CHAPTER 5 EMPIRICAL METHODOLOGY AND MODEL SPECIFICATION
6.3 Tests for Stationarity: Based on GDP Deflator Index 102
6.3.3 Dickey Fuller (DF) Test for Stationarity: Based on GDP Deflator 105
Thus, in this no constant, no trend scenario, the 1, 5 and 10 per cent critical values, in absolute terms, exceed the computed tau (τ ) value -.502. The estimated coefficient of PPPt-1
(lagged value of PPP), δ, is not statistically significantly different from zero. Thus, we fail to reject the null hypothesis that δ = 0 that implies that the process is rather non-stationary.
In the second scenario, in which there is a constant but no trend, the 1, 5 and 10 percent critical values, in absolute terms, exceed the computed tau (τ ) value -.2.236.
∆ PPPt = .002 -.209PPPt-1 + εt
The estimated coefficient of PPPt-1 (lagged value of PPP), which in this case is δ, is not statistically significantly different from 0.Thus, the null hypothesis is not rejected because of the indication of the presence of non-stationarity.
The equation in the index form has been differenced twice, in order to explore the presence of a differenced stationary. In the same scenario, in which there is a constant but no trend, the 1, 5 and 10 percent critical values, in absolute terms, exceed the computed tau (τ ) value
-2,229.
∆ PPPt = .002 -.209PPP t-1 + εt
The estimated coefficient of PPP t-1 (lagged value of PPP), δ, is not statistically significantly different from 0. Thus, the null hypothesis, once again, is not rejected because of the indication of the presence of non-stationarity.
In the third scenario in which there is a constant and trend, the 1, 5 and 10 percent critical values, in absolute terms, exceed the computed tau (τ ) value -2.011.
∆ PPPt = .002 – 2.10E-06t -.221PPP t-1 + εt
The estimated coefficient of PPP t-1 (lagged value of PPP), δ, is not statistically significantly different from zero, as a result of which the null hypothesis is not rejected.
Comparing the informal and formal tests for stationarity of the PPP, the realisations or outcomes are similar. In the index form of variables, the non-stationarity nature of the PPP has been depicted by graphical method as well as by the residual values in the sample
correlogram. However, the unit root test has shown that PPP is not a difference stationary.
Similar results were obtained in the earlier section when the calculation was based on equilibrium exchange rates.
PPP, as it has been the case in level form, may be a trend stationary rather than difference stationary. In order to avoid over differencing i.e. treating a trend-stationary process (TSP) as difference-stationary process, it is imperative that the time series in regressed on time. The time series is regressed on time in order to examine whether the residuals from this regression are stationary i.e. trend stationary. Accordingly, the regression runs as follows:
PPPt= α+ α2 t + εt
- where PPPt is the time series under study and where t is the trend variable measured chronologically.
- εˆt = (PPPt – βˆ1– βˆ2 t )
∆εˆt = δεˆt-1
= -.227εˆt-1
In the above regression, the computed tau (τ ) value -2.111 exceeds, in absolute terms, the 5 and 10 per cent critical values (-1.95 and -1.61) respectively . The estimated coefficient of PPPt-1 (lagged value of PPP), δ, is statistically significantly different from zero. Thus, the null hypothesis that implies that the process is rather not stationary ( δ = 0) is rejected.
The regression result has shown that PPPt is a trend stationary. Thus, εˆt is a (linearly) detrended time series. It can be concluded that the informal and formal tests for stationarity of the PPP have similar outcome realisations.
6.3.3.2 Dickey Fuller (DF) Test for Productivity
Following similar procedure, equations for the productivity variable are formulated for the three versions. To proceed to estimation and interpretation, the first difference of Productivity, (DIFPROD) is regressed on its lagged value (LGPROD) and it has provided the following result:
∆ Prodt = -.001 Prod t-1 +εt
The null and alternative hypotheses applied earlier on the PPP are also used in treating the productivity variable in the three scenarios i.e. a pure random, random walk with drift, and a random walk with deterministic trend. Thus, in this no constant, no trend scenario, the 1, 5 and 10 per cent critical values, in absolute terms, exceed the computed tau (τ ) value -.140.
The estimated coefficient of PPP t-1 (lagged value of PPP), δ, is not statistically significantly different from zero. Thus, the null hypothesis that implies that the process is rather non- stationary is rejected.
In the second scenario, in which there is a constant but no trend, the 1, 5 and 10 percent critical values, in absolute terms, exceed the computed tau (τ ) value -.2.016.
∆ Prodt = .184 -.191 Prodt-1 +εt
The estimated coefficient of Prodt-1 (lagged value of Productivity), δ, is not statistically significantly different from 0. Thus, the null hypothesis is not rejected because of the indication of the presence of non-stationarity.
In order to explore the presence of a differenced stationary, the equation in the index form, as it was the case in the equilibrium exchange rate in level form, has been differenced twice.
The 1, 5 and 10 percent critical values, in absolute terms, exceed the computed tau (τ ) value - 2.066.
∆2 Prod t = .0195 -.201Prod t-2 +εt
The estimated coefficient of Prod t-1 (lagged value of Productivity), δ, is not statistically significantly different from 0. Thus, the null hypothesis is not rejected because of the indication of the presence of non-stationarity.
In the third scenario, in which there is a constant and trend, the 1, 5 and 10 percent critical values, in absolute terms, exceed the computed tau (τ ) value -1.657.
∆ Prod t = .174- .001t -.167Prod t-1 +εt
The estimated coefficient of PPP t-1 (lagged value of Productivity), δ, is not statistically significantly different from zero, as a result of which the null hypothesis is not rejected.
When comparing the informal and formal tests for stationarity of the productivity variable in the index form, it can be seen that the realisations or outcomes are similar to that of the level form. In the index form of variables, the non-stationarity nature of the productivity variable has been depicted graphically and by sample correlogram. However, the unit root test has shown that productivity is not a difference stationary.
Nevertheless, it is possible that productivity is a trend stationary rather than difference stationary. In order to avoid over-differencing i.e. treating a trend-stationary process (TSP) as difference-stationary process, the procedure applied earlier on PPP has been employed here as well. The time series has been regressed on time, and the residuals saved from this regression have been tested to determine their (non) stationarity. Accordingly, the regression runs as follows:
Prod t= α + α2 t + εt
- where Prodt is the time series under study and where t is the trend variable measured chronologically.
- εˆt = (Prodt – βˆ1– βˆ2 t ).
∆εˆt = δεˆt-1
= -.178 εˆt-1
In the above regression, the computed tau (τ ) value -1.782 exceeds, in absolute terms, the 10 per cent critical value (-1.61). The estimated coefficient of Prodt-1 (lagged value of productivity), δ, is statistically significantly different from zero, implying that the process is rather stationary. Thus, the null hypothesis δ = 0 is rejected. Thus, εˆt is a (linearly) detrended time series. It can be concluded that, similar to the circumstances in PPP, the informal and formal tests for stationarity of the productivity have similar outcome realisations.
6.3.4 Engle Granger (EG) Test for Cointegration : Based on GDP Deflator index