Other Unit Root Tests

Two other unit root tests are commonly used, in addition to or instead of the Augmented Dickey-Fuller Test, namely:

• Phillips-Perron (PP) test
• Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test

While the ADF test uses a parametric autoregression to estimate the errors, the PP test uses a non-parametric approach.

The KPSS test uses yet a different approach. Unlike the other tests, the null hypothesis for the KPSS test is that the time series is stationary, while the alternative hypothesis is that there is a unit root.

Real Statistics Functions: The Real Statistics Resource Pack provides the following array functions where R1 contains a column of time series data.

PPTEST(R1, lab, lags, type, alpha) – an array function that returns a column range for the PP test consisting of tau-stat, tau-crit, stationary (yes/no), lags and the autocorrelation coefficient and p-value.

KPSSTEST(R1, lab, lags, type, alpha) – an array function that returns a column range for the KPSS test consisting of test-stat, crit-value, stationary (yes/no), lags and p-value.

As usual, if lab = TRUE (default is FALSE), the output consists of two columns whose first column contains labels. type = the test type (0, 1, 2, default is 1). The default value for alpha is .05.

To specify the test type, you can use “” or “none” instead of 0, you can use “drift” or “constant” instead of 1 and you can use “trend” or “both” instead of 2.

Note too that the KPSS test does not support the case where there is no constant and no trend. Thus, type for KPSSTEST is restricted to 1 and 2. If type = 0 is used, then it is assumed that type = 1.

You can either specify the number of lags to test or use the values “short” or “long”. If “short” is specified then lags is calculated to be =Round(4*(n/100)^.25,0) where n = the number of elements in the time series, while if lags = “long” then the value =Round(12*(n/100)^.25,0) is used.

In Figure 1, we repeat the analysis for Example 1 of Augmented Dickey-Fuller Test using the PP and KPSS tests, specifying lags = “short” (which is equivalent to lags = 3).

Figure 1 – PP and KPSS Tests