# How to Perform Unit Root Test in EViews

##
**What
is Stationarity in Time Series Analysis?**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.

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**How to detect unit root in a series?**

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*(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*(0), otherwise it is*I*(*d*) 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*(0).
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*(*d*) where*d*is the number of differencing to render the series stationary. Hence, a stationary series at level,*d*= 0 is a series with an*I*(0) process. Although, for any non-stationary series,*‘d’*can assume any value greater than zero, however, in applied research, only the unit root process of*I*(1) process is allowed, otherwise such series with higher order of integration (*d*> 1) should be excluded in the model as no meaningful policy implications or relevance can be drawn from such series.
Here is an example of a bivariate linear
regression model:

*Y*= 𝛂₀ +

_{t }*b*

*X*+

_{t}*u*

*[1]*

_{t}
Assume

*Y*and_{t}*X*are two random walk models that are_{t}*I*(1) processes and are independently distributed as:*Y*= ρ

_{t }*Y*+

_{t-1}*v*

*, -1 ≤ ρ ≤ 1 [2]*

_{t}*X*= ղ

_{t }*X*+

_{t-1}*e*

*, -1 ≤ ղ ≤ 1 [3]*

_{t}
and

*v*and_{t}*e*have zero mean, a constant variance and are orthogonal (these are_{t}*white noise*error terms).
We also assumed that

*v*and_{t}*e*are serially uncorrelated as well as mutually uncorrelated. As stated in [2] and [3], both these time series are nonstationary; that is, they are_{t}*I*(1) or exhibit stochastic trends. Suppose we regress*Y*on_{t}*X*. Since_{t}*Y*on_{t}*X*are uncorrelated_{t}*I*(1) processes, the*R*^{2}from the regression of*Y*on*X*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*Y*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_{t}*Y*and_{t}*X*are stationary. In practice, then, it is important to ﬁnd out if a time series possesses a unit root._{t}
Given equations [2] and [3], there
should be no systematic relationship between

*Y*and_{t}*X*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_{t}*b*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,*b*takes any value randomly, and its*t*-statistic indicates significance of the estimate.**But how can unit root be detected?**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:

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?

2. Or
carry out a regression analysis on two series and observe the

*R*^{2}. If it is above 0.9, it may suggest that the variables are nonstationary.
3. The rule-of-thumb: if the

*R*^{2}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.
Using Gujarati and Porter Table 21.1
quarterly data of 1970q1 to 1991q4, examples of nonstationary series and
spurious regression can be seen from the

*pce*,*pdi*and*gdp*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.**Nonstationary series:**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.

EViews - Example of nonstationary series Source: CrunchEconometrix |

**What is a spurious regression?**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

*lnpce*on

*lnpdi*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.

But how do we know this? Take a look at
the

*R*^{2}the value of**0.9944**is higher than the Durbin Watson statistic of**0.57**. So, whenever the*R*^{2}> DW, a spurious regression has occurred because the variables are nonstationary.EViews - Example of a spurious regression Source: CrunchEconometrix |

As you can see, the coefﬁcient of

*lnpdi*is highly statistically signiﬁcant, and the*R*^{2}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*a priori*there may or may*not*be none. This is simply the phenomenon of**spurious or nonsense regression**, 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,*R*^{2}> DW is a good rule of thumb to suspect that the estimated regression is spurious, as in the given example.**Why is it important to test for stationarity?**

We usually consider a nonstationary
series for the following reasons:

1. 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.

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.

**How to correct for nonstationarity?**

What can be done with nonstationarity in
a time series knowing that performing OLS on such a model yields spurious
regression?

**The Unit Root Test**

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

*Y*on its (one-period) lagged value_{t}*Y*and ﬁnd out if the estimated ρ is statistically equal to 1? If it is, then_{t−1}*Y*is nonstationary. Repeat same for the_{t}*X*series. This is the general idea behind the unit root test of stationarity._{t}
For theoretical reasons, equation [2] is
manipulated as follows: Subtract

*Y*from both sides of [2] to obtain:_{t−1}*Y*-

_{t }*Y*= ρ

_{t-1 }*Y*-

_{t-1}*Y*+

_{t-1 }*v*

*[4]*

_{t}
= (ρ - 1)

*Y*+_{t-1 }*v*_{t}
and this can be stated alternatively as:

⃤

*Y*= δ_{t}*Y*+_{t-1 }*v**[5]*_{t}
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.

