions from Fuel Combustion<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Electricity information<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Energy Prices and Taxes<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Energy Technology Research and Development
Database<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Natural Gas Information<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Oil Information<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA Renewables Information<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IEA World Energy Statistics and Balances<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">ILO Key Indicators of the Labour Market<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IMF Balance of Payment Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IMF Direction of Trade Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IMF Government Finance Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">IMF International Financial Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IMF World Economic Outlook</span><span lang="EN-US" style="font-family: "times new roman" , "serif";"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Education Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Globalisation<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD International Development<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD International Direct Investment Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD International Migration Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD International Trade by Commodities Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Main Economic Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Main Science and Technology Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD National Accounts<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Quarterly Labour Force Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Services Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Social Expenditure Database<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">OECD Structural Analysis<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">UNIDO Industrial Demand Supply<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">UNIDO Industrial Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">World Bank Global Development Finance<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">World Bank World Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">World Bank Africa Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Other sources of international data include but not
limited to:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">World Bank - </span><span lang="EN-US"><a href="http://data.worldbank.org/data-catalog/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://data.worldbank.org/data-catalog/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">International Monetary Fund - </span><span lang="EN-US"><a href="http://www.imf.org/external/data.htm#data"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://www.imf.org/external/data.htm#data</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">United Nations - </span><span lang="EN-US"><a href="http://data.un.org/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://data.un.org/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span lang="EN-US"><a href="http://unstats.un.org/unsd/databases.htm"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://unstats.un.org/unsd/databases.htm</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Data on aid flows complied by OECD - </span><span lang="EN-US"><a href="http://www.oecd.org/dac/stats/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://www.oecd.org/dac/stats/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">NBER data sets - </span><span lang="EN-US"><a href="http://www.nber.org/data/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://www.nber.org/data/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">For information from over 256 and regions since 1960,
the accessible databases are:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">World Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Global Development Finance<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">The African Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Doing Business<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Education Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Enterprise Surveys<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Gender Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Health Nutrition and Population Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-weight: normal;">Millennium Development Goals<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Worldwide Governance Indicators</span><span lang="EN-US" style="font-family: "times new roman" , "serif";"><o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Endeavour to
check out those sites that are relevant to your study. <o:p></o:p></span></div>
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<span style="font-weight: normal;"><span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Note: it is expected that you state your data source in
your thesis/dissertation and the years of coverage say 1980 to 2016, or 1970 to
2015 etc.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>What
is a Panel Data?<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">The panel data
approach pools time series data with cross-sectional data. Depending on the
application, it can comprise a sample of individuals, firms, countries, or
regions over a specific time period. The general structure of such a model
could be expressed as follows:<o:p></o:p></span></div>
<div class="Default" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: bold;">Y<sub>it</sub>
= </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13pt; font-weight: bold;">a</span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: bold;">
+ </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13pt; font-weight: bold;">b</span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>X</b><sub style="font-weight: bold;">it</sub><b>
+ u</b><sub style="font-weight: bold;">it</sub> </span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">where u<sub>it</sub> ~ IID(0, </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13pt;">s</span><sup><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">2</span></sup><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">),<i style="font-weight: normal;">i</i> = 1,2,…,<i style="font-weight: normal;">N</i>
individual-level observations, and <i style="font-weight: normal;">t</i>
= 1, 2,…,<i style="font-weight: normal;">T</i> time series observations.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">In this
application, it is assumed that Y<sub>it</sub>
is a continuous variable. The panel data model is simply where the observations
of each individual, firm or country over time are stacked on top of each
another. This is the standard pooled model where intercepts and slope
coefficients are homogeneous across all <i>N</i>
cross-sections and through all <i>T</i> time
periods. The application of ordinary least squares (OLS) to this model ignores
the temporal and spatial dimension inherent in the data and thus throws away
useful information. It is important to note that the temporal dimension
captures the ‘<i>within’</i> variation in
the data while the spatial dimension captures the ‘<i>between’</i> variation in the data. The pooled OLS estimator exploits
both ‘between’ and ‘within’ dimensions of the data but does not do so
efficiently. Thus, in this procedure each observation is given equal weight in
estimation. In addition, the unbiasedness and consistency of the estimator
requires that the explanatory variables are uncorrelated with any omitted
factors. The limitations of OLS in such an application prompted interest in
alternative procedures. There are a number of different panel estimators but
the most popular is the fixed effects (or ‘within’) estimator and this will be
reviewed extensively here. Lastly, the generalized methods of moments (GMM)
estimator will be discussed given its relevance to dynamic panel modelling.<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Some Advantages of Panel Data Analysis</b></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Panel data
analysis has quite a number of distinct advantages over time series and cross-section analysis:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Panel
(or longitudinal) data allows a researcher to analyse a number of important
economic questions not readily answerable by either a cross-section or a
time-series dataset alone.<o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The
availability of panel data increases the number of data points available and
reduces collinearity among the explanatory variables thus improving the
efficiency of the econometric estimates.<o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Panel
data captures the heterogeneity that is related to the individuals, firms,
states, countries etc. over time.<o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">By
combining time series of cross-sectional observations, panel data gives “more
informative data, more variability, less collinearity among variables, more
degrees of freedom and more efficiency”.<o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Dynamic
effects cannot be estimated using cross-sectional data. Even time series data
are imprecise in this regard as there is generally limited change or variation
in the data to identify such effects. For instance, in estimating a distributed
lag model using only time series data, <a href="http://cruncheconometrix.blogspot.com.ng/2018/01/multicollinearity.html" target="_blank">multicollinearity</a> lowers the precision
of the estimates. Hence, panel data models can provide greater variation in the
explanatory variable for a given year thus reducing the degree of
multicollinearity and improving the precision of the estimates. This clearly
renders panel data better suited to the study of dynamic change However, it
should be emphasised that the estimation procedures required for dynamic models
which include a lagged dependent variable are not straightforward and this
issue is the subject of discussion in later sections. <o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Panel
data models can take into account a greater degree of the heterogeneity that
characterize individuals, states, and firms over time.<o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">(Detailed discussion on the rudiments of panel data
analysis will be done in the next tutorial).</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">(Here is the
<a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos?view_as=subscriber" target="_blank">link</a> to video clip on converting wide-format data to long-format in Stata).<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Model
Framework and Specification<o:p></o:p></b></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">This section
focusses on the theoretical framework and model specification. I will also
touch on description of variables in a model, the <i>a priori</i> expectations and finally, the method of analysis (or the
estimation technique(s) to be used in testing the research hypothesis). <o:p></o:p></span></div>
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<br /></div>
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<b><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">Theoretical Framework</span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Before you
specify the empirical model, you <u>must</u>
first state the <a href="http://cruncheconometrix.blogspot.com.ng/2018/01/a-step-by-step-tutorial-on-research-and_14.html" target="_blank">theoretical model</a>. That is, let your readers know where your
empirical model is linked to. The theoretical model is that model supporting
the theory you are using to undertake your research because no research can be
done in isolation without an underlying theory. For instance, if my study is on
the effect of exchange rate on output for 30 countries from 2000 to 2016 (that is, 17years), then I must look for a
suitable theory which I can adapt to my research. </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Hence, I may decide to use the “monetary model of exchange
rate” which is one of the earliest models used to determine the exchange rate.
It is used as a measure to study the other approaches that are used in
determining exchange rate. The monetary model approach assumes a simple demand
for money curve, the purchasing power parity or the law of one price and a
vertical aggregate supply curve. </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">The theoretical
framework can be built as follows: <span style="background: yellow; mso-highlight: yellow;">(remember that this is just an example, and should not to be copied
literarily!)</span><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">From the
absolute purchasing power parity (</span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>P = EP*</b></span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">), the exchange rate is
obtained by dividing the price of the domestic currency by the foreign price
for that domestic currency. That is: </span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>E<sup>ppp</sup> = P/P*</b></span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">. The demand
for money assumption: since real money balance depends on real income, demand
for money is given as </span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>M<sub>d</sub> = <i>k</i>PY</b></span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">, where <i>k</i> is constant and Y is the real income level. Hence, in
equilibrium, money demand (M<sub>d</sub>) equals money supply (M<sub>s</sub>)
and at the point of intersection of the aggregate demand and the aggregate
supply curve:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">k</span></i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">PY<i>=</i>M<sub>s</sub><o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">P<i> = </i>M<sub>s</sub><i>/k</i>Y<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">EP*<i> = </i>P<i>
= </i>M<sub>s</sub><i>/k</i>Y <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">and E<i> = </i>M<sub>s</sub><i>/</i>P<i>*k</i>Y<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">From the stated framework, it is theorised that if the money
supply within an economy increases, it will result in appreciation of the
domestic currency. Hence, if it is generalised for the 30 countries in the data, the same assumption must be made, <i>ceteris paribus</i>. Likewise, foreign price level and the output level are
inversely related to the exchange rate. If fixed money supply rises in the
domestic economy, since prices are held constant, excess money supply leads to
higher demand for goods and services within the economy.<a href="https://www.blogger.com/null" name="_Toc479270173"><o:p></o:p></a></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">Model
Specification</span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">So, having
stated the theoretical framework, I can now go ahead to modify it to suit my
research and form there formulate my empirical model. For instance, in using a
Cobb-Douglas production from the neo-classical growth mode, I will attempt to explain
output growth in the context of capital accumulation, labour and productivity,
usually referred to as technological progress. The Cobb- Douglas production
model is implicitly stated as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Y
= f(AL<sub>β</sub>K<sub>α</sub>) [1]<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">where, Y is output;
K is capital stock; L is labour and A is productivity of
labour which grows at an exogenous rate. As a result of constant returns to
scale, if all inputs are increased by the same amount, then there would be an
increase in output. The production function,<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Y
= K<sup>α</sup>L<sup>1-α </sup> [2]<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-weight: normal;"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">where (1 - </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">a</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> = </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">b</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">) is
mainly used by economists and researchers due to the following reasons:
firstly, there is a constant return to scale and secondly, the two exponents α
and (1 - </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">a</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">),
sum up to one. <o:p></o:p></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Next, is to tie
up the empirical model to the theoretical framework. That is given the
relationship between exchange rate and output, the model is implicitly specified
as: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Y<sub>it</sub>
= f (Exchrate<sub>it</sub>, X<sub>1it</sub>, X<sub>2it</sub>, …, X<sub>nit</sub>) [3]<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">where Y<sub>it</sub>
= output (the dependent variable, state the measurement either gross output, or
% of GDP, or growth rate etc.) <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Exchrate<sub>it</sub>
= real exchange rate (main explanatory variable)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">X<sub>1it</sub>,
X<sub>2it</sub>, …, X<sub>nit</sub> = control variables (state their individual
measurements either gross output, or % of GDP, or growth rate etc.)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">On the basis of
the theoretical framework and using the Cobb-Douglas production, the explicit
model is stated as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Y<sub>it</sub>
= β<sub>0</sub> + β<sub>1</sub>Exchrate<sub>it</sub> + β<sub>2</sub>X<sub>1it</sub>
+ β<sub>3</sub>X<sub>2it</sub> + … + β<sub>n</sub>X<sub>nit </sub>+ u<sub>it</sub> [4]<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">where, u<sub>t</sub>
= white noise error term <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"> <b><i>A Priori</i>
Expectation</b></span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>s<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Always know that
the expected <i>a priori </i>is directly
related to what theory says. It is from that you know what signs of the
coefficients are expected from the main regressor and other covariates. For
instance, from the theory, it is expected that currency depreciation will have
a positive impact on domestic output, hence, a negative sign of the coefficient
is expected. That is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">β<sub>1
</sub><i>< 0<o:p></o:p></i></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Therefore, the
expected signs of the control variables must be in line with their respective
theories which must be related to your study.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;"><b>Estimation
Technique<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">At this point,
the researcher may not know the exact technique or estimator to adopt between
the fixed-effects within-group (fixed effects model) or the random effects
estimator. The choice between these two is subject to the outcome of the <i>Hausman test</i>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">That is, to determine
which model is the more appropriate to adopt, a statistical test is implemented.
The Hausman test compares the random effects estimator to the ‘within’
estimator. The null hypothesis of the test is that the composite error term is
not correlated with the explanatory variables in the model. If the null is
rejected, then the fixed effects estimator is applicable (i.e., it favours the
fixed effects but only relative to the random effects). The use of the test in
this case is to discriminate between a model where the omitted heterogeneity is
treated as fixed and correlated with the explanatory variables, and a model
where the omitted heterogeneity is treated as random and independent of the
explanatory variables.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">Variables</span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">, Measurement and Description<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Lastly, tabulate
your variables detailing their names, short description, measurement and
sources. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Here’s an
example:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">Table xxx: Variables
Description and Measurement<o:p></o:p></span></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; mso-yfti-tbllook: 1184; width: 595px;">
<tbody>
<tr>
<td style="border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 95.4pt;" valign="top" width="127"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Variables<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 157.5pt;" valign="top" width="210"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Short
Definition<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 94.15pt;" valign="top" width="126"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Measurement<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 99.25pt;" valign="top" width="132"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Source<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 95.4pt;" valign="top" width="127"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Output<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 157.5pt;" valign="top" width="210"><div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 94.15pt;" valign="top" width="126"><div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 99.25pt;" valign="top" width="132"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">World Bank (2016)<o:p></o:p></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 95.4pt;" valign="top" width="127"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Real exchange
rate<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 157.5pt;" valign="top" width="210"><div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 94.15pt;" valign="top" width="126"><div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 99.25pt;" valign="top" width="132"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">World Bank (2016)<o:p></o:p></span></div>
</td>
</tr>
</tbody></table>
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal; line-height: 115%;">If you have any
comments or question in relation to what have been discussed in this post, do
not hesitate to post them in the comment section below….</span><br />
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 10pt; font-weight: normal;">Source:
Researcher’s compilation <span style="background: yellow; mso-highlight: yellow;">(always put this at the bottom
of the Table)</span><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b style="font-family: "times new roman", serif; font-size: 13pt;">Conclusion</b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt; font-weight: normal;">I have taken you
through the steps required on how to source for your data, in addition to a brief on panel data
analysis and its relevance over time series and cross-sectional data. I also briefly explained how to
formulate a theoretical framework, adapting the framework to align with the
research, how to construct the empirical model, stating the expected <i>a priori</i>, having an idea about the
estimation technique with a brief on the Hausman test, tabulating your data
showing the brief description of your variables, their measurements and data
sources.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 13pt;">From next
lecture, I will begin analysing the data using both Stata and EViews analytical
packages. So, endeavour to follow these tutorials by getting the most of it to
ease the dissertation pressure. Make sure you follow me on the next lecture
series which is: <i><b>Panel Data Analysis (Lecture 2): Setting up panel data model and the
Hausman Test.</b><o:p></o:p></i></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">If you have any
comments or question in relation to what have been discussed, do
not hesitate to post them in the comment section below….</span><br />
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-47841598402035932022018-02-07T07:30:00.000+01:002018-02-08T11:54:44.615+01:00Time Series Analysis (Lecture 1): Sourcing Data, Theoretical Framework and Model Specification<h2 style="text-align: justify;">
<b><span lang="EN-US" style="background: yellow; color: red; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Caution: This tutorial is
only a guide and should not be adopted in its entirety. Endeavour to consult your
tutor and other resource materials for proper guidance!</span></b></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<h2 style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Introduction</span></b></h2>
<h2 style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The dissertation
fervor is heating up with the usual twists and turns. In view of these and in
response to readers’ requests, I will be starting a series of lectures on how to run time
series and panel data analyses. These will be in parts and supported with short
video tutorials posted to YouTube (so ensure to hook up to get the hands-on
training). In order not to leave anyone out, these practical lectures will be
carried out using three (3) analytical packages that is common among final-year
students – <b>Stata</b>, <b>EViews</b> and <b>Excel</b>. Also, real country-level data will be used (but subject to
my modifications to prevent unethical conduct from readers). Lastly, only
quantitative research will be addressed. </span></h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;">For </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: red;">time series analysis</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;">, the lectures will only cover:
data sourcing, model specification, lag selection, unit root testing,
cointegration test, vector autoregressive model (VAR), autoregressive
distributed lag model (ARDL), vector error correction mechanism (VECM), Granger
causality tests, CUSUMSQ test and other post-estimation tests. While for </span><b style="font-family: "Times New Roman", serif; font-size: 13pt;"><span style="color: red;">panel data analysis</span></b><span style="font-family: "times new roman" , serif; font-size: 13pt;">,
the lectures will only cover: setting up a panel data in Stata and EViews, data
sourcing, model specification, Hausman test, fixed effects (FE) model, random
effects (RE) model and generalised methods of moments (GMM).</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span>
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">So, in order to get prompt tutorials, the moment I click
the “post” button, I will encourage you to subscribe for these blog posts. Use
the <b><span style="background: yellow; mso-highlight: yellow;">“Follow by Email”</span></b> menu on my blog <i><a href="https://cruncheconometrix.blogspot.com.ng/">https://cruncheconometrix.blogspot.com.ng</a></i>,
activate the link once you receive the notification in your email (check your
spam box too) and you are good to go! Likewise, follow that up by subscribing
to my YouTube videos for those short hands-on video clips. Click on this link:<i> <b><a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos?view_as=subscriber" target="_blank">CrunchEconometrix</a></b></i><b><a href="https://www.youtube.com/channel/UCK9hD254JKbCZ4Bf8Iz1s7g/videos?view_as=subscriber" target="_blank"> YouTube videos</a></b> and
subscribe!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Data
Sourcing<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">“I can’t get
data!!!”, “what’s a proxy?”, “I have data but not for 30 years”,“ how do I go
about modeling my theoretical framework?”, “how do I construct my empirical
model?”, “in fact, I’m confused!”…so many questions and believe me the
chattering seems endless. First, I always tell students to relax! Secondly, I
tell them that the moment the research area has been identified, and the topic
streamlined, the next thing to do is to go on data-search. Okay, think about this:
of what use is an empirical research if there is no data (or you have
insufficient data to test your hypothesis)? So before, you proceed to writing
chapter 1 (that is, the study background), make certain that you have the data
handy.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Primary
Data Sources<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Regardless of
the field of study or research discipline, primary data gathering requires the
use of questionnaires, interviews, focus group discussions etc. It may require
one of these or a combination of 2 or 3 data-gathering methods. So, if you are using
primary data, ensure to get out these materials and distribute to the
respondents in order to harvest responses within the shortest time frame. Getting
a good number of responses is a precursor to having a quality research and unbiased
results. However, these structured tutorials will not be extended to analysing
primary data….my sincere apologies!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Secondary
Data Sources</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Since, research
is not limited to those in the field of <a href="http://cruncheconometrix.blogspot.com.ng/2018_01_21_archive.html" target="_blank">economics</a>, it is important that
researchers identify those databases hosting the relevant data required for
their work. As an economist, I will indicate some databases/sources where
students can go source for their data. </span><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Here are some
which can be accessed (for macro and micro datasets):</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Coal Information<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA CO2 Emissions from Fuel Combustion<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Electricity information<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Energy Prices and Taxes<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Energy Technology Research and Development
Database<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Natural Gas Information<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Oil Information<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA Renewables Information<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IEA World Energy Statistics and Balances<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">ILO Key Indicators of the Labour Market<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IMF Balance of Payment Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IMF Direction of Trade Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IMF Government Finance Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IMF International Financial Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">IMF World Economic Outlook</span><span lang="EN-US" style="font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Education Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Globalisation<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD International Development<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD International Direct Investment Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD International Migration Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD International Trade by Commodities Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Main Economic Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Main Science and Technology Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD National Accounts<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Quarterly Labour Force Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Services Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Social Expenditure Database<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">OECD Structural Analysis<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">UNIDO Industrial Demand Supply<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">UNIDO Industrial Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">World Bank Global Development Finance<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">World Bank World Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">World Bank Africa Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<br /></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Other sources of international data include but not
limited to:<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">World Bank - </span><span lang="EN-US"><a href="http://data.worldbank.org/data-catalog/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://data.worldbank.org/data-catalog/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">International Monetary Fund - </span><span lang="EN-US"><a href="http://www.imf.org/external/data.htm#data"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://www.imf.org/external/data.htm#data</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">United Nations - </span><span lang="EN-US"><a href="http://data.un.org/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://data.un.org/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span lang="EN-US"><a href="http://unstats.un.org/unsd/databases.htm"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://unstats.un.org/unsd/databases.htm</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Data on aid flows complied by OECD - </span><span lang="EN-US"><a href="http://www.oecd.org/dac/stats/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://www.oecd.org/dac/stats/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">NBER data sets - </span><span lang="EN-US"><a href="http://www.nber.org/data/"><span lang="EN-GB" style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">http://www.nber.org/data/</span></a></span><span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;"><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<br /></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">For information from over 256 and regions since 1960,
the accessible databases are:<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">World Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Global Development Finance<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">The African Development Indicators<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Doing Business<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Education Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Enterprise Surveys<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Gender Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Health Nutrition and Population Statistics<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Millennium Development Goals<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span style="font-family: "times new roman" , "serif"; mso-ansi-language: EN-GB;">Worldwide Governance Indicators</span><span lang="EN-US" style="font-family: "times new roman" , "serif";"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Endeavour to
check out those sites that are relevant to your study. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="background: yellow; font-family: "times new roman" , "serif"; font-size: 13.0pt;">Note: it is expected that you state your data source in
your thesis/dissertation and the years of coverage say 1980 to 2016, or 1970 to
2015 etc.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Model
Framework and Specification<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">This section
focusses on the theoretical framework and model specification. I will also
touch on description of variables in a model, the <i>a priori</i> expectations and finally, the method of analysis (or the
estimation technique(s) to be used in testing the research hypothesis). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Theoretical Framework</span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Before you
specify the empirical model, you <b><u>must</u></b>
first state the <a href="http://cruncheconometrix.blogspot.com.ng/2018/01/a-step-by-step-tutorial-on-research-and_14.html" target="_blank">theoretical model</a>. That is, let your readers know where your
empirical model is linked to. The theoretical model is that model supporting
the theory you are using to undertake your research because no research can be
done in isolation without an underlying theory. For instance, if my study is on
the impact of exchange rate on output, then I must look for a suitable theory which
I can adapt to my research. </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Hence,
I may decide to use the “monetary model of exchange rate” which is one of the earliest
models used to determine the exchange rate. It is used as a measure to study
the other approaches that are used in determining exchange rate. The monetary
model approach assumes a simple demand for money curve, the purchasing power
parity or the law of one price and a vertical aggregate supply curve. </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">The theoretical framework can be built like this: (remember that this is
just an example, and not to be copied literarily!)<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">From the absolute purchasing power parity (<b>P =
EP*</b>), the exchange rate is obtained by dividing the price of the domestic
currency by the foreign price for that domestic currency. That is: <b>E<sup>ppp</sup>
= P/P*. </b>The demand for money assumption: since real money balance depends
on real income, demand for money is given as <b>M<sub>d</sub> = <i>k</i>PY</b>, where <i>k</i> is constant
and Y is the real income level. Hence, in equilibrium, money demand (M<sub>d</sub>)
equals money supply (M<sub>s</sub>) and at the point of intersection of the
aggregate demand and the aggregate supply curve:<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">k</span></i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">PY<i>=</i>M<sub>s</sub><o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">P<i> = </i>M<sub>s</sub><i>/k</i>Y<o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">EP*<i> = </i>P<i> = </i>M<sub>s</sub><i>/k</i>Y <o:p></o:p></span></div>
<div class="MsoNoSpacing">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">and E<i> = </i>M<sub>s</sub><i>/</i>P<i>*k</i>Y<b><o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">From the stated framework, it is theorised that if the money
supply within an economy increases, it will result in appreciation of the
domestic currency. Likewise, foreign price level and the output level are
inversely related to the exchange rate. If fixed money supply rises in the
domestic economy, since prices are held constant, excess money supply leads to
higher demand for goods and services within the economy.<a href="https://www.blogger.com/null" name="_Toc479270173"><o:p></o:p></a></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Model
Specification</span></b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">So, having
stated the theoretical framework, I can now go ahead to modify it to suit my
research and form there formulate my empirical model. For instance, in using a
Cobb-Douglas production from the neo-classical growth mode, I will attempt to explain
output growth in the context of capital accumulation, labour and productivity,
usually referred to as technological progress. Focusing on a closed economy, I
can implicitly express the Cobb- Douglas production model as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Y
= f(AL<sub>β</sub>K<sub>α</sub>) [1]<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">where, Y is output;
<b>K </b>is capital stock; <b>L </b>is labour and <b>A </b>is productivity of
labour which grows at an exogenous rate. As a result of constant returns to
scale, if all inputs are increased by the same amount, then there would be an
increase in output. The production function,<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Y
= K<sup>α</sup>L<sup>1-α </sup> [2]<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">where (1 - </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">a</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> = </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">b</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">) is
mainly used by economists and researchers due to the following reasons:
firstly, there is a constant return to scale and secondly, the two exponents α
and (1 - </span><span lang="EN-US" style="font-family: "symbol"; font-size: 13.0pt;">a</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">),
sum up to one. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Next, is to tie
up my empirical model to the theoretical framework. That is given the
relationship between exchange rate and output, I can specify the model
implicitly as: <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Y<sub>t</sub>
= f (Exchrate<sub>t</sub>, X<sub>1t</sub>, X<sub>2t</sub>, …, X<sub>nt</sub>) [3]</span></b><br />
<span style="font-family: "times new roman" , serif; font-size: 13pt;">where Y = output
(the dependent variable, state the measurement either gross output, or % of
GDP, or growth rate etc.)</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Exchrate = real
exchange rate (main explanatory variable)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">X<sub>1</sub>, X<sub>2</sub>,
…, X<sub>n</sub> = control variables (state their individual measurements
either gross output, or % of GDP, or growth rate etc.)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">On the basis of
the theoretical framework and using the Cobb-Douglas production, the explicit
model is stated as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Y<sub>t</sub>
= β<sub>0</sub> + β<sub>1</sub>Exchrate<sub>t</sub> + β<sub>2</sub>X<sub>1t</sub>
+ β<sub>3</sub>X<sub>2t</sub> + … + β<sub>n</sub>X<sub>nt </sub>+ u<sub>t</sub> [4]</span></b><br />
<span style="font-family: "times new roman" , serif; font-size: 13pt;">where, u = error
term</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> <i>A Priori</i>
Expectation</span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">s<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Always know that
the expected <i>a priori </i>is directly
related to what theory says. It is from that you know what signs of the
coefficients are expected from the main regressor and other covariates. For
instance, from the theory, it is expected that currency depreciation will have
a positive impact on domestic output, hence, a negative sign of the coefficient
is expected. That is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">β<sub>1
</sub><i>< 0<o:p></o:p></i></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Therefore, the
expected signs of the control variables must be in line with their respective
theories which must be related to your study.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Estimation
Technique<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">At this point,
the researcher may not know the exact technique to adopt between the vector
autoregressive model (VAR) and the autoregressive distributed lag (ARDL) model.
