A SIMPLE USE TO UNIT ROOT TEST AND ITS APPLICATION USING EVIEWS · Definition of Unit root test · Stationary...

A SIMPLE USE TO UNIT ROOT TEST AND ITS APPLICATION USING EVIEWS

·

**Definition of Unit root test**

**· Stationary and Non stationary**

**· Unit root test statistics**

**· Importance of unit root test**

**· Application in Eviews using DGP model**

**The unit root test is one way to ascertain the stationarity of a data series. It test whether a time series variable is non stationary and possesses unit root. Unit root test enable us to ascertain the order of integration of a series.**

**Recent advances in econometrics have however shown that oftentimes economic data (mostly time series data) are not as well behaved. Consequently, different data series may not exhibit the same characteristics. It is thus possible to have some data series that exhibit the characteristics of diverging away from their mean with the passage of time, while others may converge to the mean overtime.**

**Stationary and Non Stationary**

**Covariance stationarity of a variable (Z) implies that overtime, Z has ;**

**-Constant mean**

**-Constant variance**

**-Covariance between different observation do not depend on time (t), only on the distance on lag between them (j)**

**Cov[Z**

_{t}, Z_{t+j}]=Cov[Z_{s}, Z_{s+j}]

**A time series with the characteristics is known as weakly or covariance stationary. This means the order of integration is at level or I(0).**

**Z is said to be non stationary if any of this condition do not hold. We can say that data series that diverge from their mean are said to be non stationary. A series is said to be non stationary when it possesses a unit root. The divergence away can be in one or two possible directions; an upward direction and a downward direction, in any case of which the data series is said to exhibit a trend. To utilize such data in any meaningful regression analysis, then we proceed by purging the trend. This is more technically referred to as detrending of the data series.**

**Detrending of data can be accomplished using one of two possible ways;**

**· Incorporating an explicit trend variable in the model specified. A process with deterministic trend is stationary but not a unit root and it is called Trend stationary process (TSP). In TSP, stationary is achieve via: the explicit inclusion of trend variable. For example**

**Z**

_{t}= a_{0}+a_{1}t+U_{t }[1]

**t is a time trend and U**

_{t}is the error term. After running a regression, we obtain.

**Ũ=Z**

_{t}- b_{0}– b_{1}t [2]

**Ũ represent the detrended Z**

_{t}.

**· Taking successive difference of the variable in the model such that, we then run our regression using data on the variables in the specification not in their level form, but in the their first difference form. This is called Difference stationary process (DSP).**

**A non stationary process that can be differences to achieve stationary is called INTEGRATED NON STATIONARY SERIES. The number of times the series need to be differenced to achieve stationarity on the other hand defines the order of integrated series. If a series is difference once, i.e I(1), it is integrated of order 1. If it is achieved at differencing twice, it is integrated of order 2 or I(2).**

**Where the problem come from is that most time series economic data are non stationary and if we run a non stationary data in a regression, we get a result called SPURIOUS Regression.**

**UNIT ROOT TEST**

**1. Dickey Fuller(DF) Test**

**Dickey and Fuller have proposed the use of a likelihood ratio (LR) test for unit root. This is basis of the so called Dickey Fuller test**

**- Intercept: ∆Z**

_{t}= µ+ b Z_{t }+e_{t}[3]

**-intercept and time trend: ∆Z**

_{t}= µ+ bZ_{t }+ ɑ t +e_{t}[4]

**The null hypothesis Ho: b=0; Z**

_{t}has a unit root. Rejecting the unit root test implies the Z_{t}series is stationary.

**Interpretation:**

**If Z is integrated at I(0), then it is stationary at level. No differencing .**

**If Z is integrated at I(1), then its difference once to be stationary .**

**If Z is integrated at I(2), then it is the difference of the difference of Z will be stationary.**

**Note: DF test assumes that the errors are white noises. To get a better test we need to move beyond white noise disturbance and use different version of test allowing for higher order lag.**

**2. Augmented Dickey Fuller (ADF) Test**

**∆Z**

_{t}= µ +ɑ t + bZ_{t -1}+_{t-1}+e_{t}[5]

**The Augmented Dickey Fuller for unit root takes the form of adding the lagged value of a series. e**

_{t}is a pure white noise error, is the maximum length of lag dependent variable. The essence of the ADF is to improve the statistical fitness of the models. As with DF, ADF test if the coefficient Z_{t-1 }(b)≠ 0.