Before we proceed to estimate [5], it
may be noted that if δ = 0, [5] will become:

⃤

*Y*=_{t}*Y*-_{t-1}*Y*_{t-1}*=*_{ }*v**[6]*_{t}
(Remember to do the same for

*X*series)_{t}
Since

*v*_{t}is a white noise error term, it is stationary, which means that**the ﬁrst difference of a random walk time series is stationary**.
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.

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

*Y*and regress on_{t}*Y*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_{t−1}*Y*is nonstationary. But if it is negative, we conclude that_{t}*Y*is stationary._{t}**Note:**Since δ = (ρ − 1), for stationarity ρ must be less than one. For this to happen δ must be negative!

The only question is which test do we
use to ﬁnd out if the estimated coefﬁcient of

*Y*in [5] is zero or not? You might be tempted to say, why not use the usual_{t−1}*t*test? Unfortunately, under the null hypothesis that δ = 0 (i.e., ρ = 1), the*t*value of the estimated coefﬁcient of*Y*does not follow the_{t−1}*t*distribution even in large samples; that is, it does not have an asymptotic normal distribution.
What is the alternative? Dickey and
Fuller (DF) have shown that under the null hypothesis that δ = 0, the estimated

*t*value of the coefﬁcient of*Y*in [5] follows the_{t−1}**statistic. These authors have computed the critical values of the***τ (tau)**tau statistic*on the basis of Monte Carlo simulations.**Note:**Interestingly, if the hypothesis that δ = 0 is rejected (i.e., the time series is stationary), we can use the usual (Student’s)

*t*test.

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:

1.
random
walk (that is, model has no constant, no trend)

2.
random
walk with drift (that is, model has a constant)

3.
random
walk with drift and a trend (that is, model has a constant and trend)

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

*Y*is a stationary time series with zero mean in the case of [5], that_{t}*Y*is stationary with a nonzero mean in the case of a random walk with drift model, and that_{t}*Y*is stationary around a deterministic trend in the case of random walk with drift around a trend._{t}
It is extremely important to note that
the critical values of the

*tau test*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*tau statistic*(|τ|) exceeds the DF or MacKinnon critical*tau values*, 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*tau*value, we fail to reject the null hypothesis, in which case the time series is nonstationary.**Note:**Students often get confused in interpreting the outcome of a unit root test. For instance, if the calculated

*tau*statistic is -2.0872 and the MacKinnon

*tau*statistic is -3.672, you cannot reject the null hypothesis. Hence, the conclusion is that the series is nonstationary. But if the calculated

*tau*statistic is -5.278 and the MacKinnon

*tau*statistic is -3.482, you reject the null hypothesis in favour of the alternative. Hence, the conclusion is that the series is stationary.

*Always use the appropriate
critical τ values for the indicated model specification.

**How to Perform Unit Root Test in EViews (See here for Stata)**

Example
dataset is from Gujarati and Porter T21.1

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.

**How to Perform the Augmented Dickey-Fuller (ADF) Test**

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

*Y*series, in conducting the DF test, it is assumed that the error term_{t}*v*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._{t}
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.

1.
Load
workfile into EViews.

2.
We
are considering only

*lnpce*and*lnpdi*in natural logarithms.**The Augmented Dickey-Fuller (ADF) Test**

Unit root test for

*lnpce*:
·
Double
click the

*lnpce*series to open it.
· Go
to

**View**>>**Unit root test**>> dialog box opens >> Under**Test Type**, select**Augmented Dickey Test**
· Decide
whether to test for a unit root in the

**level**,**1st difference**, or**2nd difference**of the series. Ideally, always start with the**level**and if we fail to reject the test in levels then continue with testing for the first difference. Hence, we first click on**'Level'**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.
· 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

**intercept**,**trend and intercept**, or**none**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:**“none”**,**“constant”**,**“constant****and****trend”**.
·
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.