The choice between these two is subject to the outcome of the unit root test
(URT) on each of the variables used in the model. This implies that, until the
URT is carried out, one cannot know whether to use VAR or ARDL model. This is
because if the variables are integrated of the same order, the VAR model is
applicable but the ARDL model suffices if otherwise. I will discuss these two
in detail in subsequent tutorials using both Stata and EViews analytical
softwares in addition to video clips on how to perform them.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Variables</span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">, Measurement and Description<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Lastly, tabulate
your variables detailing their names, short description, measurement and
sources. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Here’s an
example:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Table xxx: Variables
Description and Measurement<o:p></o:p></span></b></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-table-layout-alt: fixed; mso-yfti-tbllook: 1184; width: 595px;">
<tbody>
<tr>
<td style="border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 95.4pt;" valign="top" width="127"><div align="center" class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Variables<b><o:p></o:p></b></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 157.5pt;" valign="top" width="210"><div align="center" class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Short
Definition<b><o:p></o:p></b></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 94.15pt;" valign="top" width="126"><div align="center" class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Measurement<o:p></o:p></span></div>
</td>
<td style="border-left: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-themecolor: text1; mso-border-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 99.25pt;" valign="top" width="132"><div align="center" class="MsoNoSpacing" style="text-align: center;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Source<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 95.4pt;" valign="top" width="127"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Output<b><o:p></o:p></b></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 157.5pt;" valign="top" width="210"><div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 94.15pt;" valign="top" width="126"><div class="MsoNoSpacing">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 99.25pt;" valign="top" width="132"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">World Bank (2016)<b><o:p></o:p></b></span></div>
</td>
</tr>
<tr>
<td style="border-top: none; border: solid black 1.0pt; mso-border-alt: solid black .5pt; mso-border-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 95.4pt;" valign="top" width="127"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Real exchange
rate<o:p></o:p></span></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 157.5pt;" valign="top" width="210"><div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 94.15pt;" valign="top" width="126"><div class="MsoNoSpacing">
<br /></div>
</td>
<td style="border-bottom: solid black 1.0pt; border-left: none; border-right: solid black 1.0pt; border-top: none; mso-border-alt: solid black .5pt; mso-border-bottom-themecolor: text1; mso-border-left-alt: solid black .5pt; mso-border-left-themecolor: text1; mso-border-right-themecolor: text1; mso-border-themecolor: text1; mso-border-top-alt: solid black .5pt; mso-border-top-themecolor: text1; padding: 0cm 5.4pt 0cm 5.4pt; width: 99.25pt;" valign="top" width="132"><div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">World Bank (2016)<b><o:p></o:p></b></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 10.0pt;">Source:
Researcher’s compilation <b><span style="background: yellow; mso-highlight: yellow;">(always put this at the bottom
of the Table)</span><o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Conclusion</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">I have taken you
through the steps required for sourcing your data, formulating your theoretical
framework, adapting the framework to align with your research, constructing
your empirical model, stating the expected <i>a
priori</i>, having an idea about the estimation technique and tabulating your
data showing the brief description of your variables, their measurements </span><span style="font-family: "times new roman" , serif; font-size: 13pt;">and
data sources.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 13pt;"><br /></span>
<span style="font-family: "times new roman" , serif; font-size: 13pt;">From next
lecture, I will begin analysing the data using both Stata and EViews analytical
packages. So, endeavour to follow these tutorials by getting the most of it to
ease the dissertation pressure. Make sure you follow me on the next lecture
series which is: </span><b style="font-family: "times new roman", serif; font-size: 13pt;"><i>Time Series Analysis (Lecture 2): Optimal lag selection.</i></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<span style="font-family: "times new roman" , serif; font-size: 13pt; text-align: justify;">If you have any
comments or question in relation to what have been discussed in this post, do
not hesitate to post them in the comment section below….</span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-52365650214732257992018-02-03T09:30:00.000+01:002018-02-14T14:56:38.054+01:00Interpreting Regression Output from EViews<h2 style="text-align: justify;">
<span style="font-family: "times new roman" , serif;"><span style="font-size: 12pt;">The </span></span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">dissertation
semester</i><span style="font-family: "times new roman" , serif;"><span style="font-size: 12pt;"> is here for undergraduate students in most tertiary institutions,
at least for those whose academic calendar is </span><span style="font-size: 16px;">uninterrupted</span><span style="font-size: 12pt;"> </span></span><span style="font-family: "wingdings"; font-size: 12pt;">J</span><span style="font-family: "times new roman" , serif; font-size: 12pt;">. The students are in different
stages of their <i>project</i>, as it is
commonly called. Some are yet to wrap up their chapter one which gives the
“study background” and the framing of research hypotheses, objectives and
questions. Some have moved on to chapter two reviewing relevant literature
related to their scope of study. Others have gone further in developing both
the theoretical and empirical frameworks for chapter three, but not without the
usual teething lags…but they’ll get around it, somehow </span><span style="font-family: "wingdings"; font-size: 12pt;">J</span><span style="font-family: "times new roman" , serif; font-size: 12pt;">. A handful have made tremendous
progress in hitting chapter four attempting to analyse their data.</span></h2>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Because chapters one to three are
relative to each students’ scope of work, while a regression output is common
to all (although actual outcomes differ), I decided to do this tutorial in
explaining the basic features of a regression output. Again, this write-up is
in response to requests received from readers on (1) what some specific figures
in a regression output are and (2) how to interpret the results. Let me state
here that regardless of the analytical software whether Stata, EViews, SPSS, R,
Python, Excel etc. what you obtain in a regression output is common to all
analytical packages (howbeit with slight changes).</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">For instance, in undertaking an ordinary
least squares (OLS) estimation using any of these applications, the regression
output will give the ANOVA (analysis of variance) table, <i>F</i>-statistic, <i>R</i>-squared,
prob-values, coefficient, standard error, <i>t</i>-statistic,
sum of squared residuals and so on. These are some common features of a
regression output. However, the issue is: what do they mean and how can they be
interpreted in relation to your study?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Hence, the essence of this tutorial is to
teach students the significance of these features and how to interpret their
results. I will be using <b>EViews</b>
analytical package to explain a regression output, but you can practise along
using any analytical package of your choice. (See "How-to-interpret regression output" here for <b><a href="http://cruncheconometrix.blogspot.com.ng/2018/01/how-to-interpret-regression-output-in.html" target="_blank">Stata</a></b> and <b><a href="http://cruncheconometrix.blogspot.com.ng/2018/02/how-to-interpret-regression-output-in.html" target="_blank">Excel</a></b> users).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">An Example: Use <a href="https://drive.google.com/drive/u/0/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table7_12.xlsx dataset</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note: I will not be discussing
stationarity or cointegration analysis in this contest, just doing a <span style="background: yellow; mso-highlight: yellow;">simple linear regression</span>
analysis (a bi-variate analysis) with only one explanatory variable. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The dataset is on the United States from
1960 to 2009 (50 years data). The outcome variable is consumption expenditure (<i>pce</i>) and the explanatory variable is
income (<i>income</i>).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">First step: Load data in Excel format into EViews<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Here is the data in excel format:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmPD4WBy3j5EC1eQql5y566h6hI7MR13Jse8XhsISqEAf1EikCpOlsW0_H0Ed-FigYakfge61k1wg1uP4qAO7LKbVV-kHpsWEy7sM5D9IvQjmWcnY5iAAmkykpwe7D832ee4hKEKVSYFg/s1600/Data+in+Stata+format.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Data in excel format from cruncheconometrix.com.ng" border="0" data-original-height="711" data-original-width="705" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmPD4WBy3j5EC1eQql5y566h6hI7MR13Jse8XhsISqEAf1EikCpOlsW0_H0Ed-FigYakfge61k1wg1uP4qAO7LKbVV-kHpsWEy7sM5D9IvQjmWcnY5iAAmkykpwe7D832ee4hKEKVSYFg/s640/Data+in+Stata+format.png" title="Data in excel format" width="634" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Data in Excel file format<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shapetype coordsize="21600,21600" filled="f" id="_x0000_t75" o:preferrelative="t" o:spt="75" path="m@4@5l@4@11@9@11@9@5xe" stroked="f">
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<v:formulas>
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</v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:formulas>
<v:path gradientshapeok="t" o:connecttype="rect" o:extrusionok="f">
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<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">To import the Excel file into EViews, go
to: </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">File</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> >> </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Import</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> >> </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Import from file</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> >> </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Next</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">
>> </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Finish</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">. If it is correctly
done, you obtain:</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_1" o:spid="_x0000_i1037" style="height: 250.5pt; mso-wrap-style: square; visibility: visible; width: 278.25pt;" type="#_x0000_t75">
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</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJeQCCPGgchpS14VyztSUAE10Zsz_wvLryzWVA83aOzugx8XWzOKLgdL-Ox9KRKgKotSsqp3Fe6HxV2AUEYxyeVTd-LLzQGp0GRCqUI46qjaPB0eWxdWzsm1YxIo05BeIg5HAP_svTyjw/s1600/EViews+-+Import+Excel+File.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Import Excel from into EViews from cruncheconometrix.com.ng" border="0" data-original-height="448" data-original-width="497" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJeQCCPGgchpS14VyztSUAE10Zsz_wvLryzWVA83aOzugx8XWzOKLgdL-Ox9KRKgKotSsqp3Fe6HxV2AUEYxyeVTd-LLzQGp0GRCqUI46qjaPB0eWxdWzsm1YxIo05BeIg5HAP_svTyjw/s1600/EViews+-+Import+Excel+File.png" title="Import Excel file into EViews" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Import Excel file into EViews<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="background-color: yellow; font-family: "times new roman" , serif; text-align: justify;">Note: In EViews almost everything can
be done either by typing commands or by choosing a menu</span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="background-color: yellow; font-family: "times new roman" , serif; text-align: justify;">item (the Guide User
Interface, GUI). The choice is a matter of personal preference.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Second step: Visualise the relationship between the variables<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before analysing the data, it is good to
always graph the dependent and key explanatory variable (using a scatter plot) in
order to observe the pattern between them. It sorts of gives you what to expect
in your actual analysis.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since we want to see the relationship
between <i>pce</i> and <i>income </i>over the 50-year period, it means that we want to look at
the variables <i>pce</i> and <i>income</i> together. In EViews a collection
of series dealt with together is called a <b>Group</b>. Thus, to create a group including
<i>pce</i> and <i>income</i>, first click on <i>income</i>.
Now, while holding down the Ctrl-key, click on <i>pce</i>. Then right-click anywhere on the interface highlighting <b>New
Object</b>, bringing up the context menu as shown below:</span><span style="font-family: "times new roman" , serif; font-size: 12pt;"> </span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLns9sZX95KzR0sU8Dvrt5G1JZpy3poWufGExpzfVaAckEIHwtPCqStTNhMJ4zHGpReEn7rjMi6lrTVflewqPYXoJKsqJiWidUWnGq9_2SxmCJlxf00VRuqbLK6RT05ENa3EW7TKBNlVI/s1600/EViews+-+Creating+Group.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Creating Group data in EViews from cruncheconometrix.com.ng" border="0" data-original-height="450" data-original-width="507" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLns9sZX95KzR0sU8Dvrt5G1JZpy3poWufGExpzfVaAckEIHwtPCqStTNhMJ4zHGpReEn7rjMi6lrTVflewqPYXoJKsqJiWidUWnGq9_2SxmCJlxf00VRuqbLK6RT05ENa3EW7TKBNlVI/s1600/EViews+-+Creating+Group.png" title="Creating Group data in EViews" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Creating Group data in EViews<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Click <b>New Object</b> and the dialogue box opens:<o:p></o:p></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><v:shape id="Picture_x0020_10" o:spid="_x0000_i1035" style="height: 171.75pt; visibility: visible; width: 230.25pt;" type="#_x0000_t75"><v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image004.png"></v:imagedata></v:shape></span><span style="font-family: "times new roman" , serif; font-size: 12pt;"><o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIzd5R20ssP1BFaCMpV6w4SPVJ0oYdFA4gb6cjn-hAvI04OpMATUpCJpx2ZoCjP46_8JJr7W5I_fpfS0O47pxVUq4mJ3FWQH9nSRv8AdIrGUNJmvH8p4iTrNltWrFlqwult9CDk6pgS_s/s1600/EViews+-+New+Object+Dialogue+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: New Object dialogue box from cruncheconometrix.com.ng" border="0" data-original-height="392" data-original-width="307" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIzd5R20ssP1BFaCMpV6w4SPVJ0oYdFA4gb6cjn-hAvI04OpMATUpCJpx2ZoCjP46_8JJr7W5I_fpfS0O47pxVUq4mJ3FWQH9nSRv8AdIrGUNJmvH8p4iTrNltWrFlqwult9CDk6pgS_s/s640/EViews+-+New+Object+Dialogue+Box.png" title="EViews: New Object dialogue box" width="501" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: New Object dialogue box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Click </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">OK </b><span style="font-family: "times new roman" , serif; font-size: 12pt;">to open the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Series List </b><span style="font-family: "times new roman" , serif; font-size: 12pt;">dialogue
box and type in </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">income pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">:</span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqPQDk4Cwj66sF-dRCAAoI4gxVGKRya_eVX8Col4tgaGBUluIs1OVdRf0mZWl1ETGz-dB8OVLIFniX_QsdbaogbwSn7lwFI69-wnOD1gtImi8btjToQarUmDNWO2jzjErd-Ewy67jfc90/s1600/EViews+-+Series+List+Dialogue+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Series List dialogue box from cruncheconometrix.com.ng" border="0" data-original-height="400" data-original-width="507" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqPQDk4Cwj66sF-dRCAAoI4gxVGKRya_eVX8Col4tgaGBUluIs1OVdRf0mZWl1ETGz-dB8OVLIFniX_QsdbaogbwSn7lwFI69-wnOD1gtImi8btjToQarUmDNWO2jzjErd-Ewy67jfc90/s1600/EViews+-+Series+List+Dialogue+Box.png" title="EViews: Series List dialogue box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Series List dialogue box<br />
Source: CrunchEconomterix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_11" o:spid="_x0000_i1034" style="height: 243pt; mso-wrap-style: square; visibility: visible; width: 308.25pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image005.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">Click </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">OK </b><span style="font-family: "times new roman" , serif; font-size: 12pt;">and your data should look like this:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqbVF4KvD4d3qUa_sGRsXTbVtpIhk0WyaKE-Z3NGWreR5EYtZUV880Rw1ysNhVK6-sEz8u4d0CfsFD65y11zV5oMduFwQQ-0YO4mi8umB2UDrstyRWA4YtobzUaiPOUmC1-sqNcn07w_0/s1600/EViews+-+Data+format.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Group data from cruncheconometrix.com.ng" border="0" data-original-height="445" data-original-width="506" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqbVF4KvD4d3qUa_sGRsXTbVtpIhk0WyaKE-Z3NGWreR5EYtZUV880Rw1ysNhVK6-sEz8u4d0CfsFD65y11zV5oMduFwQQ-0YO4mi8umB2UDrstyRWA4YtobzUaiPOUmC1-sqNcn07w_0/s1600/EViews+-+Data+format.png" title="EViews: Group data" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Group data<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_12" o:spid="_x0000_i1033" style="height: 248.25pt; mso-wrap-style: square; visibility: visible; width: 282.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image006.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">At this point it is important to save
your data file. Click on </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Name</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> and under
</span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Name to identify objec</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">t change </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">group01</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> to the desired the file name:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzCm7oxlQ17bQIsnXPqvNmH_xX7RokJ-cP8G_pqArIl0qHnLS_Hdd2qEMtfRD1nv89j9vghuzGjrOEI1YGe2yKneXQaaygLPs6vdqLAJyhYJn8qFxcjxMKt_0YntXL6pxOyqvFM8mF2J4/s1600/EViews+-+Saving+Data+File.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Object Name dialogue box from cruncheconometrix.com.ng" border="0" data-original-height="216" data-original-width="336" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzCm7oxlQ17bQIsnXPqvNmH_xX7RokJ-cP8G_pqArIl0qHnLS_Hdd2qEMtfRD1nv89j9vghuzGjrOEI1YGe2yKneXQaaygLPs6vdqLAJyhYJn8qFxcjxMKt_0YntXL6pxOyqvFM8mF2J4/s1600/EViews+-+Saving+Data+File.png" title="EViews: Object Name dialogue box" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Object Name dialogue box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: left;">
<span style="background-color: yellow; font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">Note:
Spaces are not allowed when naming an object in EViews.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">I will save this file as <b>pce_income</b>. Click <b>OK</b> and the file appears as <b>G
pce_income</b> like this:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB9KA-v82qZisDohp6Kelq6-kAfeqtXTNzAnotLamygB2j9JVEwowHGsmkeB_H9hddrogsB7Tck4j-7WC8AS-aXdk99dstNwgAvGmwXrPdqVGBYN0-8wxi57fRSYfn3sqaLaF7Re9a_xA/s1600/EViews+-+File+Name.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Naming a file from cruncheconometrix.com.ng" border="0" data-original-height="228" data-original-width="268" height="340" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjB9KA-v82qZisDohp6Kelq6-kAfeqtXTNzAnotLamygB2j9JVEwowHGsmkeB_H9hddrogsB7Tck4j-7WC8AS-aXdk99dstNwgAvGmwXrPdqVGBYN0-8wxi57fRSYfn3sqaLaF7Re9a_xA/s400/EViews+-+File+Name.png" title="EViews: Naming a file" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Naming a file<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_3" o:spid="_x0000_i1031" style="height: 171pt; mso-wrap-style: square; visibility: visible; width: 201pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image008.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">Now we have finished with all the data
prepping. It’s time to observe the relationship between two series. To do that,
we will use the scatter diagram. Click on </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">G
pce_income</b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> to open the file. Then click on </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">View >> Graph >></b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">Scatter
>> OK</b></div>
<div class="MsoNoSpacing" style="text-align: left;">
<b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;"><br /></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The scatter diagram indicates a positive relationship between the two variables:</span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ-Fxj0J0FNcCrwP4YigBbts8R3xaxzzb8EN0M-UliOTBKOfpA2jnBVXq8S7LbJMFcVYUihB3Z-XbcGROPAtqyQw69mmjAuz9ExcIyVeGVKjxBHmDG6hUc5FOdX7RBL1i1IZGO1I-6wUg/s1600/EViews+-+Scatter+Plot+%2528pce+and+income%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Scatter plot (pce and income) from cruncheconometrix.com.ng" border="0" data-original-height="463" data-original-width="510" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ-Fxj0J0FNcCrwP4YigBbts8R3xaxzzb8EN0M-UliOTBKOfpA2jnBVXq8S7LbJMFcVYUihB3Z-XbcGROPAtqyQw69mmjAuz9ExcIyVeGVKjxBHmDG6hUc5FOdX7RBL1i1IZGO1I-6wUg/s1600/EViews+-+Scatter+Plot+%2528pce+and+income%2529.png" title="EViews: Scatter plot (pce and income)" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Scatter plot (pce and income)<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">This positive relationship seems
plausible because the more income you have, the more you’ll want to consume,
except you are very economical </span><span style="font-family: "wingdings"; font-size: 12pt; text-align: justify;">J</span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">To graph the model (<i>pce</i>) with the linear prediction (<i>pce<sub>hat</sub></i>), Click on <b>G
pce_income</b> to open the file. Then click on <b>View >> Graph >></b> <b>Scatter
>> </b>on the left-hand side of the dialog that pops up >> select <b>Regression
line </b>from the <b>Fit lines </b>dropdown menu. The default options for a
regression line are fine, so hit to dismiss the dialog.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Or, simply right click inside the graph:
<b>Fit lines</b> >> select <b>Regression line</b> >> <b>OK</b><o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSmQWSRYYjjRqHX1qiOX27Y3BmMFWRLQYDUdPU8EaOcFJKVeoSV1bU0Ctya9jsR6JFe8Nq5t5tx4c7UeRnMnMkJah6Zp4rt8ns7vrBUaTcmFT6o-zkb4DNvrDTSbgt3z9NdAUFe4adQBs/s1600/EViews+-+Scatter+Plot+%2528pce_income_fit+line%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Scatter plot with fit line from cruncheconometrix.com.ng" border="0" data-original-height="475" data-original-width="701" height="271" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSmQWSRYYjjRqHX1qiOX27Y3BmMFWRLQYDUdPU8EaOcFJKVeoSV1bU0Ctya9jsR6JFe8Nq5t5tx4c7UeRnMnMkJah6Zp4rt8ns7vrBUaTcmFT6o-zkb4DNvrDTSbgt3z9NdAUFe4adQBs/s400/EViews+-+Scatter+Plot+%2528pce_income_fit+line%2529.png" title="EViews: Scatter plot with fit line" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Scatter plot with fit line<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><b><br /></b></span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_24" o:spid="_x0000_i1029" style="height: 249pt; mso-wrap-style: square; visibility: visible; width: 368.25pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image010.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">As observed from the graph, all the
points do not fall on the predicted line. Some lie above, while some are
beneath the line. These are all the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;"><span style="color: blue;">residuals</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> (in order words, the remnants obtained
after the regression analysis).</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Third step: The scientific investigation<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Now we want to scientifically
investigate the relationship between <i>pce</i>
and <i>income</i>. </span><span style="font-size: 12.0pt;">I</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">n EViews you specify a regression with the <b><i>ls</i></b>
command followed by a list of variables. (“<b><i>LS</i></b>” is the name for the EViews
command to estimate an ordinary <b><i>L</i></b>east <b><i>S</i></b>quares regression.) The
first variable is the <b><i>dependent
variable</i></b>, the variable we’d like to explain <i>pce </i>in this case. The rest of the list gives the <b><i>independent variables</i></b>, which are
used to predict the dependent variable. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Also, one can “run a regression” either
by using the <b><i>menu</i></b> or <b><i>type-command</i></b> approach. Using the <b><i>menu
approach</i></b>, from the Tool Bar, pick the menu item <b>Quick >> Estimate
Equation </b>and a dialog box opens:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnnISsaYyYTK0r7G384fUvUcv_pndJamIDyRuB0cRBuEq0cOxAWp11lMWcvDjE0o5oqnXGMGUvrDpf0BRwvqsChZffGcvObOnTmosX9tjrChXy5EuY5OtrFJEbUwVZ1LSDCWPieXSzWGg/s1600/EViews+-+Estimate+Equation+DB.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Equation Estimation dialogue box from cruncheconometrix.com.ng" border="0" data-original-height="429" data-original-width="471" height="363" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnnISsaYyYTK0r7G384fUvUcv_pndJamIDyRuB0cRBuEq0cOxAWp11lMWcvDjE0o5oqnXGMGUvrDpf0BRwvqsChZffGcvObOnTmosX9tjrChXy5EuY5OtrFJEbUwVZ1LSDCWPieXSzWGg/s400/EViews+-+Estimate+Equation+DB.png" title="EViews: Equation Estimation dialogue box" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Equation Estimation dialogue box<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_27" o:spid="_x0000_i1028" style="height: 180pt; mso-wrap-style: square; visibility: visible; width: 197.25pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under <b>Equation
specification</b>, </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">type
“<i>pce c income</i>” click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Hold
on a bit. </span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If
<i>pce </i>is the dependent variable and <i>income </i>is the explanatory variable
so, where does the “<i>C</i>” in the
command come from? “<i>C</i>” is a special
keyword telling EViews to estimate the equation with an <b><i>intercept</i></b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">And if you prefer to use the <i>type-command approach</i>, go to the command
section and type in:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">ls
pce c income<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">(You
have simply told EViews to regress the dependent variable, <i>pce</i>, on the explanatory variable, <i>income </i>and a <i>constant</i>).</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Therefore, whether you use the menu or type
a command, EViews churns out the regression results shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLXKgEh588fHx8yDJFyNCu6Lu_NOT6iES1-vFI97et6gyKo_0O9KMiSA8NCTRAGMOFUvnFQ_hox8p627STMXpUHa9N9V3jeGaaEtnyF49CvyKDUeg34BzV0TQxjtklXOOm0UUn3lRiazY/s1600/EViews+-+Regression+Output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Regression Output from cruncheconometrix.com.ng" border="0" data-original-height="339" data-original-width="454" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLXKgEh588fHx8yDJFyNCu6Lu_NOT6iES1-vFI97et6gyKo_0O9KMiSA8NCTRAGMOFUvnFQ_hox8p627STMXpUHa9N9V3jeGaaEtnyF49CvyKDUeg34BzV0TQxjtklXOOm0UUn3lRiazY/s1600/EViews+-+Regression+Output.png" title="EViews: Regression Output" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Regression Output<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fourth step: The features of a regression output</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So what do these figures mean? I will
explain each feature in turns.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Dependent
variable: </span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">this
is <i>pce </i>and it is clearly defined. It
is also the outcome variable.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Method:
</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">this
is the estimation technique. In this example, it is ordinary least squares<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Date:
</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">captures
the exact time you are carrying out the analysis<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Sample:
</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">must
be in line with your scope of research; that is 1960 to 2009<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Included
observations</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
since the data span is from 1960 to 2009, observations = 50<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Variable</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: includes both
the intercept and slope<b><o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Coeff</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: these captures
the estimates for intercept and slope. The sign of the coefficient also tells the
direction of the relationship. A positive (negative) sign implies a positive
(negative) relationship.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Std.
error</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
this is the standard deviation for the coefficient. That is, since you are not
so sure about the exact value for <i>income</i>,
there will be some variation in the prediction for the coefficient. Therefore,
the standard error shows how much deviation occurs from predicting the slope
coefficient estimate.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></i></b>
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">t</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-stat</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this measures
the number of standard errors that the coefficient is from zero. It is obtained
by: <i>coefficient/std.error</i>. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">A <i>t</i>-stat above 2 is sufficient
evidence against the null hypothesis</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Prob.</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: there are
several interpretations for this. (1) it is smallest evidence required to
reject the null hypothesis, (2) it is the probability that one would have
obtained the slope coefficient value from the data if the actual slope
coefficient is zero, (3) the <i>p</i>-value
looks up the <i>t</i>-stat table using the
degree of freedom (df) to show the number of standard errors the coefficient is
from zero, (4) tells whether the relationship is significant or not.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, if the <i>p</i>-value is 0.35, then it means that you are only 65% (that is,
(100-35)%) confident that the slope coefficient is non-zero. This is not good
enough. This is because a very low <i>p</i>-value
gives a higher level of confidence in rejecting the null hypothesis. Hence, a <i>p</i>-value of 0.01, implies that you are 99%
(that is, (100 - 1)%) confident that the slope coefficient is non-zero. This is
very comforting! </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">R</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: the value of
0.999273 gives the variation in <i>pce</i>
that is explained by <i>income</i>. The
higher the <i>R</i><sup>2</sup>, the better
the model and the more predictive power the variables have. Although, an <i>R</i><sup>2</sup> that equals 1 will elicit
some suspicion. The R is actually the correlation coefficient between the 2
variables. That implies that: </span><!--[if gte msEquation 12]><m:oMath><m:rad><m:radPr><m:degHide
m:val="on"/><span style='font-size:12.0pt;mso-ansi-font-size:12.0pt;
mso-bidi-font-size:12.0pt;font-family:"Cambria Math","serif";mso-ascii-font-family:
"Cambria Math";mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:
"Times New Roman";font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>R</m:r></span></i></m:e><m:sup><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;
font-family:"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>2</m:r></span></i></m:sup></m:sSup></m:e></m:rad></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><v:shape id="_x0000_i1025" style="height: 16.5pt; width: 21pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image014.png">
</v:imagedata></v:shape></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh0OfhRWqSwA5ke58UmnMrU38LXEucDF0DvyDq52ls318ookQZG54EOyMbozWBkRSURwlRDoGqqFL5F88McCKBWbJmVQrkwxBlIOU_D2Uf41SmDecQ0wxir4W8uiThhK2CxU71D5C_Dv4/s1600/R2.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="21" data-original-width="37" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh0OfhRWqSwA5ke58UmnMrU38LXEucDF0DvyDq52ls318ookQZG54EOyMbozWBkRSURwlRDoGqqFL5F88McCKBWbJmVQrkwxBlIOU_D2Uf41SmDecQ0wxir4W8uiThhK2CxU71D5C_Dv4/s1600/R2.PNG" /></a> </div>
= the correlation coefficient.<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span><br />
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Adjusted R</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the <i>R</i><sup>2</sup> adjusted as you increase your explanatory variables. It
(0.999257) reduces as more explanatory variables are added.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">S.E of regression</span></i></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the
summary measure based on the estimated variance of the residuals.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Sum
squared resid</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
implies <span style="color: blue;">sum of squared residuals</span> for the Model
(explained variation in <i>pce</i>) and
Residuals (unexplained variation in <i>pce</i>).
