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Unit Root Test

Contents

1: What is unit root?2: How to check unit root?3: Types of unit root test4: Dickey fuller5: Augmented dickey fuller6: Phillip perron7: Testing Unit Root on E-views

Unit Root Test

What is unit root?

A unit root test is a statistical test for the proposition that in a autoregressive statistical model of a time series, the autoregressive parameter is one. A unit root is an attribute of a statistical model of a time series whose autoregressive parameter is one. It is as:

yt=ρyt-1+ut

where -1≤ρ≤1

If ρ is in fact 1, we face what is known as the unit root problem that is a situation of non stationary .

How to check Unit Root?We start with Yt = ρYt−1 +ut

−1≤ ρ ≤1

where ut is a white noise error term. We know that if ρ =1, that is the case of the unit root, which we know is a non-stationary stochastic process. Then we simply regress Yt on its lagged value Yt−1 and find out if the estimated ρ is statistically equal to 1? If it is, then Yt is nonstationary. This is the general idea behind the unit root test of stationarity.

Steps to check unit root test

Step 1: Subtract Yt−1 from both sides of equation.to obtain

Yt −Yt−1 = ρYt−1 −Yt−1 +ut

Yt= (ρ −1) Yt−1 +ut where δ =(ρ −1)

Step 2: Now we 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. It may be noted that if δ =0 then

Yt =(Yt −Yt−1)=ut

Since ut is a white noise error term, it is stationary, which means that the first differences of a random walk time series are stationary.

Types of Unit Root Test

There are three types of Unit root test1: Dickey fuller2: Augmented Dickey Fuller3: Phillip perron

Dickey fuller testDickey and Fuller ( 1979, 1981) devised a procedure to formally test for non-stationarity. The key insight of their test is that testing for non-stationarity is equivalent to testing for the existence of a unit root. Thus the obvious test is the following which is based on the simple AR(1) model of the form:

Yt = ρYt−1 + ut

What we need to examine here is whether ρ is equal to 1 ('unit root').

Ho: ρ = 1 (null hypothesis )

H1: ρ < 1 (Alternative hypothesis )

By subtracting both sides Yt-I with

Yt −Yt−1 = ρYt−1 −Yt−1 + ut

Yt= (ρ −1) Yt−1 +ut

Yt=ФYt−1 +ut

where of course Ф = (ρ -1).

The Dickey-Fuller test for stationarity is then simply the normal 't' test on the coefficient of the lagged dependent variable Yt-I. The DF-test statistic is the t statistic for the lagged dependent variable.

If the DF statistical value is smaller in absolute terms than the critical value then we reject the null hypothesis of a unit root and conclude that Yt is a stationary process.

Augmented Dickey FullerAs the error term is unlikely to be white noise, Dickey and Fuller extended their test procedure suggesting an augmented version of the test which includes extra lagged terms of the dependent variable in order to eliminate autocorrelation.

The testing procedure for the ADF test is the same as for the Dickey–Fuller test but it is applied to the model

Where α is a constant, β the coefficient on a time trend and ρ the lag order of the autoregressive process. Imposing the constraints and corresponds to modelling a random walk and using the constraint corresponds to modelling.

Phillip perron

Phillips and Perron ( 1988) developed a generalization of the ADF test procedure that allows for fairly mild assumptions concerning the distribution of errors. The test regression for the Phillips-Perron (PP) test is the AR(l) process:

Yt= αₒ+ФYt−1 +ut

While the ADF test corrects for higher order serial correlation by adding lagged differenced terms on the right-hand side, the pp test makes a correction to the t statistic of the coefficient y from the AR(1) regression to account for the serial correlation in ut. So, the PP statistics are just modifications of the ADF t statistics that take into account the less restrictive nature of the error process.

Testing Unit root in e-views

Step 1: Open the file in EViews by clicking File/Open/Workfile and then choosing the file name from the appropriate path.

Step 2: Let's assume that we want to examine whether the series named GDP contains a unit root. Double click on the series named 'gdp' to open the series window and choose View/Unit Root Test .In the unit-root test dialog box that appears, choose the type of test (i.e. the' Augmented Dickey-Fuller test) by clicking on it.

Step 3: We then have specify whether we want to test for a unit root in the level, first difference, or second difference of the series. We first start with the level.

Step 4: We also have to specify which model of the three ADF models we wish to use (i.e. whether to include a constant, a constant and linear trend, or neither in the test regression).

Step 5: Finally, we have to specify the number of lagged dependent variables to be included in the model in order to correct for the presence of serial correlation . (For the PP test we specify the lag truncation to compute the Newey- West heteroskedasticity and autocorrelation (HAC) consistent estimate of the spectrum at zero frequency).

Step 6: Having specified these options, click <OK>: to carry out the test. EViews reports the test statistic together with the estimated test regression.

Step 7: We reject the null hypothesis of a unit root against the alternative if the ADF statistic is less than the critical value, and we conclude that the series is stationary.

References

1: Applied Econometrics

(Dimitrios Asterius and stephen)

2: Basic econometrics

(Damodar N. Gujarati)

3:http://economics.about.com/od/economicsglossary/g/unitroottest.htm4: http://en.wikipedia.org/wiki/Phillips%E2%80%93Perron_test.htm