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

*tau statistic*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.
Another way of stating this is that in
failing to reject the null hypothesis of a unit root, the computed τ value
should be

**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.**__more__
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.

· Having
specified the

**“none”**option where both the intercept and trend are excluded in the test regression, the unit root test dialog box is shown thus:EViews - Unit root test dialog box Source: CrunchEconometrix |

·
The
ADF unit root test results for the selected regression option, “

**none**” appears as follows:EViews - Augmented Dickey-Fuller test ("none") option Source: CrunchEconometrix |

Following similar procedures, the select
“

**intercept**” for the ADF unit root test yields:EViews - Augmented Dickey-Fuller test ("intercept") option Source: CrunchEconometrix |

The result for the “

**trend and intercept**” options for the ADF unit root test is shown below:EViews - Augmented Dickey-Fuller test ("trend and intercept") option Source: CrunchEconometrix |

·
Do same for the

*lnpdi*series.
The three ADF specifications all confirm
that

*lnpce*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**“1**option, and if the series is still nonstationary, the^{st}difference”**“2**option is conducted.^{nd}difference”
·
1

^{st}difference with**“none”**option result:EViews - Augmented Dickey-Fuller test 1st difference ("none") option Source: CrunchEconometrix |

·
1

^{st}difference with**“intercept”**option result:EViews - Augmented Dickey-Fuller test 1st difference ("intercept") option Source: CrunchEconometrix |

·
1

^{st}difference with**“trend and intercept”**option result:EViews - Augmented Dickey-Fuller test 1st difference ("trend and intercept") option Source: CrunchEconometrix |

The three ADF specifications all confirm
that

*lnpce*is stationary at 1^{st}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**because it is only at that specification that the null hypothesis of a unit root is rejected. Hence, carrying out a***lnpce*is stationary at 1^{st}difference with a constant**“2**test is unnecessary.^{nd}difference”
·
Again, do same for the

*lnpdi*series.**How to Perform the Phillips-Perron (PP) Test**

Phillips and Perron use nonparametric statistical methods to take
care of the serial correlation in the error terms

**adding lagged difference terms. Procedures for testing for unit root using PP test are similar to that of ADF earlier discussed except for the**__without__**Test Type**options.**Note:**The asymptotic distribution of the PP test is the same as the ADF test statistic.

**After unit root testing, what next?**

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.

**Scenario 1: When series under scrutiny are stationary in levels?**

If

*pce*and*pdi*are stationary in levels, that is, they are*I*(0) series (integrated of order zero). In this situation, performing a cointegration test is**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:**__not__*pce*

*= 𝛂₀ +*

_{t }*b*

*pdi*+

_{t}*u*

_{t}
In essence, the estimation of short run
model is not necessary if series are

*I*(0).**Scenario 2: When series are stationary in first differences?**

· Under
this scenario, the series are assumed to be non-stationary.

· One
special feature of these series is that they are of the same order of
integration.

· 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?

· 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).

· 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.

· Note
that both long run and short run models must be estimated when there is
cointegration.

· The
estimation will require the use of vector autoregressive (VAR) model analysis
and VECM models.

· 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!

· There
are however, two prominent cointegration tests for

*I*(I) series in the literature. They are Engle-Granger cointegration test and Johansen cointegration test.
· The
Engle-Granger test is meant for single equation model while Johansen is
considered when dealing with multiple equations.

**Scenario 3: The series are integrated of different order?**

· Should
in case

*lnpce*and*lnpdi*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.
· The
appropriate cointegration test to apply is the Bounds test for cointegration
and the estimation technique is the autoregressive distributed lag (ARDL) model.

· 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.

· However,
both the long run and short run models are valid if there is cointegration.
That is run both ARDL and ECM models.

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*(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.**[Watch video clip on performing ADF stationarity test in EViews]**

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 datasets uploaded to this bog
to practise in order to get more hands-on knowledge.

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