After doing the regression analysis, all the points on <i>pce<sub>ha</sub></i><sub>t</sub> do not fall on the regression line.
Those points outside the line are known as <b><i>residuals</i></b>. Those that can be
explained by the model are known as <b>Explained
Sum of Squares</b> (ESS) while those that are due to random nature, which are
outside the model are known as <b>Residual
Sum of Squares</b> (RSS).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Having seen the plot of the scatter
diagram, it is pretty clear that the predicted line does an almost-accurate job
of giving a 50-year summary of <i>pce</i>.
In regression analysis, the amount by which the right-hand side of the equation
misses the dependent variable is called the <b><i>residual</i></b><i>. </i>Calling the residual <b><i>e</i></b><i> </i>(“<b><i>e</i></b>”
stands for “<b>error</b>”), we can write an
equation that really is valid in each and every year, that is: <b><i>pce =
</i>-31.88<i> + </i>0.819<i>income + e</i></b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since the residual is the part of the
equation that’s left over after we’ve explained as much as possible with the
right-hand side variables, one approach to getting a better fitting equation is
to look for patterns in the residuals.<b><span style="color: blue;"><o:p></o:p></span></b></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">To obtain the table showing the
predicted and residual values, go to <b>View
>> Actual, Fitted, Residual</b> >> <b>Actual, Fitted, Residual Table</b>
and you get:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdCeBEh0193wc9xWSXMtkTLbOpeJAx3IlKN9D1j1B4HnXVt5Cg_8dy3-wA10bUXigJvpy7Rg_XRWEiHty4fX9ek0oG-ilRIhFQgsz57dIxT_Ykk-MV60md9MqkN2f0bzCYAb0KIfTSoZ8/s1600/EViews+-+Table+of+actual%252C+pred+and+residual+values.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Table of actual, predicted and residual values from cruncheconometrix.com.ng" border="0" data-original-height="461" data-original-width="461" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdCeBEh0193wc9xWSXMtkTLbOpeJAx3IlKN9D1j1B4HnXVt5Cg_8dy3-wA10bUXigJvpy7Rg_XRWEiHty4fX9ek0oG-ilRIhFQgsz57dIxT_Ykk-MV60md9MqkN2f0bzCYAb0KIfTSoZ8/s1600/EViews+-+Table+of+actual%252C+pred+and+residual+values.png" title="EViews: Table showing actual, predicted and residual values" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Table of actual, predicted and residual values<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">If the predicted line falls above a
point, it means that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is
over-predicted (that is, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce – pce<sub>hat</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">
is negative) and if it is beneath a point, it implies that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is under-predicted (that is, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce – pce<sub>hat</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is positive). The sum and mean of the
residuals equals zero.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Likewise, to obtain the plot of the
predicted and residual values, go to <b>View
>> Actual, Fitted, Residual</b> >> <b>Actual, Fitted, Residual Graph</b>
and you get:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKIOvN23ahw9MqFfqVcpjE3jTuL_Rx44ESSI6freK1KK-r6UkQbtdwCqK0_c9sqVsUT_0c9TWYR1p1eAtVVa2oKse3mlr7P7d1LjyE_I38xn-Asohw2tpSH0o38sQeFyKvVvaTNSzi00k/s1600/EViews+-+Plot+of+actual%252C+pred+and+residual+values.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="EViews: Graph of actual, predicted and residual values" border="0" data-original-height="393" data-original-width="448" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKIOvN23ahw9MqFfqVcpjE3jTuL_Rx44ESSI6freK1KK-r6UkQbtdwCqK0_c9sqVsUT_0c9TWYR1p1eAtVVa2oKse3mlr7P7d1LjyE_I38xn-Asohw2tpSH0o38sQeFyKvVvaTNSzi00k/s1600/EViews+-+Plot+of+actual%252C+pred+and+residual+values.png" title="EViews: Graph showing actual, predicted and residual values" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">EViews: Graph of actual, predicted and residual values<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Log
likelihood</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;">:
this the difference between the log likelihood values of the restricted and
unrestricted versions of the model.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">F</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-statistic</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: captures
whether the explanatory variable, <i>income</i>
is significant in explaining the outcome variable, <i>pce</i>. The higher the <i>F</i>-stat,
the better for the model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Prob
(<i>F</i>-statistic)</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: the
probability value of 0.0000 is the probability value that indicates the
statistical significance of the <i>F</i>
statistic. You will prefer to have a <i>prob</i>-value
that is less than 0.05.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Mean
dependent var</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
the figure of 3522.160 indicates the average value of <i>pce</i> in the data.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">S.
D. dependent var</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
the figure of 3077.678 indicates the deviation from the average value of <i>pce</i> in the data<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Akaike/Schwartz/Hannan-Quinn
info criterion</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
these are often used to choose between competing models. The lower the value of
these criteria, the better the model is. From this example, the Akaike info
criterion (AIC) figure of 11.73551 is the lowest of the three and therefore
indicates that it is the best model to adopt in this case.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Durbin-Watson
stat</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
is used to find out if there is first-order serial correlation in the error
terms. <i>Rule of thumb:</i> if DW < 2
equals evidence of positive serial correlation. So, from our example, the DW
value of 0.568044 indicates serial correlation in the residuals.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Assignment:</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><b>Use <a href="https://drive.google.com/drive/u/0/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter</a> Table7_12.xlsx dataset.<o:p></o:p></b></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(1)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> With <i>pce</i>
as the dependent variable and <i>gdpi</i> as
the explanatory variable, plot the graph of <i>pce</i>
and <i>gdpi</i>, what do you observe?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(2)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Run your regression. Can you interpret the
table and the features?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(3)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Plot the predicted line. What are your
observations?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span>
<div style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><b>[Watch video on how to interpret regression output in EViews]</b></span></div>
<div style="text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/LNA_8p9087E/0.jpg" src="https://www.youtube.com/embed/LNA_8p9087E?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<div style="text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><b><br /></b></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">I have taken you through the basic
features of a regression output using EViews analytical package on ordinary
least squares (OLS) model in a simple linear regression. Practice the
assignment and if you still have further questions, kindly post them below…..</span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-24916390841618988092018-01-30T14:40:00.000+01:002018-02-06T17:09:37.719+01:00How to Interpret Regression Output in Stata<h2 style="text-align: center;">
How to Interpret Regression Output in Stata</h2>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">This period happens to be the <i>dissertation semester</i> for undergraduate
students in most universities, at least for those with undisrupted academic
calendar </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">. The students
are in different stages of their <i>project</i>,
as it is commonly called. Some are yet to wrap up their chapter one which gives
the “study background” and the framing of research hypotheses, objectives and
questions. Some have moved on to chapter two reviewing relevant literature
related to their scope of study. Others have gone further in developing both
the theoretical and empirical frameworks for chapter three, but not without the
usual teething lags…but they’ll get around it, somehow </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">. A handful have even done better
by progressing to chapter four attempting to analyse their data.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since, chapters one to three are
relative to each students’ scope of research, but a regression output is common
to all (although actual outcomes differ), I decided to do this tutorial in
explaining the basic features of a regression output. Also, this write-up is in
response to requests received from readers on (1) what some specific figures in
a regression output are and (2) how to interpret the results. Let me state here
that regardless of the analytical software whether Stata, EViews, SPSS, R,
Python, Excel etc. what you obtain in a regression output is common to all
analytical packages.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">For instance, in undertaking an ordinary
least squares (OLS) estimation using any of these applications, the regression
output will churn out the ANOVA (analysis of variance) table, <i>F</i>-statistic, <i>R</i>-squared, prob-values, coefficient, standard error, <i>t</i>-statistic, degree of freedom, 95%
confidence interval and so on. These are the basic features of a regression output regardless of your model and/or estimation technique. However,
the issue is: what do they mean and how can they be interpreted and related to
your study.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Hence, the essence of this tutorial is to
teach students the relevance of these features and how to interpret their
results. I will be using <b><span style="color: blue;">Stata</span></b>
analytical package to explain a regression output, but you can practise along
using any analytical package of your choice. <o:p></o:p></span><span style="font-family: "times new roman", serif; font-size: 16px;">(See "How-to-interpret regression output" here for <a href="http://cruncheconometrix.blogspot.com.ng/2018/02/how-to-interpret-regression-output-in_3.html" target="_blank"><b>EViews</b></a></span><span style="font-family: "times new roman", serif; font-size: 16px;"> and </span><b style="font-family: "times new roman", serif; font-size: 16px;"><a href="http://cruncheconometrix.blogspot.com.ng/2018/02/how-to-interpret-regression-output-in.html" target="_blank">Excel</a></b><span style="font-family: "times new roman", serif; font-size: 16px;"> users)</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">An Example: Using Gujarati and Porter Dataset Table7_12.dta
or Table7_12.xlsx dataset</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note: In this tutorial I will not be discussing
stationarity or cointegration analysis (those topics will be covered in subsequent tutorials). Since the purpose is simply to explain the basic features of a regression output, I will only be doing a <span style="background: yellow; mso-highlight: yellow;">simple linear regression</span>
analysis (a bi-variate analysis) with only one explanatory variable. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The dataset is on the United States from
1960 to 2009 (50 years data). The outcome variable is consumption expenditure (<i>pce</i>) and the explanatory variable is
income (<i>income</i>).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">First step: load data in excel format into Stata<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Here is the data in excel format:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimc14LM8ga5mUa1DSXSciMVf4aY9xMCWOchQH06Q60yhYXlmw6iqxylXho3PeZyYtKiL1syFVWqrj3iny8iA99NqYHjMY4VCuQGgVYVeSpfI_-Lhnb02CSCdRYBn4PMB9Ws3_Q5RVNxfg/s1600/Data+in+excel+format.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Data in excel file from http://cruncheconometrix.com.ng" border="0" data-original-height="650" data-original-width="665" height="624" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimc14LM8ga5mUa1DSXSciMVf4aY9xMCWOchQH06Q60yhYXlmw6iqxylXho3PeZyYtKiL1syFVWqrj3iny8iA99NqYHjMY4VCuQGgVYVeSpfI_-Lhnb02CSCdRYBn4PMB9Ws3_Q5RVNxfg/s640/Data+in+excel+format.png" title="Data in excel file" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Data in Excel format<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">And here is the data in Stata format:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaI0CRTFPx-HMfA8JlFF65gK4if2sGSdoMLop26Rz6dGKWxhlYLT3omE2oDs5QOHKbsiMJyuf0igV59hVfYJtzUdi-5zoaJkjkDiN4dFI38q2kVnj7CCnuMPhJ738DU3kNKrBBZIYqGnc/s1600/Data+in+Stata+format.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Data in Stata format from http://cruncheconometrix.com.ng" border="0" data-original-height="711" data-original-width="705" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaI0CRTFPx-HMfA8JlFF65gK4if2sGSdoMLop26Rz6dGKWxhlYLT3omE2oDs5QOHKbsiMJyuf0igV59hVfYJtzUdi-5zoaJkjkDiN4dFI38q2kVnj7CCnuMPhJ738DU3kNKrBBZIYqGnc/s640/Data+in+Stata+format.png" title="Data in Stata format" width="634" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Data in Stata format<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Second step: Set the time variable in Stata for analysis</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before analysing the data, you must set
up the time variable in readiness for the regression. The general code is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">tsset
timevar<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">in my case, the time variable is <i>obs</i>, and my code becomes:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">tsset
obs<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">and Stata responds with:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhX7zwgh-siCfYZE2cHFq_E46rjkJFnvPj6J1u8JX_A4tX3aD6njfFHe_ax2ZQQsn9rOuEKtqPdoGgyxOdOyZeTgPEXQ-kIHM4od9PU2U0YfWWEIV5hW0eQJeJTYW7L3JIolUYeHgPN9X8/s1600/Setting+time+variable.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Time set command in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="51" data-original-width="309" height="65" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhX7zwgh-siCfYZE2cHFq_E46rjkJFnvPj6J1u8JX_A4tX3aD6njfFHe_ax2ZQQsn9rOuEKtqPdoGgyxOdOyZeTgPEXQ-kIHM4od9PU2U0YfWWEIV5hW0eQJeJTYW7L3JIolUYeHgPN9X8/s400/Setting+time+variable.png" title="Time set command in Stata" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Time set command in Stata<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">tsset</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">
implies </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">“time series set”</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> and as you
can see, the begin year is 1960 and the end year is 2009. You must always do
this after loading your data and before you begin your regressions.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Third step: Visualise the relationship between the variables<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Before analysing the data, it is good to
always graph the dependent and key explanatory variable (using a scatter plot) in
order to observe the pattern between them. It kind of gives you what to expect
in your actual analysis.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, to graph <i>pce</i> and <i>income</i>, the Stata
code is:<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 107%;">twoway
(scatter pce income)<o:p></o:p></span></i></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 107%;"><br /></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 107%;">The scatter diagram
indicates a positive relationship between the two variables:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEC_RIxzzAbwZdJXzB1EDxUTJoqtLcXICFniRarcQn4Fc_YFO8PxljdqodUWdv6XRt1X-dAi98pF8NUM-6HA1q4jJFR63llwc7EY2te7XDh8ZWmSR3M4018NHKx20ktTFB-5pDIl3sJLQ/s1600/Graph+-+pce+and+income.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Scatter plot of the variables from http://cruncheconometrix.com.ng" border="0" data-original-height="1129" data-original-width="1600" height="450" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEC_RIxzzAbwZdJXzB1EDxUTJoqtLcXICFniRarcQn4Fc_YFO8PxljdqodUWdv6XRt1X-dAi98pF8NUM-6HA1q4jJFR63llwc7EY2te7XDh8ZWmSR3M4018NHKx20ktTFB-5pDIl3sJLQ/s640/Graph+-+pce+and+income.png" title="Scatter plot of the variables" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Scatter plot of the variables<br />
Source: CrunchEconomterix</td></tr>
</tbody></table>
<div class="MsoNormal" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">This positive relationship seems
plausible because the more income you have, the more you’ll want to consume, except
you are very frugal </span><span style="font-family: "wingdings"; font-size: 12pt;">J</span><span style="font-family: "times new roman" , serif; font-size: 12pt;">.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fourth step: The scientific investigation<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Now we want to scientifically
investigate the relationship between <i>pce</i>
and <i>income</i>. The Stata code is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">regress
pce income<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(You have simply told Stata to regress
the dependent variable, <i>pce</i>, on the
explanatory variable, <i>income</i>), and
the output is shown as:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzStJIbzPzqeoWOnCePW7crwL_tQlG4oTQuMpTrMgGdzYr7MM9dFH_Wa7iv2lnNMGfWFeUktCP6u9ZP3mIRomkX8yKRYWuTwAepqQRr2T3fn0yDFlDYJz9wpAAEsFCrcQN8MIacn8cS0o/s1600/Regression+output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Regression output in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="248" data-original-width="582" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzStJIbzPzqeoWOnCePW7crwL_tQlG4oTQuMpTrMgGdzYr7MM9dFH_Wa7iv2lnNMGfWFeUktCP6u9ZP3mIRomkX8yKRYWuTwAepqQRr2T3fn0yDFlDYJz9wpAAEsFCrcQN8MIacn8cS0o/s640/Regression+output.png" title="Regression output in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Regression output in Stata<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Fifth step: The features of a regression output</span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So what do these figures mean? I will
explain each feature in turns.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Source</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: there are two
sources of variation in the dependent variable, <i>pce</i>. Those explained by the regression (i.e, the <b><span style="color: blue;">Model</span></b>)
and those due to randomness (<b><span style="color: blue;">Residuals</span></b>)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">SS</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: implies <span style="color: blue;">sum of squared residuals</span> for the Model (explained
variation in <i>pce</i>) and Residuals
(unexplained variation in <i>pce</i>). After
doing the regression analysis, all the points on <i>pce<sub>ha</sub></i><sub>t</sub> do not fall on the regression line.
Those points outside the line are known as residuals. Those that can be
explained by the model are known as <b>Explained
Sum of Squares</b> (ESS) while those that are due to random nature, which are outside
the model are known as <b>Residual Sum of
Squares</b> (RSS).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">To graph the model (<i>pce</i>) with the linear prediction (<i>pce<sub>hat</sub></i>), the Stata code is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">scatter
pce income || lfit pce income<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlsBFbusr81uIn3uVAclFiD2WTZ89L0c9urQ737RbGjtliO_H3pLjSQE4f4Vdj6nvQJ8hGkXzsZKrdVz_TIzR_4SZVGslcxZVc67MQt6YC_6P2GqUCOXOHlzZVYs4fhJGHuATW8BBIMF8/s1600/Residuals.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Scatter plot of the linear prediction from http://cruncheconometrix.com.ng" border="0" data-original-height="433" data-original-width="603" height="457" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlsBFbusr81uIn3uVAclFiD2WTZ89L0c9urQ737RbGjtliO_H3pLjSQE4f4Vdj6nvQJ8hGkXzsZKrdVz_TIzR_4SZVGslcxZVc67MQt6YC_6P2GqUCOXOHlzZVYs4fhJGHuATW8BBIMF8/s640/Residuals.PNG" title="Scatter plot of the linear prediction" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Scatter plot of the linear prediction<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</span></div>
<div align="center" class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_6" o:spid="_x0000_i1026" style="height: 218.25pt; mso-wrap-style: square; visibility: visible; width: 304.5pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image006.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">As observed from the graph, all the
points do not fall on the predicted line. Some lie above, while some are
beneath the line. These are all the <b><span style="color: blue;">residuals</span></b> (in order words, the remnants obtained
after the regression analysis). <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">To obtain the predicted value, the Stata
command is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">predict<i> pce_hat</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">and to obtain the residual value, the
Stata command is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">predict<i> pce_resid<o:p></o:p></i></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><v:shape id="Picture_x0020_8" o:spid="_x0000_i1025" style="height: 184.5pt; mso-wrap-style: square; visibility: visible; width: 309pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image007.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEht90PA4jOZZZr4RaaiZw4GvlTVs1i51XvyYedmhySxYZQvEKFzR-GqW9jPMZgro05GjSmlGsAMIxcUiGyust5tXTOefKdARhye343Lj-otrWW52XLKRjUgl_dZEwTvbuqhBuqUW70WINk/s1600/Stata+-+Predicted+value+and+Residuals.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Predicted and residual value from http://cruncheconometrix.com.ng" border="0" data-original-height="497" data-original-width="833" height="380" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEht90PA4jOZZZr4RaaiZw4GvlTVs1i51XvyYedmhySxYZQvEKFzR-GqW9jPMZgro05GjSmlGsAMIxcUiGyust5tXTOefKdARhye343Lj-otrWW52XLKRjUgl_dZEwTvbuqhBuqUW70WINk/s640/Stata+-+Predicted+value+and+Residuals.png" title="Predicted and residual value" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Predicted and residual value of the dependent variable<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">If the predicted line falls above a
point, it means that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is
over-predicted (that is, <i>pce</i></span><i style="font-family: "Times New Roman", serif; font-size: 12pt;"> – pce<sub>hat</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;">
is negative) and if it is beneath a point, it implies that </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pce</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is under-predicted (that is, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;"><span style="font-family: "times new roman" , serif; font-size: 12pt; font-style: normal;"><i>pce</i></span><i style="font-size: 12pt;"> – pce</i><sub>hat</sub></i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is positive). The sum and mean of the residuals
equals zero.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">df</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is <span style="color: blue;">degree of freedom</span> calculated as <i>k - 1</i> (for the model)
and <i>n - k</i> (for the residuals). <i>n</i> = number of observations; <i>k</i> = number of restrictions on the model<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">MS</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: implies <span style="color: blue;">mean sum of squared residuals</span> and obtained by dividing <i><b>SS</b></i> by <i><b>df</b></i> i.e. <b><i>SS/df</i></b></span><!--[if gte msEquation 12]><m:oMath><m:f><m:fPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math","serif";mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>SS</m:r></span></i></m:num><m:den><i
style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;font-family:
"Cambria Math","serif";mso-bidi-font-family:"Times New Roman"'><m:r>df</m:r></span></i></m:den></m:f></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 8.5pt;"><v:shape id="_x0000_i1025" style="height: 21.75pt; width: 10.5pt;" type="#_x0000_t75"><v:imagedata chromakey="white" o:title="" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image008.png"></v:imagedata></v:shape></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">No.
of obs</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
the data span is from 1960 to 2009 = 50 years<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">F</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-stat</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: captures
whether the explanatory variable, <i>income</i>
is significant in explaining the outcome variable, <i>pce</i>. The higher the F-stat, the better for the model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Prob>F</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the
probability value that indicates the statistical significance of the <i>F</i> ratio.You will prefer to have a <i>prob</i>-value that is less than 0.05.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">R</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: gives the
variation in <i>pce</i> that is explained by
<i>income</i>. The higher the <i>R</i><sup>2</sup>, the better the model and
the more predictive power the variables have. Although, an <i>R</i><sup>2</sup> that equals 1 will elicit some suspicion. The R is
actually the correlation coefficient between the 2 variables. This implies that: </span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDwrDTg__hE4NO4WztxMczJOOXtYuP0gtypZFiDQsXo4oJXRYoiKMXCXCA4d9wbpk-TmlUeD84A7Ueo9FQbfZ8m5Ygm_e0VVC4cnbeg8p2WXATMcTP9xuCXBP1Smxe7PKmGqWef5ptGVQ/s1600/R2.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="21" data-original-width="37" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDwrDTg__hE4NO4WztxMczJOOXtYuP0gtypZFiDQsXo4oJXRYoiKMXCXCA4d9wbpk-TmlUeD84A7Ueo9FQbfZ8m5Ygm_e0VVC4cnbeg8p2WXATMcTP9xuCXBP1Smxe7PKmGqWef5ptGVQ/s1600/R2.PNG" /></a></div>
= the correlation coefficient.<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span><br />
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></i></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Adjusted R</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span></b><span style="font-family: "times new roman" , serif; font-size: 12pt;">: this is the <i>R</i><sup>2</sup> adjusted as you increase your explanatory variables.
It reduces as more explanatory variables are added.</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Coeff</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the
slope coefficient. The estimate for <i>income</i>.
The sign of the coefficient also tells you the direction of the relationship. A
positive (negative) sign implies a positive (negative) relationship.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">_cons</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this is the
hypothetical outcome on <i>pce</i> if <i>income</i> is zero. It is also the intercept
for the model.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Std.
error</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
this is the standard deviation for the coefficient. That is, since you are not
so sure about the exact value for <i>income</i>,
there will be some variation in the prediction for the coefficient. Therefore,
the standard error shows how much deviation occurs from predicting the slope
coefficient estimate.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">t</span></i></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-stat</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: this measures
the number of standard errors that the coefficient is from zero. It is obtained
by: </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> <b><i>coeff/std. error</i></b>. A <i>t</i>-stat above 2 is sufficient
evidence against the null hypothesis</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">P>|t|</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: there are
several interpretations for this. (1) it is smallest evidence required to
reject the null hypothesis, (2) it is the probability that one would have
obtained the slope coefficient value from the data if the actual slope coefficient
is zero, (3) the p-value looks up the <i>t</i>-stat
table using the degree of freedom (df) to show the number of standard errors
the coefficient is from zero, (4) tells whether the relationship is significant
or not.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, if the <i>p</i>-value is 0.4, then it means that you are only 60% (that is,
(100-40)% ) confident that the slope coefficient is non-zero. This is not good
enough. This is because a very low <i>p</i>-value
gives a higher level of confidence in rejecting the null hypothesis. Hence, a <i>p</i>-value of 0.02, implies that you are
98% (that is, (100 - 2)% ) confident that the slope coefficient is non-zero.
