The ADF and PP unit root tests are for the null hypothesis that a time series Yt is I(1). Stationarity tests, on the other hand, are for the null that Yt is I(0). The most commonly used stationarity test, the KPSS test, is due to Kwiatkowski, Phillips, Schmidt and Shin (1992) (KPSS). They derive their test by starting with the model

where ut is I(0) and may be heteroskedastic.

The null hypothesis that Yt is I(0) is formulated as H0: a2E = 0, which implies that ßt is a constant. Although not directly apparent, this null hypothesis also implies a unit moving average root in the ARMA representation of AYt.

The KPSS test statistic is the Lagrange multiplier (LM) or score statistic for testing a2 = 0 against the alternative that o > 0 and is given by

t

where St = Yl uj, ut is the residual of a regression Yt on t and A£.

j=i

Critical values from the asymptotic distributions must be obtained by simulation methods. The stationary test is a one-sided right-tailed test so that one rejects the null of stationarity at the a level if the KPSS test statistic is greater than the 100(1 — a) quantile from the appropriate asymptotic distribution.

4.4. Example: Purchasing Power Parity

It is very easy to perform unit root and stationarity tests in EViews. As an example, consider a Purchasing Power Parity condition between two countries: USA and UK. In efficient frictionless markets with internationally tradable goods, the law of one price should hold. That is,

where st is a natural logarithm of the spot exchange rate (price of a foreign currency in units of a domestic one), pt is a logarithm of the aggregate price index in the domestic country and is a log price in the foreign country. This condition is referred to as absolute purchasing power parity condition.

This condition is usually verified by testing for non-stationarity of the real exchange rate qt = st + pi — pt. Before we perform this let us look at properties of the constituent series.

We consider monthly data for USA and UK over the period from January 1989 to November 2008.

Although plots of both consumer price indices and the exchange rate indicate non-stationarity, we perform formal tests for unit root and stationarity.

In the dataset PPP.wf1, there are levels of the exchange rate and consumer price indices are given, so we need to create log series to carry out tests. This is done as usually,

Let us start with the UK consumer price index. We can find the option Unit Root Tests... in the View section of the series object menu (double click on lcpi_uk icon). In the Test Type box there is a number of tests available in EViews. We start with Augmented-Dickey-Fuller test. As we are interested in testing for unit roots in levels of log consumer price index, we choose Test for unit root in levels in the next combo-box, and finally we select testing with both intercept and trend as it is the most general case.

Figure 4.1: Augmented Dickey-Fuller test dialog window

EViews will also select the most appropriate number of lags of the residuals to be included in the regression using the selected criteria (it is possible to specify a number of lags manually is necessary by ticking User specified option).

Click OK and EViews produces the following output

Figure 4.2: Output for the Augmented Dickey-Fuller test

The absolute value of the t-statistic does not exceed any of the critical values given below so we cannot reject the null hypothesis of the presence of unit root in the series.

Unfortunately, EViews provides only the test of the null hypothesis H0: ( = 0. One can perform more general test by estimating Dickey-Fuller regression (4.2.6). In the command line type the following specification

to run the ADF regression with intercept and trend component. As you noted, the function @trend allows to include the time trend component that increases by one for each date in the workfile. The optional date argument 1989M01 is provided to indicate the starting date for the trend. We did not include any MA components in the regression since based on the previous results (see Figure ??) zero lag is optimal according to the Schwartz selection criterium.

The regression output is identical with that produced by the Augmented Dickey-Fuller test

Figure 4.3: Output of the regression-based procedure of the Augmented Dickey-Fuller test

However, the approach enables us to perform the Wald test of linear restrictions and specify the null hypothesis H0 : (ft 0) = (0 0) (or more general, H0 : (a ft 0) = (0 0 0)).

Figure 4.4: Wald test results for the Augmented Dickey-Fuller test specifications

The value of Wald test statistic in the case of the null H0: (ft 0) = (0 0) is 1.7648; this has to be compared with the critical values tabulated in MacKinnon (1996).

As the test statistic is smaller than all of the critical values, we cannot reject the null hypothesis, which confirms non-stationarity of the log consumer price index series.

This conclusion is also confirmed by other stationarity tests. For example, in Kwiatkowski-Phillips-Schmidt-Shin test (specification with the intercept and trend), the test statistic is 0.4257 which is higher than the critical value at 1% significance level (which is 0.216). Thus, we reject the null hypothesis of stationarity of the series.

If the Purchasing Power Parity condition holds one would expect the real exchange rate qt = st — pt + Pt to be stationary and mean reverting. The presence of unit root in the deviations series would indicate the existence of permanent shocks which do not disappear in a long run.

We create a series of deviations

Augmented Dickey-Fuller does not reject the null hypothesis of the presence of unit root in the deviations series. Also, the value of Wald test statistic is 1.8844 indicates which confirms nonstationarity of the deviations from the Purchasing Power Parity condition.

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