**The ADF statistics; used in the test, is a negative number. The more negative number it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence (Wikipedia)**

**Interpretation:**

**The H**

_{0}: b=0; Z has a unit root

**If the ADF t-statistics less than any and/or all the t-critical value, we accept the null hypothesis but if its greater than all critical value , we reject the null hypothesis. Note: Putting ADF t-stat and all critical value in absolute form.**

**/t-stat/</t-crit/, we do not reject H**

_{0}. Therefore, Z is non stationary.

**/t-stat/>/t-crit/, we reject the H**

_{0}. Therefore, Z is stationary.

**DF and ADF have been found to have low power in certain circumstances :**

**-it is difficult to distinguish between b=0.97 and b=1, especially in small sample.**

**- ADF has a low power in the case of trend stationary process. They fail to reject.**

**3. Phillips-Perron (PP) Test**

**Phillips-Perron correct for any serial correlation and heteroscedasticity in the errors by directly modifying the test statistics. In Phillips-Perron test, the lag lengthened not to be specify.**

**∆Z**

_{t}= µ^{*}_{t}+ð^{*}_{t }+ bZ_{t -1}+e_{t}[6]

**The PP test builds on the DF of null hypothesis. Like ADF, PP addresses the issue that process generating data for Z**

_{t}might have a higher order autocorrelation than is admitted in the equation ,making Z_{t-1 }exogenous and thus invalidating the DF test.

**Davison and Mackinon (2004) report that the Phillip-Perron test performs worse in finite sample than ADF test.**

**Interpretation:**

**The H**

_{0}:b=0; Z has a unit root

**If the PP t-statistics less than any and/or all the t-critical value, we accept the null hypothesis but if its greater than all critical value , we reject the null hypothesis. Note: putting PP t-stat and all critical value in absolute form.**

**/t-stat/</t-crit/, we do not reject H**

_{0}. Therefore, Z is non stationary.

**/t-stat/>/t-crit/, we reject the H**

_{0}. Therefore, Z is stationary.

**4. Kwiatkowski- Phillips- Schmidt-Shin(KPSS) Test**

**The KPSS test is a stationary test, the null hypothesis implies that Z**

_{t}is I(0).

**Unlike DF,ADF, and PP that are called unit root test, KPSS is a stationary test**

**Interpretation:**

**H**

_{0}: b=0, Z is trend stationary.

**H**

_{0}: Variance e_{t}=0. Therefore , mean is constant and Z is trend stationary .

**H**

_{1}: Variance e_{t}>0. Therefore , non stationary exist in Z series.

**If KPSS t-stat is greater than all critical value, we reject the H**

_{0}but if its less than all critical value we accept the H_{0}.

**t-stat>t-crit, we reject the H**

_{0}. Therefore, Z is non stationary

**t-stat <t-crit, we do not reject H**

_{0}. Therefore, Z is stationary.

**NOTE:**

**Checking the graph of a series is important before testing for unit root. The graph tells us the correct specification i.e whether to include intercept or trend and intercept or none in the test equation.**

**IMPORTANCE OF UNIT ROOT TEST**

**Whether a data series has unit root or not, it has a far reaching implications for economic analysis and policy formation and interpretation. Where data series has unit root b=1, any shock to the series is long lasting / permanent. Thus, there will be a cumulative divergence away from the mean or trend of the series. The instability in such series will tend to render any policy formulated and implemented on the basis of the model estimated using such series impotent. This is because underlying any policy formation and implementation is the implicit assumption of stability of the series on which the formulated policies are based.**

**If the hypothesis that b<1 is upheld, the effect of any shock to the series will fade away progressively overtime. This too has implications for any policy formulated on the basis of model estimated with such series. In the light of the forgoing, the critical point of interest in our analysis is the magnitude b. The closer b is to unity, the stronger chance that shocks to the series will be long lasting and vice versa. For most practical purposes therefore, what is important in our analysis is the size of b- the autoregressive parameter which bears critically on the duration and effect of a shock.**

**References**

**Davidson, Russell; Mackinnon, James,G. (2004).Econometric Theory and Methods. New york: Oxford University Press. P. 623.**

Iganinga, B.O. (2009). Introductory Econometrics: Made Easy. Anitop Books. P.151-158.

Iganinga, B.O. (2009). Introductory Econometrics: Made Easy. Anitop Books. P.151-158.

Phillips,P.C.B; Perrron, P.(1988). "Testing for Unit Root in Time Series Regression. Biometrika.75(2);345-346.

Phillips,P.C.B; Perrron, P.(1988). "Testing for Unit Root in Time Series Regression. Biometrika.75(2);345-346.

## COMMENTS