This is very comforting! </span><span style="font-family: "wingdings"; font-size: 12.0pt;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">95%
confidence interval</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">:
if the coefficient is significant, this interval will contain that slope
coefficient but it will not, if otherwise.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Assignment:</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Use <a href="https://drive.google.com/drive/folders/1n9hISlqAPpgdl_G7Gs-w8_rkNybloQ0j" target="_blank">Gujarati and Porter datasets</a> Table7_12.dta or
Table7_12.xlsx dataset.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(1)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> With <i>pce</i>
as the dependent variable and <i>gdpi</i> as
the explanatory variable, plot the graph of <i>pce</i>
and <i>gdpi</i>, what do you observe?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(2)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Run your regression. Can you interpret the
table and the features?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(3)<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Plot the predicted line. What are your
observations?<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">I have taken you through the basic
features of a regression output using Stata analytical software on ordinary
least squares (OLS) model in a simple linear regression. Hence, you now have the
basic idea of what the <i>F</i>-stat, <i>t</i>-stat, df, SS, MS, prob>F, p>|t|,
confidence interval, <i>R</i><sup>2</sup>,
coefficient, standard error stand for. <o:p></o:p></span></div>
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<br /></div>
<br />
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Practice the assignment and if you still
have further questions, kindly post them below…..<o:p></o:p></span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-18717828835570990392018-01-22T13:47:00.003+01:002018-02-06T17:15:51.563+01:00Data Handling: Interpretation and Discussion of Results in Scientific Economic Research<h2 style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjy7xO0_iBDTfYRtuTf8S9DyIIbKu7OOpZBWWdcCThaDlJMTsopkWGPdb5hI3lqYJ5q1DQsMNQkd8A85f3kMJ0wjUbgdgVAXpINzwFRD_eswLM7g3ThdLm97qUJFqQ8dOjIS74isyQyHfU/s1600/Prof.+Alege.jpg" imageanchor="1" style="font-family: "Times New Roman", serif; font-size: 12pt; margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1094" data-original-width="813" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjy7xO0_iBDTfYRtuTf8S9DyIIbKu7OOpZBWWdcCThaDlJMTsopkWGPdb5hI3lqYJ5q1DQsMNQkd8A85f3kMJ0wjUbgdgVAXpINzwFRD_eswLM7g3ThdLm97qUJFqQ8dOjIS74isyQyHfU/s200/Prof.+Alege.jpg" width="147" /></a></h2>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;"><b>Philip O. Alege, Ph.D</b></span></div>
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<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Professor of Economics</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> Department of Economics and
Development Studies <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Covenant University, Ota, Ogun State<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><a href="mailto:philip.alege@covenantuniversity.edu.ng">philip.alege@covenantuniversity.edu.ng</a></span><span class="MsoHyperlink"><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></span></div>
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<br /></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Introduction</span></b><b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The tools available to modern economists in
the discharge of functions as an analyst are very many simply because of
various infiltrations of knowledge from other sciences into the discipline of
economics such as physics, biology, mechanical engineering and particularly
mathematics and statistics. Today, modern economies will be difficult to
analyse, understand and predict without the tools of mathematics, statistics
and in particular <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/tell-me-what-is-econometrics.html" target="_blank">econometrics</a>. This can be explained by virtue of the growing
number of economic activities and interactions among the different agents in a
given country and between/among countries. There is a school of thought that
believes in more of economics and little of mathematics. There is also another school
of thought that believes in substantial application of the tools of mathematics
in economics as necessary to get the “useful” results from our analysis. Though
I belong to the latter school, I do also contend that things must be done
properly. <o:p></o:p></span></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Basics of Econometric Modeling</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Basically, econometrics is to provide
empirical support for economic data. Its main purpose is to estimate the
parameter(s) of a model that capture the behaviour of economic agent(s) as
described by the theory and the model. Since the estimated parameters may be
useful in understanding the economic theory, for policy analysis and
forecasting, it becomes necessary on the econometrician to obtain parameters
that are efficient. In order to achieve this, we must adhere to <b>some
principles of model building</b> that can generate results whose
interpretations and discussions will be useful for policy analysis as well as
decision making. These are listed as follows:</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Economic theory applicable to the specific area of the research</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l10 level1 lfo29; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Design of the mathematical model and the hypotheses of the study</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l10 level1 lfo29; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The quest to obtain the right economic statistics i.e. the collection,
collation and analysis of requisite data for the research, and</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l10 level1 lfo29; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Interpretation/discussion of the findings/results</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The researcher should keep in mind that econometric
models are tools and therefore means to some desired ends. That is, our
professional calling is to provide plausible parameter estimates that should be
useful for policy analysis and decision making. Therefore, any mathematical
and/or statistical model must be able to deliver these objectives of the
researcher in an efficient manner. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Model Specification and Estimation Techniques</span></b><b><span style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Consequently, model specification is the
nucleus/DNA of any scientific economic research. I usually call it the economics
of the study. It shows the depth of the researcher in the knowledge of theoretical
economics as well as ability to state clearly the contribution(s) to knowledge
as envisaged in the study. The latter may come as:<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Additional variable to existing theoretical model</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">A single equation now specified as system of equations in order to
capture a phenomenon hitherto not considered, or </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Application of a technique not commonly used in our own environment. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="tab-stops: list 1.0cm; text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Once the model is correctly
specified, the next step is to consider the estimation technique that will
produce the most efficient estimates of the parameters of the model. It is
important to use a technique of estimation that will deliver the objective(s)
of the study. It is apposite to mention some estimation techniques at this
stage. It should, however, be noted that the list is not exhaustive. Some of
these are as follows: ordinary least squares (OLS), indirect least squares
(ILS), instrumental variables (IL), two stage least squares (2SLS): in the case
of system of simultaneous equation, three stage least squares (3SLS), error
correction model (ECM) which examines short-run dynamics, cointegration
regression, generalised method of moments (GMM), vector autoregressive method
(VAR) which examines the effect of shocks on a system, structural vector autoregressive
method (SVAR), panel data method, panel vector autoregressive method (PVAR), panel
structural vector autoregressive method (PSVAR), vector error correction (VECM),
panel cointegration, panel vector error correction (PVECM) and so on.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Some learning resource materials are, but not limited to:</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l17 level1 lfo6; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Gujarati D. N. (2013). Basic
Econometrics, Eight Edition, McGraw-Hill International Editions Economic
Series, Glasgow </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l17 level1 lfo6; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Maddala, G. S. and Lahiri,
K. (2009). Introduction to Econometrics. Fourth Edition. John Wiley</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l17 level1 lfo6; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Wooldridge J. M. (2009).
Introductory Econometrics, Fourth Edition, South-Western Cengage Learning,
Mason, U.S.A </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<b><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Dynamic General Equilibrium (DGE) Models</span></i></b><b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></i></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">There are lots of other techniques of
estimation that should be of interest to the younger generations of economists.
The basic framework is the <b>dynamic general equilibrium</b> (DGE) theories.
Models built around this method are solved using the <b>DYNARE</b> codes in the
<b>MATLAB</b> environment or directly using the Matlab codes written for such
models. As part of the estimation is the need to calibrate the model. This
consists of finding values for some parameters in the model though theoretical
knowledge, calculating long-run averages as well as micro-econometric studies.
The statistics often used are derived from the <b>Bayesian inference</b> as
against the classical statistics referred to in the preceding paragraphs. Some
of these models are: real business cycle (RBC), New Keynesian models (NKM), dynamic
stochastic general equilibrium (DSGE), over-lapping generation (OLG), computable
general equilibrium (CGE), dynamic computable general equilibrium (DCGE), Bayesian
vector autoregression (BVAR), Bayesian structural vector autoregression (BSVAR),
dynamic macro panels (DMP), augmented gravity models (AGM) and multicounty New
Keynesian (MCNK) models.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Some learning resource materials are:</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l26 level1 lfo8; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Wichens, M. (2008).
Macroeconomic Theory: A Dynamic General Equilibrium Approach. Princeton
University Press, Princeton</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l26 level1 lfo8; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Canova, F. (undated).
Methods for applied Macroeconomic Research</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l26 level1 lfo8; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Dejong, D. N. and Dave, C.
(2007). Structural Macroeconometrics, Princeton University Press, Princeton.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l26 level1 lfo8; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Cooley, T. F. (ed.) (1995).
Frontiers of Business Cycle Research. Princeton University Press, Princeton.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l26 level1 lfo8; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">McCandless, G. (2008). The
ABCs of RBCs: An Introduction to Dynamic Macroeconomic Models. Harvard
University Press; and</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l26 level1 lfo8; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Lucas, R. E. (1991). Models
of Business Cycles.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">It is apposite to state that researchers must
have a working understanding of the tests that must be carried out under each
technique of estimation. I need to also draw the attention of interested
researcher in the area of dynamic general equilibrium because it requires
adequate knowledge of computational economics. Specifically, you need sound
working knowledge of the following: dynamic optimization, method of Lagrange
multipliers, continuous-time optimization, dynamic programming, stochastic
dynamic optimization, time-consistency and time-inconsistency and linear
rational-expectation models.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<b><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Some learning resource materials</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Dadkhah, K. (undated)
Foundation of Mathematical and Computational Economics, Thomson South-Western.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Interpretation of Results</span></b><b><span style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In interpreting the results of an econometric
model, you have the choice of the most appropriate method for your work either
the classical or Bayesian statistics as mentioned above. This aspect of the
work constitutes the scientific content emanating from economic statistics and
mathematical economics. In this case, we should be addressing statistics such
as:</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">R</span></i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-squared</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Adjusted <i>R</i>-squared (“goodness
of fit” test)</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">F</span></i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">-statistics</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Durbin-Watson statistic</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">These, in addition to the test of <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/heteroscedasticity.html" target="_blank">heteroscedasticity</a>
constitute the “diagnostic tests”. Once they fail to fall within the zones of
acceptance, we cannot go ahead to test for the significance of each variable. There
may be the need for: model re-specification, detection and correction of
autocorrelation, and/or detection and correction of <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/multicollinearity.html" target="_blank">multicollinearity</a>. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We may also need to test for
heteroscedasticity. The occurrence of any of this is an evidence of the
violation of assumption(s) of the technique being applied. This is followed by
the statistics to test the significance of the individual variables included in
the model. This was the standard during the époque of almighty OLS. Later in
the history of applied econometrics, it was observed that certain time-series
are non-stationary, i.e. their means, variances and covariances are not
constant over time. In such situation regression results are generally
meaningless and are, therefore, termed spurious. In order to correct for the
latter, the statistics often used to examine the stationarity of time series
include the following: Dickey-Fuller test</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">, </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">“augmented” Dickey-Fuller
test in the presence of error term that is none white noise, Panel data unit
root tests, co-integration tests and error correction model (ECM), to mention a
few. </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">The use of any of these tests should be in response
to the objective of the researcher and the desired contribution(s) to
knowledge.</span></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Some Pitfalls in Econometrics<o:p></o:p></span></b></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The wrong way to go in modelling</span></i></b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></i></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">How one interprets the coefficients in
regression models will be a function of how the dependent (<i>y</i>) and independent (<i>x</i>)
variables are measured. In general, there are three main types of variables
used in econometrics: (1) continuous variables, (2) the natural logarithm of
continuous variables, and (3) dummy variables. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="font-family: "symbol"; font-size: 12pt; text-indent: -14.2pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><b style="text-indent: -14.2pt;"><span lang="EN-US" style="font-size: 12pt;">Some Specific Rules of Thumb from Statistics</span></b></div>
<br />
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<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">After performing a regression analysis:</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Look
at the number of observations:</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l25 level1 lfo17; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Is your result in line with
a priori expectation? </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l25 level1 lfo17; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If not, you should find out
why. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l25 level1 lfo17; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Remember, any observations
with missing values will be dropped from the regression.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l25 level1 lfo17; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Do not take the logarithm of
a variables whose value equals zero. The model will not run, simple. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<!--[if !supportLists]--><span lang="EN-US" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Ensure the number of observations in your model falls within the rule
i.e. sample size should be greater than or equals to 30 (the law of large
numbers).<o:p></o:p></span></div>
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<br /></div>
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<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Observe
the value of the <i>R</i><sup>2</sup></span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">: </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l18 level1 lfo18; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The <i>R</i><sup>2</sup> tells you the percentage of the total variation in
the dependent variable that the independent variables of your model “explains”.
</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l18 level1 lfo18; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">This should be less than 1.
The rest is the error term. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l18 level1 lfo18; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Suppose an estimated model
of <i>R</i><sup>2</sup> = 0.46.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">This means that 46% of the total variation in the dependent variable is
explained by the independent variables. This is not a “good fit”. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l18 level1 lfo18; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">For a regression to have a
good fit then we must have a result such that 0.5<<i>R</i><sup>2</sup><1. This is in the case of a time series
regression. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l18 level1 lfo18; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, it is considered
good for cross-section data and very good for panel data.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; text-align: justify; text-indent: -14.2pt;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Problems with R<sup>2</sup>: </span></i><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If you have a ‘very low’ <i>R</i><sup>2</sup>, have a rethink about
whether you might have omitted some important variables. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">However, be careful not to
include unnecessary variables only to increase your <i>R</i><sup>2</sup>. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">A ‘very high’ <i>R</i><sup>2</sup> could indicate several
problems. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Firstly, if a high <i>R</i><sup>2</sup> is combined with many
statistically significant variables, your independent variables might be highly
correlated amongst themselves (multicollinearity).</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">You might consider dropping
some in the interest of parsimony. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">It might be an indication
that you have mis-specified your model.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; text-align: justify; text-indent: -14.2pt;">
<i><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The adjusted R<sup>2</sup>: </span></i><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Adjust the <i>R</i><sup>2</sup> to penalize the inclusion
of more variables. i.e. correct for the degree of freedom.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l28 level1 lfo19; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Include as many variables as
you need but keep your model as parsimonious as possible. Observe the rules guiding
this.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l8 level1 lfo1; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Look
at the <i>F</i>-test</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l7 level1 lfo20; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The <i>F</i>-test aims at the “joint significance” of the</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">model. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l7 level1 lfo20; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">More formally it is a test
of whether all your coefficients are jointly equal to zero under the null
hypothesis. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l7 level1 lfo20; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If they are, effectively
your model is not really explaining anything. Hint: ideally you want a high <i>F</i>-value, and a low corresponding <i>p</i>-value
</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 36.0pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l8 level1 lfo1; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Interpret
the signs of the coefficients. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l23 level1 lfo21; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Which ones should be
positive and which should be negative from the theoretical perspective?
Interpret this!</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l23 level1 lfo21; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">A positive coefficient means
that variable has a positive impact on your dependent variable, and a negative
one has a negative impact or inverse relationship.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 36.0pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l8 level1 lfo1; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Interpret
the size of the coefficients where relevant.</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l14 level1 lfo22; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If you obtain a
statistically significant coefficient-wonderful! </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l14 level1 lfo22; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So maybe you’ve found
consumption increases with disposable income. But by how much? Is it close to 1
by which the marginal propensity to consume is high and the marginal propensity
to save is low? What would be the effect of this on the economy?</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 36.0pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 1.0cm; mso-list: l8 level1 lfo1; tab-stops: list 1.0cm; text-align: justify; text-indent: -14.15pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Look
at the significance of the coefficients (most important?).</span><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l15 level1 lfo23; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">This should in fact become
the first thing that your eyes drift towards when you get regression output. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l15 level1 lfo23; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">You should feel a little
hint of excitement as you are waiting to find out whether your model works and
whether your theory has been proved correct or not.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l15 level1 lfo23; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The test of significance is
designed to test whether a coefficient is significantly different from zero or
not. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l15 level1 lfo23; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If it is not, then you must
conclude that your explanatory variable does not, in fact, explain at all your
dependent variable.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l15 level1 lfo23; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">We use <i>t</i> - test (just like we learnt in first year statistics) to test
this so that we compare a <i>t </i>- value
taken from the table (at a given significance level, α, with <i>n - k</i>
degree of freedom) with a calculated <i>t</i>,
where <i>n = </i>number of observations and <i>k = </i>number of parameters
estimated/independent variables; <i>n – k = </i>degree of freedom.</span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l15 level1 lfo23; text-align: justify; text-indent: -14.2pt;">
<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div class="MsoNoSpacing" style="margin-left: 42.55pt; mso-list: l11 level1 lfo24; text-align: justify; text-indent: -14.2pt;">
<span style="font-family: "symbol"; font-size: 12.0pt;"><span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"></span></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.15pt;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 12pt; text-indent: -14.15pt;">Others</span></div>
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</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Other tests follow, such as testing for normality of error terms, checking
for existence of heteroscedasticity, performing specification and robustness
tests. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">But these exciting topics are to be covered if your econometric work
would have any useful output valuable for policy making and decision making.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Discussion of Results</span></b><b><span style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The essence of a scientific economic research
is to build economic models that enable us obtain plausible estimates from
given set of data. We should know that the structural parameters estimated
encapsulate our behaviour and, therefore, in discussing them, we need to go
beyond the confine of economics to locate additional means of buttressing our
results from: <o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">historical context</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">socio-political condition, and </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">psychological state as well as </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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</span></span><!--[endif]--><span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">international environment</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></div>
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<b><span lang="EN-US" style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Conclusion</span></b><b><span style="color: #0033cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">I have tried to raise some important issues in
this post. There are so many things to keep in mind when preparing a research work.
The most important of them all is the need to keep your model simple and avoid
frivolities in modeling. It is important to remember that we are first of all
economists. The tools of analysis at our disposal should not overshadow that calling.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Quite a lot has been said about how a researcher in the field of economics can handle data, interpret and discuss research findings that will be relevant for policy-making. If you still have further questions or comments in this regard, kindly post them below for the benefit of all.</span></div>
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<span lang="EN-US" style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Post your comments and questions….</span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><o:p></o:p></span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com2tag:blogger.com,1999:blog-1876221430378807805.post-56709103539056494292018-01-20T00:51:00.001+01:002018-02-06T17:18:47.545+01:00Stata: Interpreting Two-way ANOVA Procedure<h2 style="line-height: 115%; text-align: center;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 18.0pt; line-height: 115%;">Two-way
ANOVA Procedure using Stata</span></b></h2>
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<span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">This is a
follow-up to my previous post on how to analyse the <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/one-way-anova-procedure-using-stata.html" target="_blank">one-way ANOVA using Stata</a> analytical software endeavour to read it up...it also provides a good
introduction to running ANOVA.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The
essence of two-way ANOVA in data analysis<o:p></o:p></span></b></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">ANOVA simply means <b><u><span style="color: blue;">an</span></u></b>alysis <b><u><span style="color: blue;">o</span></u></b>f
<b><u><span style="color: blue;">va</span></u></b>riance
and its importance in analysing behavioural relationships between and among
variables makes its use endearing to researchers. Basically, the ANOVA procedure
is to determine if the average value (that is, the mean) of a <i>dependent</i> variable (the regressand,
outcome variable, and endogenous variable) is the same in two or more
unrelated, <i>independent</i> groups. That
is, the two-way ANOVA indicates whether the mean of a dependent variable is the
same or differs across independent unrelated categorical groups. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The
two-way ANOVA compares the mean differences between groups that have been split
on two independent variables (called <b>factors</b>).
The primary purpose of a two-way ANOVA is to understand if there is an <b>interaction</b> between the two independent
variables on the dependent variable. </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The moment you understand how to compute the two-way ANOVA and interpret
your table, you will always want to incorporate it in your study or research…after
ensuring that your data meets some salient conditions.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">For
instance, you could use a two-way ANOVA to understand whether there is an <i>interaction</i> between activity level and
diet on bmi (i.e. the dependent variable would be “<i>bmi</i>”, measured on a continuous scale, and your independent
variables would be “<i>activity</i> <i>level</i>” (which has three groups – “low”,
“moderate” and “high”) and “<i>diet</i>” (which
has three groups “vegan”, “vegetarian” and “animal-based”). Again, the two-way
ANOVA can be used understand if there is an interaction between demographic
location and types of housing on rentals (i.e. the dependent variable would be
“<i>rentals</i>”, measured on a continuous
scale, and the independent variables would be “<i>location</i>” (which has two groups – “rural” and “urban”) and “<i>housing types</i>” (which has six groups “one-room
apartment”, “two-room apartment”, “one-bedroom condo”, “two-bedroom condo”,
“mini flat”, and “standard flat”). Lastly, an agronomist may be interested in
knowing the interaction between temperate conditions and type of fertiliser on
say the crop yield of cassava (i.e. the dependent variable would be “<i>yield</i>”, measured on a continuous scale,
and the independent variables would be “<i>temperate</i>”
(which has four groups – “autumn”, “spring”, “summer” and “winter”) and “<i>fertiliser</i>” (which has two groups
“organic”, and “inorganic”). If there are three independent variables rather
than two, a three-way ANOVA will be performed, and if four independent
variables, a four-way ANOVA will be performed and so on<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Given this preamble, here is a “step-by-step”
tutorial showing you how to carry out a two-way ANOVA and some post-hoc checks
using Stata analytical package. But before I proceed, it is important for you
to understand some basic rules underlying the use of two-way ANOVA procedure.
That is, your data must meet these criteria failing which your results may be
invalidated if they are not adhered to. There are six (6) of them:<o:p></o:p></span></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rules:<o:p></o:p></span></b></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">These six "rules" represent the blueprint
guiding the use of the two-way ANOVA technique. If any is not satisfied, you
may obtain invalid results. Please note that the first three assumptions are
closely related to the nature of your data and study structure (that is,
directly related to your choice of variables), thus Stata cannot validate those
while the last three must be met using some Stata criterion. It is therefore
important that you ascertain that your study meets these conditions before
proceeding with the two-way ANOVA.<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rule #1:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> Make sure that the <b>dependent variable (regressand,
outcome variable)</b> is cardinal and measured in <b>continuous terms</b>. Some
example of variables measured in continuous terms are: time (measured in minutes,
seconds, and milliseconds), weight (measured in stone, pounds, kilogramme, and
grams); rentals (measure in local currency) and so on. These are called <b>continuous variables. <o:p></o:p></b></span></div>
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</span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rule #2:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> Both <b>explanatory variables (regressor,
independent variable)</b> ought to comprise <b>two or more categorical</b>, <b>independent
(unrelated) groups</b>. Some examples of these <b>categorical variables</b>
are income group (3 groups: high-income, middle income and low income); grade
(4 groups: excellent, very good, good, and poor); demography (2 groups: rural
and urban); banking (3 groups: investment, mortgage, microfinance) etc. So make
sure that your explanatory variable is a categorical variable.<b><o:p></o:p></b></span></div>
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<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rule #3:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> Ensure that you have <b>independence of observations</b>. That is, your
observations must not over-lap across the different groups. This simply means
that there must be no relationship between the observations in each group or
between the groups themselves. For instance, an observation in a “winter” group
must <b><u>not</u></b> be represented again
in a “spring” group. Needless to say that, participants across the groups must
be different. But where an exception is the case, the repeated measures of
ANOVA should be used rather than the two-way ANOVA.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 7.1pt; mso-list: l0 level1 lfo3; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rule #4:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> Be wary of <b>conspicuous <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/heteroscedasticity.html" target="_blank">outliers</a></b>. These are figures that are
either abnormally high or low, that is, they do not follow the typical pattern
in a particular variable. The presence of outliers can bias your results, </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">they
can have a negative effect on the two-way ANOVA, thereby reducing the results
accuracy</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">. However, they can easily be tested in Stata by
using the <b>Boxplot</b> or <i>summary</i> syntax (<i>sum</i> for short). The syntax computes the mean, standard deviation,
minimum and maximum values in each variable in your data, thus enabling you to
detect (identify) the abnormal figure.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 7.1pt; mso-list: l8 level1 lfo2; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rule #5:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> Since the two-way ANOVA is susceptible to
violations of normality, it is essential that the <b>dependent variable</b>
must be <b>approximately normally distributed for each category of the
independent variable</b>. Although, you may still obtain some valid results if
this rule is violated, that is why your data must be <b>approximately</b> and
not <b><i>100%
</i></b>normally distributed before running a two-way ANOVA. A histogram test, <b>Shapiro-Wilk</b> test or <b>Jarque-Bera</b> test can be conducted in
Stata to test for normality of residuals.<b><o:p></o:p></b></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 7.1pt; mso-list: l7 level1 lfo1; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Rule #6:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> There must be <b>homogeneity of variances</b>. This can be tested with
the <b>Levine’s test</b> for homogeneity of
variances in Stata. The Levine’s test is very vital when it comes to
interpreting the results from a two-way ANOVA guide because Stata is capable of
producing different outputs depending on whether your data meets or fails this
assumption.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Note:</span></b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> The first three rules are
specific to your data, choice of variables and nature of study which any
analytical package, like Stata, has no control and thus cannot be
scientifically verified. However, ascertaining that your data meets the last
three rules can be verified which may seem daunting, but it is important that
you do them. Moreso, these packages have really simplified these procedures.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">So let us take an example to understand the two-way
ANOVA….<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">EXAMPLE<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">From
<a href="https://drive.google.com/drive/folders/1WRZX474X8pj3Egxo51Tz0Y66me0v2JRd" target="_blank">Wooldridge’s</a> discrim2.dta or discrim2.xlsx files (if you don’t have Stata
installed on your devise, download the .xlsx file and feed into the analytical
package of your choice).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">(Note: for simplicity, I have
extracted from the initial dataset, discrim.dta, to use for this example. The
initial dataset is quite detailed such that several two-way ANOVA simulations
can be carried out)</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="color: blue; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">A researcher collected ZIP code-level data on the prices charged
for small fries at four fast-food chains – Burger King, Kentucky Fried Chicken,
Roy Rogers and Wendy’s – along with the characteristics of the ZIP-code
population in two US states – New Jersey and Pennsylvania. The idea is to
compare the prices charged by these fast-food chains to see whether the prices
are the same across the two states.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">In this example, the dependent variable is “<i>price of fries”</i> (measured in US
dollars), whilst the independent variables are “<i>state”</i> and “<i>chain</i>”. <i>state</i> has two independent groups: “<i>New Jersey</i>” and “<i>Penn</i>” and “<i>chain</i>” has
four independent groups: “<i>BK</i>”, “<i>KFC</i>”, “<i>RR</i>” and “<i>WD</i>”. Remember
that both are categorical variables whose members (observations) must not
over-lap within their groups. The two-way ANOVA in this instance, is used to
determine whether there is a statistically significant difference in prices
charged among the four fast-food chains across the two states.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">But before we begin, ensure that you set up your
data in Stata (or any analytical package of your choice)<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Setting up the data in Stata<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Ensure original data is in excel format (.xlsx,
.xls or .csv)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Have separate columns for prices of fries, state
and chain<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Open the <b>Stata</b>
application<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Go to <b>Data</b>
>> <b>Data Editor (Edit)</b> <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Highlight data to be copied from excel<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click the “<b>paste</b>”
icon in Stata<o:p></o:p></span></div>
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<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">A dialog box opens: Select “<b>Treat first row as variable names</b>”<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">8.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click “<b>OK</b>”
and <b>Save</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">These
steps (1 – 7) create your Stata dataset (that is, <i>.dta</i> file)<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">ATTENTION:</span></b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> If you are using Stata, make
sure you create a log file and a do-file.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">To
create a log file:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The log file gives a history of what you have done.
You can always revisit the log file <i>(saved
as .smcl)</i> to review the processes. So, it is advantageous to always have a
log file. To create a log file:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l13 level1 lfo10; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Go to <b>Stata</b>
>> <b>File</b> >> <b>Log</b> >> <b>Begin</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l13 level1 lfo10; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Give it a <i>filename</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l13 level1 lfo10; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>Save</b><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">To
create a do-file:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The do-file on the other-hand shows the commands
(codes) used to execute each process. Those familiar with the coding approach
will agree with me that having a do-file can speed up the time used in
executing the work. To create a do-file <i>(saved
as .do)</i>:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l5 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Go to <b>Stata</b>
>> <b>New Do-File Editor</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l5 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">New do-file opens<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l5 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click File >> <b>Save As</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l5 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Give it a <i>filename</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l5 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>Save</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Having
established that both explanatory variables are <b>categorical</b> <b>variables</b>
made up of two and four groups respectively, it is important that <b>Value Labels</b> for both explanatory
variables </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">“<i>state”</i> and “<i>chain</i>”</span><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> </span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">are created in Stata. The essence
is to create <b>values</b> for each group in
order to make estimations possible. So, the values for <i>New Jersey</i> and <i>Penn</i> under
<i>state</i> will be <b>1</b> and <b>2</b> respectively
while those for <i>BK</i>, <i>KFC</i>, <i>RR</i>
and <i>WD</i> under <i>chain</i> will be <b>1</b>, <b>2</b>, <b>3</b>
and <b>4</b> respectively.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">How to do
that? Here are the steps:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l9 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Go to <b>Stata</b>
>> <b>Data</b> >> <b>Data</b> <b>Utilities</b> >> <b>Label</b>
<b>Utilities</b> >> <b>Manage</b> <b>Value</b> <b>Labels</b> >> <b>Create</b> <b>Label</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l9 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Enter “<b>new
label name</b>”: <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l9 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Enter the appropriate values. Enter <b>1</b> for <b>Value</b>, and <b>New Jersey </b>for
<b>Label</b>, click <b>ADD</b>. Next, enter <b>2</b> for <b>Value</b>, and <b>Penn</b> for <b>Label</b> click <b>ADD</b>. Then click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l9 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Again, click “<b>Create
Label</b>”<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l9 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Enter “<b>new
label name</b>”: <i>chain</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l9 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Enter the appropriate values. Enter <b>1</b> for <b>Value</b>, and <b>BK </b>for <b>Label</b>, click <b>ADD</b>. Next, enter <b>2</b> for <b>Value</b>, and <b>KFC</b> for <b>Label</b> click <b>ADD</b>. Again, enter <b>3</b> for <b>Value</b>, and <b>RR </b>for <b>Label</b>, click <b>ADD</b>.
Lastly, enter <b>4</b> for <b>Value</b>, and <b>WD</b> for <b>Label</b> click <b>ADD</b>. Then click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">If it is correctly
done, then you should have something like this as shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6Y7CYsoThF-OGUzVdTIeud0P0ndYGsCP4vikBkPPZUv6O7y76YWEzmcGf8UAcjpOVJaeHAVE6FdglZyWOJv1N0BCrejT1zkowytg3i5xJT0fKuVH6sMiXKpWWmc4s2Uwsu-EsInxgiTE/s1600/Manage+Labels.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Creating Value Labels for Categorical Variables from http://cruncheconometrix.com.ng" border="0" data-original-height="309" data-original-width="449" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6Y7CYsoThF-OGUzVdTIeud0P0ndYGsCP4vikBkPPZUv6O7y76YWEzmcGf8UAcjpOVJaeHAVE6FdglZyWOJv1N0BCrejT1zkowytg3i5xJT0fKuVH6sMiXKpWWmc4s2Uwsu-EsInxgiTE/s1600/Manage+Labels.png" title="Creating Value Labels for Categorical Variables" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Creating Value Labels for Categorical Variables Using Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">Next is
to </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">assign value label</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> to both categorical/explanatory
variables one at a time. To do that:</span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Go to <b>Stata</b>
>> <b>Data</b> >> <b>Data</b> <b>Utilities</b> >> <b>Label</b>
<b>Utilities</b> >> <b>Assign Value</b> <b>Label to Variable</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under “<b>Variables</b>”
select <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under “<b>Value
label</b>” select <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Again, under “<b>Variables</b>”
select <i>chain</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under “<b>Value
label</b>” select <i>chain</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">You
should have something like this for both <i>state</i>
and <i>chain</i>:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_4Y1ewsOW4Jm5cHw1b0mFZZKIqy0yjpuowEoHBRQruRZ92TWCUkIq7zxdzIz94tR1KuXNKWw50n-LH2ZMdXa8_Z49066Q90UhrWX3VjWklQW7ucsLpjlrw_1J0EB2BnIffwwy2o4EZUE/s1600/Assign+labels.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Adding Value Labels to Categorical Variables from http://cruncheconometrix.com.ng" border="0" data-original-height="294" data-original-width="406" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_4Y1ewsOW4Jm5cHw1b0mFZZKIqy0yjpuowEoHBRQruRZ92TWCUkIq7zxdzIz94tR1KuXNKWw50n-LH2ZMdXa8_Z49066Q90UhrWX3VjWklQW7ucsLpjlrw_1J0EB2BnIffwwy2o4EZUE/s1600/Assign+labels.png" title="Adding Value Labels to Categorical Variables" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Adding Value Labels to Categorical Variables<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata) </td></tr>
</tbody></table>
<div align="center" class="MsoNoSpacing" style="line-height: 115%; text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"><br /></span></div>
<div style="text-align: left;">
<span style="font-size: 12pt;">With all
the steps correctly done, your dataset should look like this:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkFzcE45nNSQBPDVx6RPFY6jy7Y9VuRcY5jO7-7rJNCDXnSkiDNhyF_LMcWvgzK6bOWRyf38A4gXZxXfNMH8BMyUzxwOYBTeVk8l1nQU_VkUqw7WTc34AdeCHTLSp3SIJj4FiJjS5Bqeo/s1600/Data+Outlook.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Dataset showing dependent and explanatory variables in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="664" data-original-width="736" height="576" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkFzcE45nNSQBPDVx6RPFY6jy7Y9VuRcY5jO7-7rJNCDXnSkiDNhyF_LMcWvgzK6bOWRyf38A4gXZxXfNMH8BMyUzxwOYBTeVk8l1nQU_VkUqw7WTc34AdeCHTLSp3SIJj4FiJjS5Bqeo/s640/Data+Outlook.png" title="Dataset showing dependent and explanatory variables in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dataset showing dependent and explanatory variables in Stata<br />
Source: CrunchEconometrix from Wooldridge Dataset<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<br />
<div align="center" class="MsoNoSpacing" style="line-height: 115%; text-align: center;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">There are 410 observations, and to know the
distribution of the four fast-food chains across the two states, use the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;"><i>tabulate</i></b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">
syntax. </span><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">That is,</span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">tab</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> state chain<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">and you have this output shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKbRIZdhaoGZ3K_m3CCwvnnLhy4tsBvs5_ETj9RdNabMLCyr6tSz56LbTuJAPyhdSFtI9d6yGia0jiQvxVwVYGMSG52KOiSXHZxy13jVAxEXYyItkBuW45ibIXSeieljKgMlgjR0PA0s4/s1600/State-Chain+Distribution.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Table showing distribution, Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="152" data-original-width="507" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKbRIZdhaoGZ3K_m3CCwvnnLhy4tsBvs5_ETj9RdNabMLCyr6tSz56LbTuJAPyhdSFtI9d6yGia0jiQvxVwVYGMSG52KOiSXHZxy13jVAxEXYyItkBuW45ibIXSeieljKgMlgjR0PA0s4/s1600/State-Chain+Distribution.png" title="Table showing distribution, Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Table showing the distribution of fast-food chains across state<br />
Source: CrunchEconometrix from Wooldridge Dataset<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The above table shows how the 410 observations are
distributed among the four fast-food chains in the two US states. For instance,
Roy Rogers has 82 outlets in New Jersey and 17 in Pennsylvania, Wendy’s has 45
outlets in New Jersey and 15 in Pennsylvania and so on.</span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">We are about to dig in much further…</span><span style="font-family: "wingdings"; font-size: 12.0pt; line-height: 115%;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Please note that in Stata, you can either use the <b>code</b> (<b>command, syntax</b>) approach or the <b>graphical</b> <b>user</b> <b>interface</b> (<b>GUI</b>). Either approach is fine. If you are familiar with the coding
approach, just go ahead and use it, if otherwise use the GUI (where you just
click the applicable menus).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Having prepared our dataset, now let us run the two-way
ANOVA. This tutorial will in the <b>first part</b>
cover the two-way ANOVA analysis and in the <b>second part</b> the post-hoc checks. I will be using the <b>syntax approach</b>, but will show you later
on how to manoeuvre the GUI interface…..are you ready? On the assumption that
our dataset is in line with the six rules….we begin!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">State
the null and alternative hypotheses for the test<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">H<sub>0</sub>: the location of state will have no
effect on prices charged for small fries<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">H<sub>0</sub>: the type of fast-food chain will
have no effect on prices charged for small fries<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">H<sub>0</sub>: state and chain interaction will
have no effect on prices charged for small fries<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">H<sub>1</sub>: the null hypotheses is not true<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">All codes are typed into the <b>Command</b> window, as shown below, and you simply press the <b>ENTER</b> key:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsb3Qf1kpLEfFb7zL2fxvjGeyhK-bY__WJt2euB67wLKwChJz7KxICE9T2v4ZNjlX7wcFwlUYT9bIuMPNIOwQsiQNcDcv71WpQYuU2pyKwsV_vvcFSFhkm59VFDz0LMVZadAjwPuhF3nk/s1600/Command+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Command box in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="99" data-original-width="1001" height="63" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsb3Qf1kpLEfFb7zL2fxvjGeyhK-bY__WJt2euB67wLKwChJz7KxICE9T2v4ZNjlX7wcFwlUYT9bIuMPNIOwQsiQNcDcv71WpQYuU2pyKwsV_vvcFSFhkm59VFDz0LMVZadAjwPuhF3nk/s640/Command+Box.png" title="Command box in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Command box in Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Two-way
ANOVA Procedure</span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">I
will approach this from two angles. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">First</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">,
we may want to know the <b>main effects</b>
of each explanatory variable on the dependent variable, and the syntax is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">anova
<b><i>y
x<sub>1</sub> x<sub>2</sub></i></b><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">where
the <b><i>y</i></b>
is the dependent variable (<i>pfries</i>)
and <b><i>x<sub>1</sub></i></b>
is the categorical/explanatory variable <i>state</i>
and <b><i>x<sub>2</sub></i></b>
is the categorical/explanatory variable <i>chain.</i>
This becomes:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">anova<i> pfries</i> <i>state chain</i><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The
Stata output is shown as:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-VE_N33d-aZnh5__vbDXFkmcPZ5kl5V-mlRyCnljHMCn_xIUJsJFrsJSA1CLHEsaXoIUVYLkeM6JmOwCOAgsQ-qD-GU3liP9WYWSijpYdZ0CSA1NC67tVUrvd9R3KB27w4ZDkb68ZVyQ/s1600/Regression+%25281%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata output on the main effects from http://cruncheconometrix.com.ng" border="0" data-original-height="252" data-original-width="575" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-VE_N33d-aZnh5__vbDXFkmcPZ5kl5V-mlRyCnljHMCn_xIUJsJFrsJSA1CLHEsaXoIUVYLkeM6JmOwCOAgsQ-qD-GU3liP9WYWSijpYdZ0CSA1NC67tVUrvd9R3KB27w4ZDkb68ZVyQ/s1600/Regression+%25281%2529.png" title="Stata output on the main effects" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata output on the main effects<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
</span></div>
<div align="center" class="MsoNoSpacing" style="line-height: 115%; tab-stops: 163.05pt; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><v:shape id="Picture_x0020_24" o:spid="_x0000_i1034" style="height: 189pt; mso-wrap-style: square; visibility: visible; width: 431.25pt;" type="#_x0000_t75">
<v:imagedata o:title="Regression (1)" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image006.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">The
Stata output churns out quite a lot of information. For instance, the number of
observations is given as 393 instead of 410. Reason is because 17 observations
have missing values. The </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">F</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">-statistics
and the associated </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">p</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">-values are also
indicated. For the </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">Model</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">, the </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">F</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">-statistic (55.25) and its associated </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">p</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">-value (0.0000) shows that both
categorical variables significantly explain </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">pfries</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">.
For </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">state</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> and </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">chain</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">, their </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">F</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">-statistics
and the associated </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">p</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">-values indicate
that both have individual-significant effects on </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">pfries</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">. The </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">R</i><sup style="font-family: "Times New Roman", serif; text-align: justify;">2</sup><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">
(0.3629) shows the percentage of variation in </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">pfries</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> that is explained by </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">state</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">
and </span><i style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;">chain</i><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">.</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Second</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">,
to obtain both the individual and interactive effects of <i>state</i> and <i>chain</i> on <i>pfries</i>, the syntax is:<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<strong><span style="font-size: 12.0pt; line-height: 115%;">anova <i>pfries</i> <i>state</i> <i>chain</i> <i>state</i>#<i>chain</i></span></strong><b><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></i></b></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">and
the Stata output is as shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWQ8SOBOcojZfL_-YWplU7GMYCJJx78xHm99ZjTcQoZB9Yu2-MnxolutkFQ0UIJwxBdJuXZ0WuoNP2tF4Von2gAfPC7Qu9-Z8g6HadjrT-8gwKp0rJpz2xmj9qW_BMmkUz_iDGZtH4CQc/s1600/Regression+-+Interaction.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata output on the main and interaction effects from http://cruncheconometrix.com.ng" border="0" data-original-height="270" data-original-width="577" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWQ8SOBOcojZfL_-YWplU7GMYCJJx78xHm99ZjTcQoZB9Yu2-MnxolutkFQ0UIJwxBdJuXZ0WuoNP2tF4Von2gAfPC7Qu9-Z8g6HadjrT-8gwKp0rJpz2xmj9qW_BMmkUz_iDGZtH4CQc/s1600/Regression+-+Interaction.png" title="Stata output on the main and interaction effects" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata output on the main and explanatory effects<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
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<span style="font-family: "times new roman" , serif; font-size: 12pt;">The
explanations are similar to those stated previously except with the addition of
the interaction term </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">state#chain</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">.
Here the </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">F</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">-statistic (0.31) and its
associated </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">p</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">-value (0.8204) shows
that the joint-effect of both categorical variables </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;"><i>insignificantly</i></b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> explains </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">pfries</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">. If a statistically significant
interaction is observed, the result can be followed up by determining if there
are any “simple main effects”, and if there are, what these effects are.</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Post-hoc tests<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The
<i>F</i>-statistic tells us if there is the
need to perform a post-hoc test or not. If the statistic is significant as it
is for <i>state</i> and <i>chain</i>, then some post-hoc tests can be done but where the statistic
is not significant, then there no need to talk about the variable, act as if
the effect is <b><i>zero</i></b> as it is in the case if the interaction term <i>state#chain</i> because in actual fact, the
effect on the population is zero. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; tab-stops: 169.95pt; text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Bottom line: only discuss the results that are
significant!</span></b><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"> <o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Therefore,
since the main effect of each categorical variable is significant, post-hoc
tests can be performed as done if a one-way ANOVA procedure is conducted. In
this example, we use the Scheffe’s test. But this test will be irrelevant for <i>state</i> since we already know that there
are only two means and the <i>F</i>-statistic
has shown that the difference between them is statistically significant.
However, because we have four groups under <i>chain</i>,
the Scheffe’s test will be relevant in pointing out those combinations between
the groups that have significant differential in their mean prices. The test
can be computed using the syntax:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">oneway
<i>pfries</i> <i>chain, scheffe<o:p></o:p></i></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The
Stata output is shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKdJ9O4cqmtMJTVv2tg8fiTY5oRgbelHvsCgK87-vWa9tqz1wFaot21DS6VE6cSXNSM4xaXNGTQbAD5BhO8M6AfwrAoqaclPn3dsVFHAVA4RkblF7bP06GyaMVeNU6TE3P3kXTySKwHpc/s1600/Sceffe%2527s+Comparison.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Scheffe's post-hoc test in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="420" data-original-width="558" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKdJ9O4cqmtMJTVv2tg8fiTY5oRgbelHvsCgK87-vWa9tqz1wFaot21DS6VE6cSXNSM4xaXNGTQbAD5BhO8M6AfwrAoqaclPn3dsVFHAVA4RkblF7bP06GyaMVeNU6TE3P3kXTySKwHpc/s1600/Sceffe%2527s+Comparison.png" title="Scheffe's post-hoc test in Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Scheffe's post-hoc test in Stata<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The
Scheffe multiple comparison test tells us where the differences are between
each pair of means. Also, in a more-than-two group scenario, this test applies
corrections to the reported significance levels that take into account the fact
that multiple comparisons are being conducted. Thus, as can be seen from the
printout, the difference between the means of </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">BK</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> and </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">KFC</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> is -.053457
and the </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">t</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">-statistic is significant at
the 1% level. With all six combinations, only the difference between </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">WD</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> and </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">KFC</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> (.012284) falls just short of being statistically significant.</span></div>
<div class="MsoNoSpacing" style="line-height: 115%;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Addendum:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">By way of information,
here is how to manoeuvre the graphical user interface (GUI) to run the two-way
ANOVA.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Go to <b>Stata</b> >> <b>Statistics</b> >> <b>Linear
models and related</b> >> <b>ANOVA/MANOVA</b>
>> <b>Analysis of variance and
covariance</b> from the top menu, as shown below.<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdN9XIMNUbYevVelgAePbeuv0NPTwoZWTF6fSM_UXXVRWSelpt5DwhyphenhyphenyuolB-5plUGxARP8sC5gQKqeRZoDG2Pxh_eJ8iU4vHSQXu4cCZBvWQ3YLs3PlF8Jvsv7xWUYJF05mpipD0-t4Q/s1600/GUI+-+Anova.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Graphical user interface (GUI) for Two-way ANOVA in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="497" data-original-width="917" height="346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdN9XIMNUbYevVelgAePbeuv0NPTwoZWTF6fSM_UXXVRWSelpt5DwhyphenhyphenyuolB-5plUGxARP8sC5gQKqeRZoDG2Pxh_eJ8iU4vHSQXu4cCZBvWQ3YLs3PlF8Jvsv7xWUYJF05mpipD0-t4Q/s640/GUI+-+Anova.png" title="Graphical user interface (GUI) for Two-way ANOVA in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Graphical user interface (GUI) for Two-way ANOVA in Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: , serif; font-size: 12pt;">A dialogue box for </span><b style="font-family: proxima-nova, serif; font-size: 12pt;">anova - Analysis of variance
and covariance</b><span style="font-family: , serif; font-size: 12pt;"> opens:</span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under <b>Dependent</b> <b>variable</b>, select <i>pfries</i> from
the drop-down menu<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">You should have something like this:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGjVIwnEE434graZ7Zoa-fqkHwh3XKrTVOHFzJmgv-tpnxHedFwFdf8H1v3latwCp5eOVNwCnNyGJXnA5GumCYKnefo8CxE6L7I4B0xDGF2lkT86UIuu3dbVQ9ZiGzA12dfs-7oUNUOaA/s1600/GUI+-+Depvar.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Dialog box for dependent variable in Two-way ANOVA, Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="404" data-original-width="527" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGjVIwnEE434graZ7Zoa-fqkHwh3XKrTVOHFzJmgv-tpnxHedFwFdf8H1v3latwCp5eOVNwCnNyGJXnA5GumCYKnefo8CxE6L7I4B0xDGF2lkT86UIuu3dbVQ9ZiGzA12dfs-7oUNUOaA/s1600/GUI+-+Depvar.png" title="Dialog box for dependent variable in Two-way ANOVA, Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dialog box for dependent variable in Two-way ANOVA, Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">To analyse the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">individual
effect</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">s of both categorical variables on the dependent variables, here is
what to do:</span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">C</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">lick
on the three dot button, </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><v:shape alt="https://statistics.laerd.com/stata-tutorials/img/twa/button-three-dots.png" id="Picture_x0020_20" o:spid="_x0000_i1029" style="height: 15pt; mso-wrap-style: square; visibility: visible; width: 15pt;" type="#_x0000_t75">
<v:imagedata o:title="button-three-dots" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">, to the far right of the <span class="s-boxes1"><b><span style="background: yellow; line-height: 115%;">Model</span></b></span><span class="s-boxes1"><span style="background: yellow; line-height: 115%;">:</span></span> drop-down box and another dialog box
opens </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">where y</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">ou have <strong>Create
varlist with factor variables</strong> dialogue box:</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under <b>Type of variable</b>, leave <b>Factor variable</b> unchanged<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under <b>Specification</b>, leave <b>Main effects</b> unchanged<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Open the drop
down menu under <b>Variable</b> <b>1</b> >> select <i>state</i> >> <b>Add to
varlist</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Again, open
the drop down menu under <b>Variable 1</b>
>> select <i>chain</i> >> <b>Add to varlist</b></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Both </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">state</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">
and </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">chain</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> will be shown under </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">Varlist</b><span style="font-family: "times new roman" , serif; font-size: 12pt;">, so you should have something
like this:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1D64SF0DbWb3nbw5G34i9LRXP50SsN9FG9vnaDli7Qtqv60ji2PhBzs9NhPH5yQ0B_CiTOs5QTDgqGL_nSbrW4nv1zG4kBGRRXtUGn7N97U-JMh8Yo3PW8PHyVC-dqbwuPh9kqXVEPkk/s1600/Creating+FV.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Dialog box for factor variables in Two-way ANOVA, Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="505" data-original-width="412" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1D64SF0DbWb3nbw5G34i9LRXP50SsN9FG9vnaDli7Qtqv60ji2PhBzs9NhPH5yQ0B_CiTOs5QTDgqGL_nSbrW4nv1zG4kBGRRXtUGn7N97U-JMh8Yo3PW8PHyVC-dqbwuPh9kqXVEPkk/s1600/Creating+FV.png" title="Dialog box for factor variables in Two-way ANOVA, Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dialog box for factor variables in Two-way ANOVA, Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>OK</b> and the previous page is modified as
shown below with <i>state</i> and <i>chain</i> appearing under <b>Model</b>:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div align="center" class="MsoNoSpacing" style="line-height: 115%; text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><v:shape id="Picture_x0020_23" o:spid="_x0000_i1027" style="height: 307.5pt; mso-wrap-style: square; visibility: visible; width: 393pt;" type="#_x0000_t75">
<v:imagedata o:title="Creating Varlist (2)" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image013.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>OK</b> to obtain the same regression
outputs as in using the syntax approach.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">To analyse the <b>interactive</b>
<b>effects</b> of both categorical
variables on the dependent variables, here is what to do:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l3 level1 lfo14; tab-stops: list 14.2pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">C</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">lick
on the three dot button, </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><v:shape alt="https://statistics.laerd.com/stata-tutorials/img/twa/button-three-dots.png" id="Picture_x0020_22" o:spid="_x0000_i1026" style="height: 15pt; mso-wrap-style: square; visibility: visible; width: 15pt;" type="#_x0000_t75">
<v:imagedata o:title="button-three-dots" src="file:///C:\Users\ng\AppData\Local\Temp\msohtmlclip1\01\clip_image011.png">
</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">, to the far right of the <span class="s-boxes1"><b><span style="background: yellow; line-height: 115%;">Model</span></b></span><span class="s-boxes1"><span style="background: yellow; line-height: 115%;">:</span></span> drop-down box and another dialog box
opens </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">where y</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">ou have <strong>Create
varlist with factor variables</strong> dialogue box:</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under <b>Type of variable</b>, leave <b>Factor variable</b> unchanged<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Under <b>Specification</b>, select <b>Interaction (2-way)</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Open the drop
down menu under <b>Variable</b> <b>1</b> >> select <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 54.0pt; mso-list: l7 level2 lfo1; text-align: justify; text-indent: -11.45pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 115%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Open the drop
down menu under <b>Variable</b> <b>1</b> >> select <i>chain </i>>> <b>Add to
varlist</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l3 level1 lfo14; tab-stops: list 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">If it’s correctly done, <i>state</i> <i>chain</i> <i>state</i>#<i>chain</i> will show under <b>Varlist</b>,
so you have something like this:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHb00LX6UhB6Q-JmHipFU8QrGGDg7NkHuzA5iQ4uIxQxQIQ7T2GqXf_2H865mao9WEfx8WwXWrsYr-5IK1BgVgzixFXcOLiqVvXFMkpdnjr2TCned0-UU_oo_Or1cm6ltlzcAWNXmB8DU/s1600/Creating+FV+%25282%2529.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Dialog box for factor and interaction variables in Two-way ANOVA, Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="510" data-original-width="417" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHb00LX6UhB6Q-JmHipFU8QrGGDg7NkHuzA5iQ4uIxQxQIQ7T2GqXf_2H865mao9WEfx8WwXWrsYr-5IK1BgVgzixFXcOLiqVvXFMkpdnjr2TCned0-UU_oo_Or1cm6ltlzcAWNXmB8DU/s1600/Creating+FV+%25282%2529.png" title="Dialog box for factor and interaction variables in Two-way ANOVA, Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Dialog box for factor and interaction variables in Two-way ANOVA, Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 115%; text-indent: -14.2pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 115%; text-indent: -14.2pt;">Click <b>OK</b> and you will obtain the same regression
outputs as in using the syntax approach.</span></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Summary of points to note when
running a two-way ANOVA:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Inform
readers about the nature of your study (tell us what you are about to do)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Ensure that
your dependent variable is a continuous value<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">The
explanatory variables must be categorical variables with at least two groups<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Members in
each group must not over-lap<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">State the
null and alternative hypotheses.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Run the two-way
ANOVA before carrying out any post-hoc checks otherwise Stata will give an
error message.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Report the <i>F</i>-statistic,
degrees of freedom (df), the level of significance (the <i>prob </i>value [Prob>F])</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">8.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">A statement of whether there were statistically
significant differences between your groups and on the interaction term. Report
that of the interaction first if it is significant.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 115%; margin-left: 14.2pt; mso-list: l10 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">9.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Report the results from the post-hoc
checks and their </span><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">prob </span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">values.</span><span style="font-family: "proxima-nova" , "serif"; font-size: 12.0pt; line-height: 115%;"> </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">ASSIGNMENT</span></b></div>
<div class="MsoNoSpacing" style="line-height: 115%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 115%;">Using
<a href="https://drive.google.com/drive/folders/1WRZX474X8pj3Egxo51Tz0Y66me0v2JRd" target="_blank">Wooldridge’s</a> discrim2.dta or discrim2.xlsx show if the price of fries (<i>pfries2</i>) differ among the four
food-chains (Burger King, Kentucky Fried Chicken, Roy Rogers and Wendy’s) across
the two states – New Jersey and Pennsylvania.<o:p></o:p></span></div>
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<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/sTpeY31zcZs/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/sTpeY31zcZs?feature=player_embedded" width="320"></iframe></div>
<br />
<br /></div>
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 106%;">If you have further questions on how to run the
two-way ANOVA procedure and the post-hoc tests, kindly post your comments and
questions below….</span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-51847215587311401082018-01-18T14:54:00.000+01:002018-02-06T17:19:44.633+01:00Stata: Interpreting One-way ANOVA Procedure<div class="MsoNoSpacing" style="line-height: 150%; text-align: left;">
<h2>
<b style="text-align: justify;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">One-way ANOVA Procedure using Stata</span></b></h2>
<b style="text-align: justify;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Preamble</span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Ever wondered what the buzz about ANOVA is all
about? ANOVA simply means <b><u><span style="color: red;">an</span></u></b>alysis <b><u><span style="color: red;">o</span></u></b>f <b><u><span style="color: red;">va</span></u></b>riance. It is a statistical method in which
the <u>variation</u> in a set of observations is divided into distinct
components. It is an extension of the <i>t</i> and <i>z</i> test developed by
Roland Fisher. The ANOVA procedure is of two types – one-way and two-way- with
several dimensions. But for this tutorial, only the one-way ANOVA will be
discussed while the two-way procedure will be covered in subsequent lectures.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Why is
ANOVA useful in data analysis?<o:p></o:p></span></b></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">One importance of carrying out ANOVA is to
determine if the average value (that is, the mean) of a <i>dependent</i> variable (the regressand, outcome variable, and
endogenous variable) is the same in two or more unrelated, independent groups.
Thus, the one-way ANOVA indicates whether the mean of a dependent variable is
the same or differs across independent unrelated groups. The moment you
understand how to compute ANOVA and interpret your table, you will always want
to incorporate it in your study or research…that is, subject to data meeting
some salient conditions.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Practically, ANOVA can be used to measure the
patterns of individuals, environments, disciplines etc. across groups. For
instance, you can use a one-way ANOVA to determine whether weight loss differs
based on diet programs among women (i.e., your dependent variable would be
"weight loss", measured from 65-80kg, and your explanatory variable
would be "weight loss programmes ", which are in three groups:
"keto plan", "plant-based plan, and "vegetarian
plan"). Alternately, a one-way ANOVA could be used to understand whether
there is a difference in insurance schemes based on professions (i.e., your
dependent variable would be "insurance" and your independent variable
would be "profession", which has four categories: "mining",
"teaching", "oil drilling", "lab scientist").<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Thus, when the difference between the groups is
statistically significant, it is possible to determine which specific groups
are significantly different from each other using <i>post estimation</i> tests. These tests are necessary because the
one-way ANOVA only says that at least two groups are different without giving
information as to which specific groups were significantly different from each
other.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Given this preamble, here is a “step-by-step”
tutorial showing you how to carry out ANOVA and <a href="https://cruncheconometrix.blogspot.com.ng/2018/01/a-step-by-step-tutorial-on-research-and_14.html" target="_blank">post-estimation checks</a> using
Stata analytical package. But before I proceed, it is important for you to
understand some basic rules underlying the use of one-way ANOVA procedure. That
is, your data must meet these criteria failing which your results may be
invalidated if they are not adhered to. There are six (6) of them:<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rules:<o:p></o:p></span></b></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">These six "rules" represent the blueprint
guiding the use of the one-way ANOVA technique. If any is not satisfied, you
may obtain invalid results. Please note that the first three assumptions are closely
related to the nature of your data and study structure (that is, directly
related to your choice of variables), thus Stata cannot validate those while
the last three must be met using some Stata criterion. It is therefore
important that you ascertain that your study meets these conditions before
proceeding with the one-way ANOVA.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l1 level1 lfo1; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 150%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rule #1:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> Make sure that the <b>dependent variable
(regressand, outcome variable)</b> is cardinal and measured in <b>continuous
terms</b>. Some example of variables in measured in continuous terms are:
distance (measured in miles, kilometres), weight (measured in stone, pounds,
kilogramme, and grams); wages (measure in local currency) and so on. These are
called <b>continuous variables. </b>In the
event that you have ordinal variables, then consider doing a Kruskal-Wallis H
test.<b><o:p></o:p></b></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l3 level1 lfo2; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 150%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rule #2:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> The <b>explanatory variable (regressor,
independent variable)</b> ought to comprise <b>two or more categorical</b>, <b>independent
(unrelated) groups</b>. Some examples of these <b>categorical variables</b>
are income group (3 groups: high-income, middle income and low income); grade
(4 groups: excellent, very good, good, and poor); demography (2 groups: rural
and urban); banking (3 groups: investment, mortgage, microfinance) etc. So make
sure that your explanatory variable is a categorical variable.<b><o:p></o:p></b></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l5 level1 lfo3; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 150%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rule #3:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> Ensure that you have <b>independence of observations</b>. That is, your
observations must not over-lap across the different groups. This simply means
that there must be no relationship between the observations in each group or
between the groups themselves. For instance, an observation in a “high-income”
group must <b><u>not</u></b> be represented again in a “low-income” group. Needless to say that,
participants across the groups must be different. But where an exception is the
case, the repeated measures of ANOVA should be used rather than the one-way
ANOVA.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l0 level1 lfo4; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 150%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rule #4:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> Be wary of <b><a href="https://cruncheconometrix.blogspot.com.ng/2018/01/heteroscedasticity.html" target="_blank">outliers</a></b>. These are figures that are either
abnormally high or low, that is, they do not follow the typical pattern in a
particular variable. The presence of outliers can bias your results. However,
they can easily be tested in Stata by using the <b>Boxplot</b> or <i>summary</i> syntax
(<i>sum</i> for short). The syntax computes
the mean, standard deviation, minimum and maximum values in each variable in
your data, thus enabling you to detect (identify) the abnormal figure.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l7 level1 lfo5; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 150%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rule #5:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> Since the one-way ANOVA is susceptible to violations
of normality, it is essential that the <b>dependent variable</b> must be <b>approximately
normally distributed for each category of the independent variable</b>.
Although, you may still obtain some valid results if this rule is violated,
that is why your data must be <b>approximately</b> and not <b><i>100% </i></b>normal before
running a one-way ANOVA. A histogram test, <b>Shapiro-Wilk</b>
test or <b>Jarque-Bera</b> test can be
conducted in Stata to test for normality of residuals.<b><o:p></o:p></b></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l6 level1 lfo6; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "symbol"; font-size: 12.0pt; line-height: 150%;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Rule #6:</span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> There must be <b>homogeneity of variances</b>. This can be tested with
the <b>Bartlett’s test</b> for homogeneity
of variances in Stata. The Bartlett’s test is very vital when it comes to
interpreting the results from a one-way ANOVA guide because Stata is capable of
producing different outputs depending on whether your data meets or fails this
assumption.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><br /></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Ascertaining that your data meet the last three
rules may seem daunting, but it is important that you do them. Moreso, the
Stata package has really simplified these procedures.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">So here is an example….<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">PROBLEM:
<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">From <a href="https://drive.google.com/drive/folders/1WRZX474X8pj3Egxo51Tz0Y66me0v2JRd" target="_blank">Wooldridge’s</a> discrim1.dta<span id="goog_10496038"></span><a href="https://www.blogger.com/"></a><span id="goog_10496039"></span> or
discrim1.xlsx files (if you don’t have Stata installed on your devise, download
the .xlsx file and feed into the analytical package of your choice).<o:p></o:p></span></div>
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<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">(Note:
for simplicity, I have extracted from the initial dataset, discrim.dta to use
for this example. The initial dataset is quite detailed such that several
one-way ANOVA simulations can be carried out)</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="color: #0000cc; font-family: "times new roman" , "serif"; font-size: 12.0pt;">A researcher collected ZIP
code-level data on prices on small fries in two US states – New Jersey and
Pennsylvania. The idea is to compare the prices of small fries charged by four
fast-food chains in these states to see whether they are the same.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">In this example, the dependent variable is “<i>price of fries”</i> (measured in US
dollars), whilst the independent variable is “<i>state”</i>, with two independent groups: “<i>New Jersey</i>” and “<i>Penn</i>”.
Note that <i>state</i> is a categorical
variable split across two groups and the one-way ANOVA is used to determine
whether there is a statistically significant difference in prices charged
between the two independent groups.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Setting
up the data in Stata<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Ensure original data is in excel
format (.xlx, .xls or .csv)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Open the <b>Stata</b> application<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Data</b> >> <b>Data Editor
(Edit)</b> <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Highlight data to be copied from
excel<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click the “<b>paste</b>” icon in Stata<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">A dialog box opens: Select “<b>Treat first row as variable names</b>”<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l2 level1 lfo7; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click “<b>OK</b>” and <b>Save</b>.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">These steps (1 – 7) create your Stata dataset (that
is, <i>.dta</i> file)<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Remember that <i>state</i>
is the <b>explanatory variable</b> and a <b>categorical variable</b> that is made up of
two components – New Jersey, and Penn. Therefore, you must create <b>Value Label</b> for the variable <i>state </i>in Stata.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">How to do that? Here are the steps:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l8 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Stata</b> >> <b>Data</b>
>> <b>Data</b> <b>Utilities</b> >> <b>Label</b>
<b>Utilities</b> >> <b>Manage</b> <b>Value</b> <b>Labels</b> >> <b>Create</b> <b>Label</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l8 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Enter “<b>new label name</b>”: <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l8 level1 lfo8; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Enter the appropriate values. For
instance, enter <b>1</b> for <b>Value</b>, and <b>New Jersey </b>for <b>Label</b>,
click <b>ADD</b>. Next, enter <b>2</b> for <b>Value</b>, and <b>Penn</b> for <b>Label</b> click <b>ADD</b>. Then click <b>OK</b>.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If you did it correctly, then you should have something
like this as shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje6o1TLxnQ9hkgoJe4bHIZhD4u8aFjI_ky3EyVXU72ckfizsC8n40-AO-CfbPADBFBOhHODGqf-OjGueqJax81lUACGnim3R-Q_qxtIsqVJjvMdlD194Wd7k1UeuRzW7CGBpD2LhWmIrQ/s1600/One-way+ANOVA+-+Manage+Label+%2528State%2529.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Creating value labels for one-way ANOVA in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="284" data-original-width="460" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEje6o1TLxnQ9hkgoJe4bHIZhD4u8aFjI_ky3EyVXU72ckfizsC8n40-AO-CfbPADBFBOhHODGqf-OjGueqJax81lUACGnim3R-Q_qxtIsqVJjvMdlD194Wd7k1UeuRzW7CGBpD2LhWmIrQ/s1600/One-way+ANOVA+-+Manage+Label+%2528State%2529.jpg" title="Creating value labels for one-way ANOVA in Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Creating value labels for one-way ANOVA in Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from
StataCorp LP)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Next is to </span><b style="font-family: "Times New Roman", serif; font-size: 12pt;">assign
value label</b><span style="font-family: "times new roman" , serif; font-size: 12pt;"> to the categorical/explanatory variable </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">state</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">. To do that:</span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l10 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Stata</b> >> <b>Data</b>
>> <b>Data</b> <b>Utilities</b> >> <b>Label</b>
<b>Utilities</b> >> <b>Assign Value</b> <b>Label to Variable</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l10 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Under “<b>Variables</b>” select <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l10 level1 lfo9; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click <b>OK</b>.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">If it’s correctly done, you should have something
like this:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKjJrFbax32Bdf_JlL-d6qOMVsVGQDyxK-EIBVYGJVdqoMw2Vi8gT7CAVq2lZzv-o_WAgh8WLauzRHoQmv9cW65oIasB_Qda01lfJHqdh7lcfQq-nwch6Z3g5TRwo20z6SfuyMFBxb-4o/s1600/Assign+Value+Lbels+-+State.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Assigning value labels for one-way ANOVA in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="294" data-original-width="420" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKjJrFbax32Bdf_JlL-d6qOMVsVGQDyxK-EIBVYGJVdqoMw2Vi8gT7CAVq2lZzv-o_WAgh8WLauzRHoQmv9cW65oIasB_Qda01lfJHqdh7lcfQq-nwch6Z3g5TRwo20z6SfuyMFBxb-4o/s1600/Assign+Value+Lbels+-+State.png" title="Assigning value labels for one-way ANOVA in Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Assigning value labels for one-way ANOVA in Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="text-align: center;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"></span></div>
<div style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">With all the steps correctly done, your dataset
should look like mine shown below:</span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWKbEYX-zdwGRnjIcdnj_Jc_-jRZTF0ph7ypyCL6CDAx2ScZizkh9CyTAPYtFgANlplOUNo779gY1rS0Ut7WMZ1dNWKKYf-QJu8ZwZUQtLr6CXjiCYwZQ870afOsQBntY6TdgMbBntJ1s/s1600/Data+in+Stata+Format.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Data Editor for one-way ANOVA in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="694" data-original-width="598" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWKbEYX-zdwGRnjIcdnj_Jc_-jRZTF0ph7ypyCL6CDAx2ScZizkh9CyTAPYtFgANlplOUNo779gY1rS0Ut7WMZ1dNWKKYf-QJu8ZwZUQtLr6CXjiCYwZQ870afOsQBntY6TdgMbBntJ1s/s1600/Data+in+Stata+Format.png" title="Data Editor for one-way ANOVA in Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Data Editor for one-way ANOVA in Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<div style="text-align: left;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;">There are 410 observations, and to know the
distribution across the two groups, use the </span><b style="font-family: "Times New Roman", serif; font-size: 12pt; text-align: justify;"><i>tabulate</i></b><span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"> syntax. That is,</span></div>
<br />
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">tab</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> state<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">and you have this output shown below:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQYlx_DAdeZfGH3Z7rKxe_ZcGPkOcguqQUwtX2lHKt2RUH9YdKuA7UOPepZWamWwPGqRAyOjqU0vvHrxdC7BVGArakPj3xSeixWCr7fQc_T1hdaiczg9zBCE8u_XXlZvA4b_mpYiYJc_A/s1600/Tab+State+-+Output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Tabulate Command used for one-way ANOVA in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="135" data-original-width="373" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQYlx_DAdeZfGH3Z7rKxe_ZcGPkOcguqQUwtX2lHKt2RUH9YdKuA7UOPepZWamWwPGqRAyOjqU0vvHrxdC7BVGArakPj3xSeixWCr7fQc_T1hdaiczg9zBCE8u_XXlZvA4b_mpYiYJc_A/s1600/Tab+State+-+Output.png" title="Table showing distribution of observations for one-way ANOVA in Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Table showing distribution of observations for one-way ANOVA in Stata<span style="font-family: "times new roman" , serif;"><br />Source: </span>CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The above table shows how the 410 observations are
distributed across the two US states.</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Please note that in Stata, you can either use the <b>code</b> (<b>command, syntax</b>) approach or the <b>graphical</b> <b>user</b> <b>interface</b> (<b>GUI</b>). Either approach is fine. If you are familiar with the coding
approach, just go ahead and use it, if otherwise use the GUI (where you just
click the applicable menus).<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">ATTENTION:</span></b><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> Before now, make sure you
create a log file and a do-file.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Log
file:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">The log file gives a history of what you have done.
You can always revisit the log file <i>(saved
as .smcl)</i> to review the processes. So, it is advantageous to always have a
log file. To open a log file:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l13 level1 lfo10; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Stata</b> >> <b>File</b>
>> <b>Log</b> >> <b>Begin</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l13 level1 lfo10; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Give it a <i>filename</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l13 level1 lfo10; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click <b>Save</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Do-file:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">The do-file on the other-hand shows the commands
(codes) used to execute each process. Those familiar with the coding approach
will agree with me that having a do-file can speed up the time used in
executing the work. To create a do-file <i>(saved
as .do)</i>:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Go to <b>Stata</b> >> <b>New Do-File
Editor</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">New do-file opens<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click File >> <b>Save As</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Give it a <i>filename</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l4 level1 lfo11; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">5.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Click <b>Save</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Having prepared our dataset, now let us run the
one-way ANOVA. This tutorial will in the <b>first
part</b> cover the one-way ANOVA analysis and in the <b>second part</b> the post-estimation checks. I will be using the syntax
approach, but will show you how to manoeuvre the GUI interface…..are you ready?
On the assumption that our dataset is in line with the six rules….we begin!<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">State
the null and alternative hypotheses for the test<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">H<sub>0</sub>: the mean prices for prices in both states are equal</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">H<sub>1</sub>: the null hypothesis is not true</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Let’s begin….…</span><span style="font-family: "wingdings"; font-size: 12.0pt; line-height: 150%;">J</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">All codes are typed into the <b>Command</b> window, as shown below, and you simply press the <b>ENTER</b> key:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSdHKqcUBR8DcWOGGFxQiQ13bnJFHldy8J4zl9gflZWFRfLmX3eQBTEye2E6aAEcFwEJ_U2VtwOfPiQfMiDieqLhrqs1wUwtZ42Qz_bWxzOMJh0966ikpbMuSZQVsHh2lm1bau1wNpC2s/s1600/Command+Box.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Command Box in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="99" data-original-width="1001" height="62" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSdHKqcUBR8DcWOGGFxQiQ13bnJFHldy8J4zl9gflZWFRfLmX3eQBTEye2E6aAEcFwEJ_U2VtwOfPiQfMiDieqLhrqs1wUwtZ42Qz_bWxzOMJh0966ikpbMuSZQVsHh2lm1bau1wNpC2s/s640/Command+Box.png" title="Command Box in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: left;">The "Command" box in Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><br /></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">One-way
ANOVA</span></b></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">The basic syntax (code) of the <i>oneway</i>
command is:<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"> oneway <i>y</i> <i>x</i><o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">where the <b><i>y</i></b> is the dependent variable (<i>pfries</i>) and <b><i>x</i></b> is a
categorical/explanatory variable, in this case, <i>state</i>. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">oneway <i>pfries</i> <i>state</i><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">The Stata output is shown as:<o:p></o:p></span></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg813lmk_bGJuN_DZ3TQlOAcyHlOi7vvRg0SEwr7DKIASdJvTon_9tkODq2-mNY9bPXOULClj1Z3dZai3Uznijy_XUBYadlMn3vrxUsk276ePuZiz6VweErpn0HChB6i49gqP0SAmTqHdY/s1600/One-way+Output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata output for one-way ANOVA from http://cruncheconometrix.com.ng" border="0" data-original-height="185" data-original-width="564" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg813lmk_bGJuN_DZ3TQlOAcyHlOi7vvRg0SEwr7DKIASdJvTon_9tkODq2-mNY9bPXOULClj1Z3dZai3Uznijy_XUBYadlMn3vrxUsk276ePuZiz6VweErpn0HChB6i49gqP0SAmTqHdY/s1600/One-way+Output.png" title="Stata output for one-way ANOVA" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata output for one-way ANOVA<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">If you recall, one of the assumptions of ANOVA is that the
variances are the same across groups. The insignificant value for the Bartlett’s
statistic (0.130) confirms that this rule (#6) is not violated in this data, so
the use of ANOVA is ok.</span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">Some useful optional parameters can be included. To obtain
descriptive statistics, add the tabulate option, abbreviated <i>tab</i>. That is:<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">oneway <i>pfries</i> <i>state, tab<o:p></o:p></i></span></b></div>
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<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">The Stata output gives both the summary statistics </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">(i.e., the mean, standard deviation and Frequency) </span><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">and the Bartlett statistic, shown below:<o:p></o:p></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqUWkWWC1C6q-58OG-UBVkaehBfxEv9jUPShktGAQfVz_wa7uoeKKM2UWP76ctggtJtqb4yCkGMTRuoE1qB1XLbCzT0jXpKPv8Ab3LW26ElMzjmtwCVH0l0tjQzHfmERVIAW9ac3TrzO8/s1600/One-way+Output+%252B+Tab+Output.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata output plus summary statistics for one-way ANOVA from http://cruncheconometrix.com.ng" border="0" data-original-height="327" data-original-width="560" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqUWkWWC1C6q-58OG-UBVkaehBfxEv9jUPShktGAQfVz_wa7uoeKKM2UWP76ctggtJtqb4yCkGMTRuoE1qB1XLbCzT0jXpKPv8Ab3LW26ElMzjmtwCVH0l0tjQzHfmERVIAW9ac3TrzO8/s1600/One-way+Output+%252B+Tab+Output.png" title="Stata output plus summary statistics for one-way ANOVA" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata output plus summary statistics for one-way ANOVA<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<span style="font-family: "times new roman" , serif; font-size: 11.5pt;">The </span><b style="font-family: "times new roman", serif; font-size: 11.5pt;">Frequency</b><span style="font-family: "times new roman" , serif; font-size: 11.5pt;"> from the summary statistics table only
counts where </span><i style="font-family: "times new roman", serif; font-size: 11.5pt;">pfries</i><span style="font-family: "times new roman" , serif; font-size: 11.5pt;"> has a value. So
in this case, </span><i style="font-family: "times new roman", serif; font-size: 11.5pt;">pfries</i><span style="font-family: "times new roman" , serif; font-size: 11.5pt;"> has 393
observations with values, the remaining 17 are missing. If you add up 393 + 17,
gives you the total number of observations in the dataset which is 410.</span></div>
<div class="MsoNoSpacing">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">Post-hoc tests<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">The significant <i>F</i>
statistic (63.43) tells us that prices differ between these two states i.e. the
means are not equal. Because the explanatory variable has just two groups, carrying out any post-hoc analysis will be totally unnecessary because we already know from the <i>F</i>-ratio that the mean prices differ between the two groups. However, whenever the categorical variable has more than two groups it is necessary to carry out further pair-wise tests using
the Bonferroni, Scheffe, or Sidak multiple comparison tests to ascertain where the differences occur. Furthermore, these tests apply
corrections to the reported significance levels that take into account the fact
that multiple comparisons are being conducted and the Stata </span><span style="font-family: "times new roman" , serif; font-size: 12pt;">syntax is :</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;"><br /></span></b>
<b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">oneway <i>y x, tab bon sch sid<o:p></o:p></i></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><br /></span>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Also, note by using these tests, the likelihood of committing a Type I
error is reduced (that is, reducing the likelihood of rejecting the null hypothesis
when it is true) and ironically increases the chances of committing a Type II
error (that is, failing to reject the null hypothesis when it is false).</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br />
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Thus, in this example, no post-hoc analysis will be conducted.</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing">
<b><span style="font-family: "times new roman" , serif; font-size: 12pt;">Addendum:</span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%;">
<span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">By way of
information, here is how to manoeuvre the graphical user interface (GUI) to run
the one-way ANOVA.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Go to <b>Stata</b> >> <b>Statistics</b> >> <b>Linear
models and related</b> >> <b>ANOVA/MANOVA</b>
>> <b>One-way ANOVA</b> from the top
menu, as shown below.<o:p></o:p></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhk6kMNmS5BhRfLQ-ePXZXQxAlb7FB67RlJ9XKzHecjO3v0oWiAkru81mv2ufiCgXM9xkC9DWF_5F7sRkySw7O-Muo7nFkYicJ7OEpUNDCpfts9503GjT1r6gRWc3pkevwAt6fLmYJW9M/s1600/GUI+-+ANOVA.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata graphical user interface (GUI) for one-way ANOVA" border="0" data-original-height="347" data-original-width="918" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhk6kMNmS5BhRfLQ-ePXZXQxAlb7FB67RlJ9XKzHecjO3v0oWiAkru81mv2ufiCgXM9xkC9DWF_5F7sRkySw7O-Muo7nFkYicJ7OEpUNDCpfts9503GjT1r6gRWc3pkevwAt6fLmYJW9M/s640/GUI+-+ANOVA.png" title="Stata graphical user interface (GUI) for one-way ANOVA" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata graphical user interface (GUI) for one-way ANOVA<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<span style="font-family: , serif; font-size: 12pt;">A dialogue box for </span><b style="font-family: proxima-nova, serif; font-size: 12pt;">One-way analysis of variance</b><span style="font-family: , serif; font-size: 12pt;">
opens:</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Select <i>pfries</i> as the <b>Response variable</b> and <i>state </i>as
the <b>Factor variable </b>from the drop-down
menu.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Tick the <b>Produce summary table </b>in the <b>Output section</b><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l12 level1 lfo12; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Click <b>OK</b>.<o:p></o:p></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifBry4RbIJWq6vK4BjV0ou60rw1-tWQhRLZRUr7Ju4y9jQduiL0qol5DApq-usot0onF_FDnEWOaMJ6XYYZg_mXjKeiaGKey6dhmfeYWKV8T3up-0JIe0PHYhv3OrF4QBeUYBKlvI24Os/s1600/GUI+-+Produce+summary+table.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Stata graphical user interface (GUI) for one-way ANOVA" border="0" data-original-height="417" data-original-width="531" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifBry4RbIJWq6vK4BjV0ou60rw1-tWQhRLZRUr7Ju4y9jQduiL0qol5DApq-usot0onF_FDnEWOaMJ6XYYZg_mXjKeiaGKey6dhmfeYWKV8T3up-0JIe0PHYhv3OrF4QBeUYBKlvI24Os/s1600/GUI+-+Produce+summary+table.png" title="Stata graphical user interface (GUI) for one-way ANOVA" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Stata graphical user interface (GUI) for one-way ANOVA<br />
Source: CrunchEconometrix<br />
(Used with written permission from Stata)</td></tr>
</tbody></table>
</div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">You will obtain the same output as in using the
syntax (</span><b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">oneway <i>pfries</i> <i>state, tab</i></span></b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">) approach<a href="https://www.blogger.com/null" name="gui-post-hoc"></a>, and to obtain
the </span><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">Bonferroni, Scheffe, and Sidak statistics, simply tick the
appropriate boxes as shown in the dialog box.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">Summary of points to
note when running a one-way ANOVA:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Inform
readers about the nature of your study (tell us what you are about to do)<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Ensure that
your dependent variable is a continuous value<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">The
explanatory variable must be a categorical variable with at least two groups<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">4.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Members in
each group must not over-lap</span><br />
<span style="font-family: "times new roman" , "serif"; font-size: 12pt; line-height: 150%; text-indent: -14.2pt;">5. Check for
outliers (use </span><span style="font-family: "proxima-nova" , "serif"; font-size: 12pt; line-height: 150%; text-indent: -14.2pt;">the boxplots if there are any
significant outliers or use the summary statistics to check for the minimum and
maximum values). </span><span style="font-family: "times new roman" , "serif"; font-size: 12pt; line-height: 150%; text-indent: -14.2pt;">Here’s the
Boxplot for the example used in this tutorial:</span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnRabNiHcu26QVCnAEfmuZ3O54qgdODvD0VT65DYRHS8ND_BSmza3JvfFBxItaO9uUa22PyIBzoYXjgOndXjeu96K4T44to-HstN5W17TtzyDceJVhp6iGdqeXUwkiLUEy8JEL_80snLI/s1600/One-way+ANOVA+Boxplot.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Boxplots for one-way ANOVA using Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="345" data-original-width="474" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnRabNiHcu26QVCnAEfmuZ3O54qgdODvD0VT65DYRHS8ND_BSmza3JvfFBxItaO9uUa22PyIBzoYXjgOndXjeu96K4T44to-HstN5W17TtzyDceJVhp6iGdqeXUwkiLUEy8JEL_80snLI/s1600/One-way+ANOVA+Boxplot.png" title="Boxplots for one-way ANOVA using Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Boxplots for one-way ANOVA using Stata<br />
Source: CrunchEconometrics<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The Boxplot is in percentiles and the lines in between the boxes are not means but
medians.</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">6.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Check that
the data is <b><i>approximately</i></b> normally distributed. Below is the histogram
obtained using the syntax: <b><i>hist pfries, by(state):</i></b><o:p></o:p></span><br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_EIyBu_LTvwcyIR_L9cdvHHRcz_vndSrm5-txWsFgLHGVF4KPofqZXXH_qbJKb3ZTFawfDC947pK0WMwjVl56ERGAyr7OYue04TgZn7_qssnz4nrEzgPyo39spFapB6FMrJb8bZcKUgc/s1600/One-way+ANOVA+Histogram.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Histogram plots for one-way ANOVA using Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="345" data-original-width="474" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_EIyBu_LTvwcyIR_L9cdvHHRcz_vndSrm5-txWsFgLHGVF4KPofqZXXH_qbJKb3ZTFawfDC947pK0WMwjVl56ERGAyr7OYue04TgZn7_qssnz4nrEzgPyo39spFapB6FMrJb8bZcKUgc/s1600/One-way+ANOVA+Histogram.png" title="Histogram plots for one-way ANOVA using Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Histogram plots for one-way ANOVA using Stata<br />
Source: CrunchEconometrix<br />
(Used with written permission from StataCorp LP)</td></tr>
</tbody></table>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">The data looks approximately normally distributed,
thus fulfilling another ANOVA assumption.</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">7.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Check that
the variances are homogenous across groups (confirm from </span><span style="font-family: "proxima-nova" , "serif"; font-size: 12.0pt; line-height: 150%;">the output Stata for the Bartlett’s statistic)</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">8.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">In case, your
data fails violates any of these rules, the output obtained from the one-way
ANOVA procedure (i.e., the output we discuss above) will no longer be valid.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l9 level1 lfo13; tab-stops: list 14.2pt 36.0pt; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">9.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;">
</span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">State the
null and alternative hypotheses.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 7.1pt; mso-list: l9 level1 lfo13; tab-stops: list 7.1pt left 14.2pt 21.3pt; text-align: justify; text-indent: -7.1pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">10.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Run the
one-way ANOVA before carrying out any post-estimation checks otherwise Stata will
give an error message.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">What statistics to
report in a one-way ANOVA:<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l11 level1 lfo14; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">1.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">The <i>F</i>-statistic, degrees of freedom (df),
the level of significance (the <i>prob </i>value
[Prob>F])</span><b><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l11 level1 lfo14; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">2.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">A
statement of whether there were statistically significant differences between
your groups</span><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; margin-left: 14.2pt; mso-list: l11 level1 lfo14; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span style="font-family: "proxima-nova" , "serif"; font-size: 12.0pt; line-height: 150%;">3.<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span style="font-family: "times new roman" , serif; font-size: 12pt; line-height: 150%;">The results from the post-estimation checks and their </span><i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">prob </span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">values.</span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<b><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">ASSIGNMENT<o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;">Using
<a href="https://drive.google.com/drive/folders/1WRZX474X8pj3Egxo51Tz0Y66me0v2JRd" target="_blank">Wooldridge’s</a> discrim1.dta or discrim1.xlsx show if the price of fries (<i>pfries2</i>) differ across the two states –
New Jersey and Pennsylvania.<o:p></o:p></span><br />
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/JBT382xGD5U/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/JBT382xGD5U?feature=player_embedded" width="320"></iframe></div>
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 150%;"><br /></span></div>
<div class="MsoNoSpacing" style="line-height: 150%; text-align: justify;">
<br /></div>
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 106%;">If you have further questions on how to run the
one-way ANOVA, post your comments below….</span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com2tag:blogger.com,1999:blog-1876221430378807805.post-76355962122488905312018-01-14T18:40:00.000+01:002018-02-06T17:21:14.207+01:00A Step-by-Step Tutorial on Research and Data Analysis<h2>
<b style="text-align: justify;"><span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note: This tutorial is somewhat detailed!</span></b></h2>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12pt;">Data is essential to all disciplines, professions,
and fields of endeavours whether in social sciences, arts, technology, life
sciences or medicine. The truth is, who are we without data? Data either
qualitative or quantitative is informative. It tells us about past and current
occurrences. With data, predictions and forecasting can be made either to
forestall a negative recurring trend or improve future events. Whichever way,
knowing some rules guiding the use of data and how to make it <a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">communicate</a> </span><span style="font-family: "times new roman" , "serif"; font-size: 12pt;">is very
important since it often comes out as large voluminous tons of figures or
statements. In the same vein, undertaking a research is impossible without
data. I mean, what will be the essence of your research if you have no data. In
other words, research and data are like <i style="mso-bidi-font-style: normal;">siamese</i>
twins.</span></div>
<br />
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Everyone has different views about how research
should be undertaken and how data should be analysed. Afterall, isn’t that why
we have different <i style="mso-bidi-font-style: normal;">schools of thought</i>?
I guess, that’s why. So, what I am about to teach are just simple steps common
to all disciplines that are required to undertake any form of research and analyse
that data accordingly. Therefore, whether you are a student or a practitioner
you will find this guide very helpful. Although, I may be a bit biased towards
economics this approach is not fool proof, and regardless of what you know
already (and whatever your field is), you will learn a thing or two from this
tutorial. </span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, let us dig in…..</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">State the underlying theory.</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">You must have a theory underlying your
study or research. Theories are hypotheses, statements, conjectures, ideas, and
assumptions that someone somewhere came up with at some point in time. Such as
Darwin’s theory of evolution, Malthusian theory of food and population growth,
Keynes’ theory of consumption, McKinnon-Shaw hypothesis on financial reforms
etc. Every discipline has its fair share of theories. So make sure you have a
theory upon which your research hinges on. It is this theory you are out to
test with the available data which culminates into you undertaking a research. Right
now, I have a <i style="mso-bidi-font-style: normal;">funny</i> theory of my own
that countries that have strong and efficient institutions have lower income
inequality (…oh well, I just came up with that!). Or yours could be that richer
countries have happier citizens. Therefore, anyone can have a theory. Have a
theory <b style="mso-bidi-font-weight: normal;"><u>before</u></b> you begin that
research!</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Specify the theoretical (mathematical) model</span></b></div>
<div style="text-align: justify;">
<a href="https://www.blogger.com/" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Having established the theory within
which you are situating your research, the next thing to do is to state the
theoretical model. Remember, since theories are statements (which are somewhat
unobservable), you have to construct them in a functional mathematical form
that embodies the theory. The model to be specified is a set of mathematical
equations. For instance, given my postulated negative relationship between
effective institutions and income inequality, a <a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">mathematical economist</a></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> might specify
it as:</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<i style="mso-bidi-font-style: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> INQ</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = b<sub>1</sub> + b<sub>2</sub><i style="mso-bidi-font-style: normal;">INST</i>……………………..[1]</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, equation [1] becomes the <b style="mso-bidi-font-weight: normal;"><a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">mathematical</a></b><a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank"> </a><b style="mso-bidi-font-weight: normal;"><a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">model</a></b> of the relationship between institutions and income
inequality.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Where <i style="mso-bidi-font-style: normal;">INQ</i> = income inequality, <i style="mso-bidi-font-style: normal;">INST</i>
= institutions, <i style="mso-bidi-font-style: normal;">b<sub>1</sub></i> and <i style="mso-bidi-font-style: normal;">b<sub>2</sub></i><sub> </sub>are known as <b style="mso-bidi-font-weight: normal;">parameters</b> of the model and they are
the <b style="mso-bidi-font-weight: normal;">intercept</b> and <b style="mso-bidi-font-weight: normal;">slope</b> coefficients. According to the
theory, <i style="mso-bidi-font-style: normal;">b<sub>2</sub></i> is expected to have
a <b style="mso-bidi-font-weight: normal;">negative</b> sign. </span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The variable appearing on the left side
of the equality sign is the <b style="mso-bidi-font-weight: normal;">dependent</b>
<b style="mso-bidi-font-weight: normal;">variable</b> or <b style="mso-bidi-font-weight: normal;">regressand</b> while the one on the right side is called the <b style="mso-bidi-font-weight: normal;">independent</b> or <b style="mso-bidi-font-weight: normal;">explanatory</b> <b style="mso-bidi-font-weight: normal;">variable</b> or <b style="mso-bidi-font-weight: normal;">regressor</b>. Again, if the model has one
equation as it is in equation [1], it is known as a <b style="mso-bidi-font-weight: normal;">single-equation</b> model and if it has more than one equation, it is
called <b style="mso-bidi-font-weight: normal;">multiple-equation</b> model.
Therefore, the inequality-institutions model stated above is a single equation
model.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Specify the empirical model</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The word “empirical” connotes knowledge
derived from experimentation, investigation or verification. Therefore, the
mathematical model stated in equation [1] is of limited interest to the <a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">econometrician</a></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">. The
e<a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">conometrician</a> must modify equation [1] to make it suitable for analysis of
some sort. This is because, that model assumes that an exact relationship
exists between effective institutions and income inequality. However, this
relationship is generally inexact. This is because, if we are to obtain
institutional data on 10 countries known to have good rankings on governance,
rule of law or corruption, we would not expect all their citizens to lie
exactly on the straight line. The reason is because aside quality or effective
institutions, other variables affect income inequality. Variables such as
income level, education, access to loans, economic opportunities etc. are
likely to exert some influence on income inequality. Therefore, to capture the <i style="mso-bidi-font-style: normal;">inexact</i> relationship(s) between and
among economic variables, the <a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">econometrician</a> will modify equation [1] as:</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<i style="mso-bidi-font-style: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> INQ</span></i><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> = b<sub>1</sub> + b<sub>2</sub><i style="mso-bidi-font-style: normal;">INST + u</i> ……………………..[2]</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Thus, equation [2] becomes the <b style="mso-bidi-font-weight: normal;"><a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">econometric</a></b><a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank"> </a><b style="mso-bidi-font-weight: normal;"><a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">model</a></b> of the relationship between institutions and income
inequality. It is with this model that the econometrician verifies the
inequality-institutions hypothesis using data.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Where <i style="mso-bidi-font-style: normal;">u</i> is the <b style="mso-bidi-font-weight: normal;">disturbance term</b>
or often called the <b style="mso-bidi-font-weight: normal;">error term</b>. The
error term is a random variable that may well capture other factors that affect
income inequality but not taken into account by the model explicitly.
Technically, equation [2] is an example of a <b style="mso-bidi-font-weight: normal;">linear regression model</b>. The major difference between equations [1]
and [2] is that the econometric inequality function hypothesises that the
dependent variable <i style="mso-bidi-font-style: normal;">INQ</i> is linearly
related to the explanatory variable <i style="mso-bidi-font-style: normal;">INST</i>
but this relationship is <b style="mso-bidi-font-weight: normal;"><u>not</u></b>
exact due to individual variation represented by <i style="mso-bidi-font-style: normal;">u</i>.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">4.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Data</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Now that you have the theory and have
been able to successfully construct your model, the question is, do you have
data? To estimate the <a href="http://cruncheconometrix.blogspot.com/2018/01/tell-me-what-is-econometrics.html" target="_blank">econometric model</a> stated in equation [2], data is
essential to obtain the numerical estimates of <i style="mso-bidi-font-style: normal;">b<sub>1</sub></i> and <i style="mso-bidi-font-style: normal;">b<sub>2</sub></i>.
Your choice of data depends on the structure or nature of your research which
may determine if you will require the use of qualitative or quantitative data.
In line with that, is whether you require the use of primary or secondary data?
As a researcher, you can mix both qualitative and quantitative data to gain the
breadth and depth of understanding and corroborating what others have done.
This is known as <i style="mso-bidi-font-style: normal;">meta-data</i> analysis.
There is a growing body of researchers using this approach. At this point, you
already know whether the data is available for your research or not. </span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">When sourcing your data, identify the
dependent variable and the explanatory variables. Let me say a word or two on
the explanatory variables. They can further be broken into <b style="mso-bidi-font-weight: normal;">control variables</b>. The control variables are not directly in your
scope of research but they are often included to test if the expected <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">a
priori</i></b> on the key explanatory variable still holds with the inclusion of
control variables in the regression model. For instance, using the
inequality-institutions model, the dependent variable is <i style="mso-bidi-font-style: normal;">INQ</i>, the key explanatory variable is <i style="mso-bidi-font-style: normal;">INST</i> and I may decide to control for education, per capita income
and access to loans….the last three variables are known as the control
variables. Also, in applied research, data is often plagued by approximation
errors or incomplete coverage or omitted variables. For instance, social
sciences often depend on secondary data and usually have no way of identifying
the errors made by the agency that collected the primary data. That being said,
<span style="background: yellow; mso-highlight: yellow;">do not engage in any
research without first knowing that data is available.</span> </span></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">...So, start sourcing
and putting your data together, we are about to delve into some pretty serious
stuff! </span><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">J</span></span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">5.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Methodology</span></b></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">The next thing is knowing what
methodology to apply. This is peculiar to your research and your discipline. There
are so many methodologies, identify the one which best fits your model and use
it.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">6.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Analytical software</span></b></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Students often ask me this question: “what
analytical software should I use?” My answer has and will always be: “use the
software that you are familiar with”. Don’t be lazy! Be proficient in the use
of at least one analytical software. There are hundreds of them out there –
Stata, R, EViews, SPSS, SAS, Python, SQL, Excel, Agile, and so on. Learn how to
use any of them. There are so many tutorial videos on YouTube. For instance, I
am very proficient in the use of Stata and Excel analytical softwares with
above 60% proficiency in the usage of EViews, SAS and SQL packages. As a
researcher and data analyst, you cannot be taken seriously if you cannot lay
claim to some level of proficiency in the usage of any of these packages. I use
Stata, I love Stata and I will be giving out some periodical hands-on tutorials
on how to use Stata to analyse your data. By way of information, I currently
analyse data using Stata13.1 package.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, let us dig in further….it is getting
pretty interesting and more involving </span><b style="mso-bidi-font-weight: normal;"><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">J</span></span></b></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">7.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Estimation technique</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">This is the method of obtaining the
estimates for your model, at least an approximation. It is that method based on
finding that parameter estimate that best <i style="mso-bidi-font-style: normal;">minimises</i>
discrepancies between the observed sample(s) and the fitted or predicted model.
At this point, you already know what technique to apply that will best give <i style="mso-bidi-font-style: normal;">unbiased</i> estimates.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">8.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Pre-estimation checks</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">At this point, you are almost set to begin
analysing your data. However, before you proceed, your data must be subjected
to some pre-estimation checks. I am very sure that every discipline has these
pre-estimation checks in place before carrying out any analysis. In economics
there are several of them, such as: <a href="http://cruncheconometrix.blogspot.com/2018/01/multicollinearity.html" target="_blank">multicollinearity</a></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> test, summary
statistics (mean, standard deviation, minimum, maximum, kurtosis, skewness,
normality etc.), stationarity test, Hausman test etc. It is from these tests
that you identify and correct any abnormality in your data. You may observe the
presence of an outlier (when a figure stands out conspicuously either because
it is abnormally low or high). You will also get some information regarding the
deviation of a variable from the mean (average value), the shape of the probability
distribution is also important – is it mesokurtic, platykurtic or leptokurtic? You
may want to know whether your data is heavy- or light-tailed. Also, if you are
using a time-series data, the stationarity of each variable should be of
paramount interest and if it is a panel data (combination of time- and
cross-sectional data) the Hausman test should come handy in knowing what
estimator (whether fixed or random) to adopt. The bottom-line is that: <b style="mso-bidi-font-weight: normal;"><u>always</u></b> carry out some
pre-estimation checks before you begin your analysis!</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">9.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Functional form of the model</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Linear relationships are not often
common for all economic research, while it is general to come across several
studies incorporating many nonlinearities into their regression analysis by
simply changing the <a href="http://cruncheconometrix.blogspot.com/2018/01/heteroscedasticity.html" target="_blank">functional forms</a> of either or both the dependent
(regressand) and independent variables (regressors). More often than not,
econometricians transform variables from their level forms to functional forms </span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">using natural logarithms (denoted as <i style="mso-bidi-font-style: normal;">ln</i>).
Since variables come in different measurements, it is crucial to know how they
are measured in order to make sense of their regression estimates in an
equation. For example, using the inequality-institutions model, the inequality
variable (using the Gini index) ranges between 0 and 100 and the institution
variable is also a decimal ranging between -2.5 and +2.5, obviously these two
variables have different measurements. Therefore, an important advantage of
transforming variables into natural logarithms (<i style="mso-bidi-font-style: normal;">logs</i>, for short) is to equate the variables on the same measurement
and applying a <i style="mso-bidi-font-style: normal;">constant elasticity
relationship</i> and interpretations. It also controls for the presence of
outliers in the data amongst others. Let me state here that when the units of
measurement of the dependent and independent variables change, the ordinary
least squares (OLS) estimates change in entirely expected ways.</span></div>
<div style="text-align: justify;">
<span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">Note: changing the units of measurement
of only the regressor does not affect the intercept.</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> </span></div>
<div style="text-align: justify;">
<br /></div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3qlOSigQalmmwqGmZ3j4v6SZLrtMKT11DArruD00thHV1GajzIQWnsBv73Uh5h08C5ECaVC45nbSxV5tc2J17boBAR2xLhWprppep5SRJADtDa65_JsEtFQ6t67OKo_YZatsTeYqMZ1s/s1600/Functional+Form.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="656" data-original-width="1600" height="262" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3qlOSigQalmmwqGmZ3j4v6SZLrtMKT11DArruD00thHV1GajzIQWnsBv73Uh5h08C5ECaVC45nbSxV5tc2J17boBAR2xLhWprppep5SRJADtDa65_JsEtFQ6t67OKo_YZatsTeYqMZ1s/s640/Functional+Form.png" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Table showing different functional forms of a model</td></tr>
</tbody></table>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">So, given the inequality-institutions
model, I may decide to re-specify equation [2] in a log-linear form to obtain
an elasticity relationship. That is:</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"> ln<i style="mso-bidi-font-style: normal;">INQ</i> = b<sub>1</sub> + b<sub><span style="font-size: x-small;">2</span></sub>ln<i style="mso-bidi-font-style: normal;">INST + u</i> ……………………..[3]</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">10.<span style="font: 7.0pt "Times New Roman";">
</span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Estimate the model</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Prior to the
existence of analytical softwares, econometricians go through the cumbersome
approach of manual computation of regression coefficients. Well, I am glad to
tell you that those days are gone forever! With the advent of computerised
analytical packages like Stata, EViews, R and the rest of them all you have to
do is feed in your data into any that you are familiar with and click “<i style="mso-bidi-font-style: normal;">RUN</i>”…and voila! You have your results in
split micro-seconds! Most if not all of these packages are excel-friendly. That
is, you first have to put your data into an excel format (either .csv, .xls or.
xlx file) and then feed into any of them. This is the easiest part of the
entire data analysis process. Every researcher loves it whenever they are at
this stage. All you need do is feed in your data, click “<i style="mso-bidi-font-style: normal;">RUN</i>” and your result is churned out! However, that your
coefficients will be according to your expectations is an entirely different
story (won’t be told in this write up…hahahaha </span><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Wingdings;">J</span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">). </span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<a href="https://www.blogger.com/" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Below is an
example of a result output from Stata analytical software (see <a href="http://cruncheconometrix.blogspot.com/2018/01/heteroscedasticity.html" target="_blank">heteroscedasticity</a></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">).</span></div>
<div style="text-align: justify;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjadBHZEH7keQHsmGYwoDCOTy39oqqQeuySC-VSo7M9fkW6F_AZ9q7Q7gsCr_NbdpLN_1_etKnaJEo3phYB6-kPKgTys1krsgeqv1DMkGajWCSiTlFH7U38XAP74VHXQrQLROFWVaL9aLw/s1600/Stata+Table.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="353" data-original-width="1016" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjadBHZEH7keQHsmGYwoDCOTy39oqqQeuySC-VSo7M9fkW6F_AZ9q7Q7gsCr_NbdpLN_1_etKnaJEo3phYB6-kPKgTys1krsgeqv1DMkGajWCSiTlFH7U38XAP74VHXQrQLROFWVaL9aLw/s1600/Stata+Table.png" /></a></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">From the
regression output, Stata (just like other softwares) provides the beta coefficients,
standard errors, <i style="mso-bidi-font-style: normal;">t</i>-statistics,
probability values, the confidence intervals, R<sup>2</sup>, <i style="mso-bidi-font-style: normal;">F</i>-statistic, number of observations, the
degree of freedom, the explained (denoted as <i style="mso-bidi-font-style: normal;">Model</i>) and unexplained (denoted as <i style="mso-bidi-font-style: normal;">Residual</i>) errors. I will cover analysis of variance (ANOVA) in subsequent
tutorials.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">11.<span style="font: 7.0pt "Times New Roman";">
</span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Results
interpretation</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">In line with
model specification, you can then interpret your results. Always be mindful of
the units of measurements (if you are not using a log-linear model). The
results output shown above is for a linear (that is, a level-level) model.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 14.2pt; text-align: justify; text-indent: -14.2pt;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">12.<span style="font: 7.0pt "Times New Roman";">
</span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Hypothesis
testing</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Since your primary
goal for undertaking the research is to test a theory or hypothesis, it is at
this point you do that having stated what your null and alternative hypotheses
are. Remember any theory or hypothesis that is not verifiable by empirical
evidence is not admissible as a part of scientific enquiry. Now that you have obtained
your results, do you reject the null hypothesis in favour of the alternative? The
econometric packages always include this element in the result output so that
you don’t have to manually compute. Simply check your <i style="mso-bidi-font-style: normal;">t</i>-statistics or <i style="mso-bidi-font-style: normal;">p</i>-values
to know if you will reject the null hypothesis or not. </span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">For instance,
from the above output, the beta coefficient for <i style="mso-bidi-font-style: normal;">crsgpa</i> is 1.007817 and with the standard error of 0.1038808, you
can easily compute the <i style="mso-bidi-font-style: normal;">t</i>-statistic as
1.007817/0.1038808 = 9.70 (as given by the Stata output). Importantly, know that
a large <i style="mso-bidi-font-style: normal;">t</i>-statistic will always
provide evidence against the null hypothesis. Likewise, the <i style="mso-bidi-font-style: normal;">p</i>-value of 0.000 is indicative of the
fact that the likelihood of committing a Type I error (that is, rejecting the
null hypothesis when it is true) is very, very remote…close to zero! So, when
the null hypothesis is rejected, we say that the coefficient of <i style="mso-bidi-font-style: normal;">crsgpa</i> is <b style="mso-bidi-font-weight: normal;">statistically significant</b>. When we fail to reject the null
hypothesis, we say the coefficient is <b style="mso-bidi-font-weight: normal;"><u>not</u></b>
statistically significant. It is inappropriate to say that you “<i style="mso-bidi-font-style: normal;">accept</i>” the null hypothesis. One can
only <i style="mso-bidi-font-style: normal;">“fail to reject”</i> the null
hypothesis. This is because you <i style="mso-bidi-font-style: normal;">fail to
reject</i> the null hypothesis due to insufficient evidence against it (often
due to the sample collected). So, we don’t <i style="mso-bidi-font-style: normal;">accept</i>
the null, but simply <i style="mso-bidi-font-style: normal;">fail to reject</i>
it!</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;">(Detailed rudiments of hypothesis testing, Type I and II
errors will be covered in subsequent tutorials).</span></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="background: yellow; font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">13.<span style="font: 7.0pt "Times New Roman";"> </span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Post-estimation checks</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Having obtained
your estimates, it is advisable to subject your model to some diagnostics. Most
journals or even our supervisor will want to see the post-estimation checks
carried out on your model. It will also give some level of confidence if your
model passes the following tests: normality, stability, <a href="http://cruncheconometrix.blogspot.com/2018/01/heteroscedasticity.html" target="_blank">heteroscedasticity</a>,
serial correlation, model specification and so on. Regardless of your
discipline, empirical model and estimation technique, it is essential that your
results are supported with some “comforting” post-estimation checks. Find out those
applicable to your model and technique of estimation.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt 1cm; text-align: justify; text-indent: -1cm;">
<b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><span style="mso-list: Ignore;">14.<span style="font: 7.0pt "Times New Roman";">
</span></span></span></b><b style="mso-bidi-font-weight: normal;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Forecasting or
prediction/Submission</span></b></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">At this point, if the econometric model
does not refute the theory under consideration, it may be used for predicting
(forecast) future values of the dependent variable on the basis of the known or
expected future values of the regressors. However, if the work is for
submission, I will advise that it is proof-read as many times as possible
before doing so.</span></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">I hope this step-by-step guide gives you
some level of confidence to engage in research and data analysis. Let me know
if you have any additions or if I omitted some salient points. </span></div>
<div style="text-align: justify;">
<br /></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt; line-height: 107%;">Post your comments
and questions….</span><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com2tag:blogger.com,1999:blog-1876221430378807805.post-48757802294542544132018-01-12T08:00:00.000+01:002018-02-06T17:22:55.631+01:00Tell me, what is econometrics? <br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">This is the first question I asked the
very day this website came on live…so, I ask again: what is <b style="mso-bidi-font-weight: normal;">econometrics</b>? It simply refers to measuring
economic phenomena. The word “<i style="mso-bidi-font-style: normal;">econo</i>”
refers to the economic events while the “<i style="mso-bidi-font-style: normal;">metrics</i>”
is the measurement. So, econometrics is the process of measuring economic
scenarios. It therefore means that <b style="mso-bidi-font-weight: normal;"><u>measurement</u></b>
is the defining distinction between econometrics and other related disciplines
like statistics, mathematics and mathematical economics.</span></div>
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">I have come across other supportive
definitions of econometrics as the discipline that is more concerned with
quantitative analysis. A discipline that is wrapped around empirical testing to
either validate or refute economic theory. That means, an econometrician must
be quite proficient with the use of data and must fashion out ways of making
the data to <i style="mso-bidi-font-style: normal;">communicate</i>. Communication
is germane in the sense that a large pool of data (either quantitative or
qualitative) will not make much sense to policy makers if they (data) <i style="mso-bidi-font-style: normal;">do not say anything</i> or point out any
prevailing scenarios or assist in planning and forecasting. So, am I implying
that data, talks??? Oh yes, they do….all the time if an econometrician have a
prevailing theory and if s/he knows the appropriate model to deploy.</span><br />
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Next, I will briefly explain the pillars
that hold econometrics, so that starters will know what basic skills to
acquire. </span><br />
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Econometrics is a combination of
economic theory, mathematical economics, economic statistics and mathematical
statistics. Each of these are the benchmarks that an econometrician must be
familiar with.</span><br />
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">So, what is an <b style="mso-bidi-font-weight: normal;">economic theory</b>? These are simply hypotheses, conjectures, assumptions,
ideas that mostly describe how economies operate. Since, theories are often
qualitative in nature, it is the econometrician that validates economic theory
through empirical testing. For instance, an economic theory says that <i style="mso-bidi-font-style: normal;"><span style="background: yellow; mso-highlight: yellow;">the higher the price, the lower the quantitative demanded</span></i> of
commodities. This is an economic theory stating that price and quantity demanded
exhibit a <i style="mso-bidi-font-style: normal;">negative</i> relationship. The
econometrician then takes it from there by analysing both data on price and demand
quantities to observe what the outcome will be….sort of verifying/validating or
disproving what the economic theory says.</span><br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"><br /></span></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">A <b style="mso-bidi-font-weight: normal;">mathematical
economist</b>, on the other hand, expresses economic theories in mathematical
forms. That is, using equations. Given the economic model stated above, the
mathematical economist will express it as: <i><span style="background-color: yellow;">Q = a -bP</span></i> </span><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><v:shapetype coordsize="21600,21600" filled="f" id="_x0000_t75" o:preferrelative="t" o:spt="75" path="m@4@5l@4@11@9@11@9@5xe" stroked="f">
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</v:imagedata></v:shape></span><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">or</span><span style="font-family: "calibri" , "sans-serif"; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><v:shape id="_x0000_i1025" style="height: 15pt; width: 64.5pt;" type="#_x0000_t75">
</v:shape></span><span style="background-color: yellow; font-family: "times new roman" , "serif"; font-size: 13pt;"><i>P = a -bQ</i></span><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">. The first equation is known as demand function while the second is the inverse
demand function. The</span><span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;"> <span style="background-color: yellow;"><i>-b<span style="background-color: white;"> </span></i><span style="background-color: white; font-family: "times new roman" , "serif"; font-size: 13pt;">sign </span></span>indicates the negative slope of the
demand function which emanates from economic theory. Since the mathematical
economist is not involved in measuring the impact of price increase on quantity
demanded or in the empirical investigation of economic theory, again, this is
where the econometrician comes in using these mathematical equations to either
validate or disprove theory via empirical testing.</span></div>
<br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">What of an <b style="mso-bidi-font-weight: normal;">economic statistician</b>? Such person is only concerned with the
collection and descriptive presentation of data. The main tools used are charts,
tables and graphs. Again, from the price and quantity example, an economic
statistician goes to the field, collects data on prices and quantities and puts
them out using pie charts, bar charts, histograms, line graphs showing the
pictorial illustrations between these two variables. These <i style="mso-bidi-font-style: normal;">primary</i> level of data communication is very relevant to policy
makers as it shows the relationships between the two variables. However, since
the economic statistician is not involved in empirical testing, it is left for
the econometrician to take the data collected and subject it to tests using
several econometric tools and models at his/her disposal.</span></div>
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Lastly, the <b style="mso-bidi-font-weight: normal;">mathematical statistician</b> provides the tools used by the
econometrician. They construct the programmes and methods used in econometric
analysis.</span><br />
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">I conclude that a starting econometrician
must have some basic knowledge about each of the sub-fields explained above.
Econometrics is not difficult but interesting. Begin, by knowing the fundamentals
from the tools used, to modelling, analysing, hypothesis testing,
interpretation, forecasting. Stay with me on this platform as I patiently teach
you this subject and it will not take too long for you to gravitate into the
intermediate and complex stuff.</span><br />
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">So, let us keep it simple and take it one
step at a time 😊</span><br />
<br />
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt;">Again, I ask you: what is econometrics?
Let me know what other definitions you can come up with.</span><br />
<br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 13.0pt; line-height: 107%;">Post
your comments and questions…</span><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0tag:blogger.com,1999:blog-1876221430378807805.post-52692167633548336442018-01-10T08:00:00.000+01:002018-02-06T17:24:59.840+01:00Multicollinearity<div class="MsoNoSpacing" style="text-align: justify;">
<br />
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Multicollinearity, pronounced as <i style="mso-bidi-font-style: normal;">mul-ti-co-lli-nea-ri-ty</i> is the second
longest word in the econometrics dictionary after heteroscedasticity. It
contains 17 <i style="mso-bidi-font-style: normal;">talking</i> words! It occurs
when there exists perfect or exact linear dependence or relationships among
explanatory variables in a given model. Collinearity is when such exact
dependence is between two variables. In that wise, we say that the variables
are collinear. For instance, when an explanatory variable is 80 to 100% explained
by another explanatory variable, separating the influence of each of them on
the dependent variable (regressand) becomes difficult and interpreting the
estimated coefficients from that model will also be problematic. This is
because variation in one regressor can be completely explained by another
regressor in the same model.</span><br />
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<b style="font-family: "times new roman", serif; font-size: 12pt;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">With perfect or less than perfect multicollinearity
or collinearity:</span></b></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Regression
coefficients are indeterminate (because the collinear variables cannot be
distinguished from one another)</span></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Standard
errors are infinite (they are very large)</span></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Estimates
are biased</span></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Coefficients
cannot be estimated with precision or accuracy</span></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="background-color: yellow; font-family: "times new roman" , serif; font-size: 12pt;">Note:
multicollinearity does not violate any regression assumptions; the OLS
estimators are still BLUE (Best Linear Unbiased Estimators); it does not
destroy the property of minimum variance.</span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Multicollinearity can be detected using </span><i style="font-family: "times new roman", serif; font-size: 12pt;">“r”</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> the coefficient of correlation. So,
if </span><i style="font-family: "times new roman", serif; font-size: 12pt;">r = 1</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">, then multicollinearity or
collinearity exists. So whenever you run your correlation matrix, look out for
those relationship where </span><i style="font-family: "times new roman", serif; font-size: 12pt;">r</i><span style="font-family: "times new roman" , serif; font-size: 12pt;"> > 0.8,
that tells us that the respective variables are collinear. Multicollinearity is
ruled out when regressors in a model have non-linear relationships. </span></div>
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">
</span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
</span>
<br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">A major problem associated with
multicollinearity is that, if <i style="mso-bidi-font-style: normal;">r</i> is
high, then the standard error will be high and the computed <i style="mso-bidi-font-style: normal;">t</i>-statistic will be low making it more
likely <i style="mso-bidi-font-style: normal;">not</i> to reject the null
hypothesis when is false. Thereby committing a Type II error….that is,
incorrectly retaining a very false null hypothesis.</span></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
</span><br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;"><b style="font-size: 12pt;">How do you know if your model suffers
from multicollinearity? </b></span></div>
<span style="font-family: "times new roman" , serif; font-size: 12pt;">
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">High
<i>R</i><sup>2</sup></span></div>
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Few
significant <i style="mso-bidi-font-style: normal;">t</i>-ratios</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Wider
confidence intervals</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Contradictory
signs of beta coefficients to expected <i style="mso-bidi-font-style: normal;">a
priori</i></span></div>
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Estimates
are sensitive to even small changes in model specification</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">High
pair-wise correlation statistic among the regressors</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 36pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">From
the tolerance level and variance inflation factor (VIF). A tolerance level lower than 0.10 and a VIF of 10 are indicative of multicollinearity in a model. A higher VIF provides evidence of
multicollinearity. </span></div>
<br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<b style="font-size: 12pt;">Correcting/controlling for
multicollinearity:</b></div>
<br />
<div style="margin: 0cm 0cm 0pt 39.75pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Collect
more data</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 39.75pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Change
the scope of analysis</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 39.75pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Do
not include collinear variables in the same regression</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 39.75pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Drop
the highly collinear variable</span></div>
<br />
<div style="margin: 0cm 0cm 0pt 39.75pt; text-align: justify; text-indent: -18pt;">
<span style="font-family: "wingdings"; font-size: 12.0pt;"><span style="mso-list: Ignore;">Ø<span style="font: 7.0pt "Times New Roman";">
</span></span></span><span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;">Transform
the collinear variable through differencing (however, the differenced error
term is serially-correlated and violates OLS assumptions).</span></div>
<br />
<div style="margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-size: 12pt;">What I do often, is to drop the
collinear variable and if that variable is very important to my model, I’ll
transform my modelling structure into a step-wise fashion such that collinear
variables are not included together in the same regression.</span></div>
</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-size: 12pt;"><span style="font-family: "times new roman" , serif;"><br /></span></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-size: 12pt;"></span><br />
<div style="font-family: "times new roman", serif; margin: 0cm 0cm 0pt;">
<span style="font-size: 12pt;"><span style="font-family: "times new roman" , "serif"; font-size: 12pt;">[Watch video on multicollinearity]</span></span><br />
<div>
<span style="font-size: 12pt;"><br /></span></div>
</div>
<span style="font-size: 12pt;">
<br />
</span><br />
<div style="font-family: "times new roman", serif; margin: 0cm 0cm 0pt 21.75pt; text-align: justify;">
<span style="font-size: 12pt;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Qk1xjgnLcFE/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Qk1xjgnLcFE?feature=player_embedded" width="320"></iframe></span></div>
<span style="font-size: 12pt;">
<div style="font-family: "times new roman", serif; margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "times new roman" , "serif"; font-size: 12.0pt;"><br /></span></div>
<div style="font-family: "times new roman", serif; margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-size: 12pt;">Post your comments
and questions….</span><br />
<span style="font-size: 12pt;"><br /></span>
<span style="font-size: 12pt;"><a href="http://cruncheconometrix.com.ng/" target="_blank">Back to Home</a></span></div>
</span></div>
Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com4tag:blogger.com,1999:blog-1876221430378807805.post-52086073461934973422018-01-08T15:41:00.000+01:002018-02-06T17:26:18.462+01:00Heteroscedasticity<div class="MsoNoSpacing" style="text-align: justify;">
<span style="font-family: "times new roman" , serif; font-size: 12pt;">Heteroscedasticity,
you can try to pronounce it the way I do, </span><i style="font-family: "Times New Roman", serif; font-size: 12pt;">he-te-ro-sce-das-ti-ci-ty</i><span style="font-family: "times new roman" , serif; font-size: 12pt;">.
You see it isn’t so difficult to pronounce after all. It happens to be the
longest word in the econometrics dictionary with 18 words…yes, 18 words! Can be
written as heteroskedasticity but whichever way you choose to write it is fine,
only be consistent with your choice. I will be sticking to heteroscedasticity…</span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Perhaps you have
heard about the word, so what exactly is heteroscedasticity? It may seem like a
ton of vocals in your mouth but the concept is very simple to grasp. It refers
to disturbances (errors) whose variances are not constant in a given model. It
is when the variance of the error terms differ across observations. That is,
when a data has unequal variability (dispersion) across a given set of second
predictor variables. Again what are disturbances? You may begin to think at
this point that econometrics has tons of jargons, yes, you are absolutely
right. But relax, you will understand them as you become more involved in its
processes.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">So, again what
does heteroscedasticity mean? It means that in a given model, it is important
that error variances across observations are constant. For instance, one of the
assumptions of ordinary least squares (OLS) is that the model must be
homoscedastic. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">In
the presence of heteroscedasticity:<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">OLS
estimators, </span><!--[if gte msEquation 12]><m:oMath><m:sSub><m:sSubPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubPr><m:e><m:acc><m:accPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:accPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-size:12.0pt;
font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"'><m:r>β</m:r></span></i></m:e></m:acc></m:e><m:sub><i
style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-size:12.0pt;
font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"'><m:r>OLS</m:r></span></i></m:sub></m:sSub></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%; position: relative; top: 2.5pt;"><v:shapetype coordsize="21600,21600" filled="f" id="_x0000_t75" o:preferrelative="t" o:spt="75" path="m@4@5l@4@11@9@11@9@5xe" stroked="f">
<v:stroke joinstyle="miter">
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0">
<v:f eqn="sum @0 1 0">
<v:f eqn="sum 0 0 @1">
<v:f eqn="prod @2 1 2">
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</v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:formulas>
<v:path gradientshapeok="t" o:connecttype="rect" o:extrusionok="f">
<o:lock aspectratio="t" v:ext="edit">
</o:lock></v:path></v:stroke></v:shapetype><v:shape id="_x0000_i1025" style="height: 15pt; width: 22.5pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:/Users/pstkay/AppData/Local/Temp/msohtmlclip1/01/clip_image001.png">
</v:imagedata></v:shape></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> are still linear,
unbiased, consistent and asymptotically normally distributed. The regression
estimates and the attendant predictions remain unbiased and consistent. But the
estimators are inefficient (that is, not having minimum variance) in the class
of minimum variance estimators. Hence, OLS is not BLUE (Best Linear Unbiased Estimator),
therefore the regression predictors are also inefficient, though consistent.
What this means that the regression estimates cannot be used to construct
confidence intervals, or used for inferences.</span><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="Default">
<br /></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l0 level1 lfo1; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Heteroskedasticity causes statistical
inference based on the usual <i>t </i>and <i>F </i>statistics to be invalid,
even in large samples. As heteroskedasticity is a violation of the Gauss-Markov
assumptions, OLS is no longer BLUE.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Causes
of heteroscedasticity:<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Okay, having known
that the presence of heteroscedasticity in a model can invalidate statistical
tests of significance, it is important to know its causes. That is, what can
lead to heteroscedasticity being evident in your data?<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">The presence
of outliers can lead to your model becoming heteroscedastic. And what are
outliers? These are simply <i>bogus</i>
figures in your data that stands out. Very obvious to the prying eyes. Doing a
simple summary statistic of your data before any regression analysis, can
easily detect outliers by indicating both the minimum and maximum values of a
variable. For example, you may have a 30 years inflation data for country <i>J</i> and on average, the yearly inflation
figures for that country hovers around 9%, 7.5%, 8.2% and suddenly you observe
an inflation rate of 58.7%. Since there is no economic phenomenon to support
that outrageous figure, then 58.7% is an outlier which may cause your model to
become heteroscedastic.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Wrongly
specifying your model is another factor. This can be related to the functional
form by which your model is specified. Functional form can be a log-log model
(where the dependent variable and all or some of the explanatory variables are
in natural logarithms or logs for short); a log-level model (where only the
dependent variable is transformed into natural logarithm and the explanatory
variables are in their level forms, that is, not transformed); lastly is the
level-level form.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Wrong
data transformation. For instance over differencing a variable can be a cause.
If a variable is stationary in level at 10%, for example, I have seen cases
where students still go ahead to difference the same variable in order to
obtain stationarity at maybe 1% or 5% statistical significance. This is not
necessary. Once your variable is stationary in level, that is an <i>I</i>(0) series, just go ahead and run your
analysis. Note that further differencing the variable again, may lead to
heteroscedasticity.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Poor
data sampling method may lead to heteroscedasticity particularly when
collecting primary data.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="margin-left: 14.2pt; mso-list: l1 level1 lfo2; text-align: justify; text-indent: -14.2pt;">
<!--[if !supportLists]--><span lang="EN-GB" style="font-family: "symbol"; font-size: 12.0pt;">·<span style="font-family: "times new roman"; font-size: 7pt; font-stretch: normal; line-height: normal;"> </span></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Skewness
of one or more regressors (closely related to outliers being evident in the
data). Regressors are explanatory or independent variables.<o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<i><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Detecting
heteroscedasticity<o:p></o:p></span></i></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Having known what
heteroscedasticity is and its causes, how can it be detected? The truth is that
there is no hard and fast rule for detecting heteroscedasticity. Therefore,
more often than not, heteroscedasticity may be a case of educated guesswork,
prior empirical experiences or mere speculation. However, several formal and
informal approaches can be used in detecting the presence of heteroscedasticity
but discussions will be limited to the graphical approach (plotting the
residuals form the regression against the estimated dependent variable),
Breusch-Pagan test and White test. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">So, let us take an
<b><span style="background: yellow; mso-highlight: yellow;">example</span></b> using <a href="https://drive.google.com/open?id=1WRZX474X8pj3Egxo51Tz0Y66me0v2JRd" target="_blank">JM Wooldridge’s</a> GPA3.dta or GPA3.xls</span><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">
data to make this topic clearer. (use .xls if Stata is not installed on
your devise and run the analysis using any econometric software).</span> </div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxuFtrJ0qut-OWtOwF4pWMqoc6-a_lKNGHbdNz6jyFlzAH-FqZQxbNz7UkTR8ipNLNY9DpKaKnhnAFDySNShBDA8yc8Bz7Q-F-sXiTgrOkDnIiySXSfqgbJGc009OKRJ5GRZB1nv5EoHE/s1600/Stata+Table.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="Regression output in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="353" data-original-width="1016" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxuFtrJ0qut-OWtOwF4pWMqoc6-a_lKNGHbdNz6jyFlzAH-FqZQxbNz7UkTR8ipNLNY9DpKaKnhnAFDySNShBDA8yc8Bz7Q-F-sXiTgrOkDnIiySXSfqgbJGc009OKRJ5GRZB1nv5EoHE/s1600/Stata+Table.png" title="Regression output in Stata" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Regression output in Stata<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">From the
regression output, the <i>F</i>-statistic is
significant at the 1% level, the R<sup>2</sup> reveals that about 48% variation
are explained by the independent variables. <o:p></o:p></span><br />
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">But how do we know
if this model is heteroscedastic or not? <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">1). Start from the
informal approach which is plotting the squared residuals, </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><m:acc><m:accPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:accPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-size:12.0pt;
font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"'><m:r>u</m:r></span></i></m:e></m:acc></m:e><m:sup><i
style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-size:12.0pt;
font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"'><m:r>2</m:r></span></i></m:sup></m:sSup></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><v:shape id="_x0000_i1025" style="height: 14.25pt; width: 12.75pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:/Users/pstkay/AppData/Local/Temp/msohtmlclip1/01/clip_image003.png">
</v:imagedata></v:shape></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> against </span><!--[if gte msEquation 12]><m:oMath><m:acc><m:accPr><span
style='font-size:12.0pt;mso-ansi-font-size:12.0pt;mso-bidi-font-size:12.0pt;
font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math";
mso-hansi-font-family:"Cambria Math";mso-bidi-font-family:"Times New Roman";
font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:accPr><m:e><i
style='mso-bidi-font-style:normal'><span lang=EN-GB style='font-size:12.0pt;
font-family:"Cambria Math",serif;mso-bidi-font-family:"Times New Roman"'><m:r>Y</m:r></span></i></m:e></m:acc></m:oMath><![endif]--><!--[if !msEquation]--><span style="font-family: "calibri" , sans-serif; font-size: 11.0pt; line-height: 107%; position: relative; top: 3.0pt;"><v:shape id="_x0000_i1025" style="height: 15pt; width: 7.5pt;" type="#_x0000_t75">
<v:imagedata chromakey="white" o:title="" src="file:///C:/Users/pstkay/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png">
</v:imagedata></v:shape></span><!--[endif]--><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> using the Stata
commands <i>rvfplot</i> or <i>rvfplot, yline(0)</i> to see if there is a
definite pattern. If a definite pattern exists, then the model is
heteroscedastic. <o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="tab-stops: 140.25pt; text-align: justify;">
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;">rvfplot</span></b><br />
<b><span style="background-color: yellow;"><br /></span></b>
<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHpuovyASuoDzQikSX9tCnih6YWW3RoSa-QVRb5JnY1NAm6ocD1TTaZiYIHV9XPyazX4ofQvQsZsnjTZXeNFTScF0JPV-dHgSns9ycCEaOD470Sjt43ywSBtNtYCZ4IBXHc4vcl_zjPYY/s1600/rvfplot.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img alt="Residual plot in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="1169" data-original-width="1600" height="467" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHpuovyASuoDzQikSX9tCnih6YWW3RoSa-QVRb5JnY1NAm6ocD1TTaZiYIHV9XPyazX4ofQvQsZsnjTZXeNFTScF0JPV-dHgSns9ycCEaOD470Sjt43ywSBtNtYCZ4IBXHc4vcl_zjPYY/s640/rvfplot.jpg" title="Residual plot in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Residual plot in Stata<br />
Source: CrunchEconometrix</td></tr>
</tbody></table>
<b><span style="background-color: yellow;"></span></b><b><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p><br /></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><v:shape id="Picture_x0020_2" o:spid="_x0000_i1026" style="height: 221.25pt; mso-wrap-style: square; visibility: visible; width: 306.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:/Users/pstkay/AppData/Local/Temp/msohtmlclip1/01/clip_image005.emz">
</v:imagedata></v:shape></span><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;"><br /></span></b>
<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;">rvfplot, yline(0)</span></b><b><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p></o:p></span></b></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<br /></div>
<div class="MsoNoSpacing" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><v:shape id="Picture_x0020_3" o:spid="_x0000_i1025" style="height: 225.75pt; mso-wrap-style: square; visibility: visible; width: 291.75pt;" type="#_x0000_t75">
<v:imagedata o:title="" src="file:///C:/Users/pstkay/AppData/Local/Temp/msohtmlclip1/01/clip_image006.emz">
</v:imagedata></v:shape></span><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p></o:p></span></div>
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<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhV0pdHfQlSFF_99N2OSv1w_6dCzzPG_c_Hy4XuGaiirndTFxBZtAToc01yfXTxds_D33VSAjrgjmv-zuaBYZQLQE4z2W6jed6-jUqWEmdr4uX3EoQNcFErP2c6nRnm-QgLD80UW_O9Dko/s1600/rvfplot-yline.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img alt="Residual plot in Stata from http://cruncheconometrix.com.ng" border="0" data-original-height="1169" data-original-width="1600" height="466" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhV0pdHfQlSFF_99N2OSv1w_6dCzzPG_c_Hy4XuGaiirndTFxBZtAToc01yfXTxds_D33VSAjrgjmv-zuaBYZQLQE4z2W6jed6-jUqWEmdr4uX3EoQNcFErP2c6nRnm-QgLD80UW_O9Dko/s640/rvfplot-yline.png" title="Residual plot in Stata" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Residual plot in Stata<br />
Source: CrunchEconometrix</td></tr>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">From both plots, a
definite pattern is observed evidencing that the model is heteroscedastic.<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">2). Conduct either
the Breusch-Pagan or White heteroscedasticity test after your regression to
check if the residuals of a regression have a changing variance. The Stata
commands are: <i>estat hettest</i> and <i>estat imtest, white</i>. If the obtained <i>p</i>-values are significant, then the model
exhibits heteroscedasticity and if otherwise, then the model is homoscedastic.<o:p></o:p></span></div>
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<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;">estat hettest</span></b><b><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Breusch-Pagan/Cook-Weisberg
test for heteroscedasticity <o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> Ho: Constant variance<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> Variables: fitted values of trmgpa<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> chi2(1) =
14.12<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> Prob > chi2 =
0.0002<o:p></o:p></span></div>
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<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;">estat imtest, white</span></b><b><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">White's test for
Ho: homoscedasticity<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> against Ha: unrestricted heteroscedasticity<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> chi2(33) =
61.22<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> Prob > chi2 =
0.0020<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">//the null
hypotheses for both tests are that the model is homoscedastic. But since the <i>p</i>-values for both tests are significant,
the null hypothesis is rejected in favour of the alternative hypothesis evidencing
that the model is heteroscedastic<o:p></o:p></span></div>
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<i><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Controlling/Correcting
heteroscedasticity<o:p></o:p></span></i></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">Also, as a
pre-condition it is advisable to run your analysis using White’s heteroscedasticity-robust
standard errors by including the <i>robust</i>
option in the command line like this example:<o:p></o:p></span></div>
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<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;">reg trmgpa crsgpa cumgpa tothrs sat
hsperc female season, robust</span></b><b><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> <o:p></o:p></span></b></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">By using this
code, the problem of heteroscedasticity is controlled in comparison to if the <i>robust</i> option is not used.<o:p></o:p></span></div>
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<b><span lang="EN-GB" style="background: yellow; font-family: "times new roman" , serif; font-size: 12.0pt;">Assignment:</span></b><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> Using <a href="https://drive.google.com/open?id=1WRZX474X8pj3Egxo51Tz0Y66me0v2JRd" target="_blank">Wooldridge’s</a>
hprice1.dta or hprice1.xls data,</span><span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;"> how
can you detect if the model is heteroscedastic and how will you correct it? Compare
the usual standard errors with the obtained heteroscedasticity-robust standard
errors. What do you observe?<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "times new roman" , serif; font-size: 12.0pt;">So, with this
brief and practical tutorial, you can confidently run your regressions and test
if your model suffers from heteroscedasticity or not….good luck!<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 12pt; text-align: justify;"><b>Post your comments
and questions….</b></span>Bosede Ngozi ADELEYEhttp://www.blogger.com/profile/10017829417122250012noreply@blogger.com0