The KPSS and Phillips-Perron Tests

While probably the most popular, the ADF is not the only test to detect a unit root. In this lesson, we will see two alternatives that are also commonly used by academics and practitioners alike: the KPSS and the Phililps-Perron test.

KPSS test

The KPSS test (named after its proponents Kwiatkowski, Phillips, Schmidt, and Shin) flips the definitions of the null and alternative hypotheses. Contrary to the ADF, KPSS tests the null hypothesis that the series is stationary $I(0)$, against the alternative that it is not, $I(1)$.

The model for $y_t$ is slightly twisted: it is a combination of both a stationary process, $u_t$, and a random walk, $x_t$. Remember that a random walk is composed of a lagged observation of itself plus white noise, $\epsilon_t$. The KPSS test exploits this structure by testing whether or not the white noise component has a variance greater than 0, i.e., whether it actually exists.

Formally, the hypothesis test is $H_0: \sigma_{\epsilon}^2=0$vs. $H_1 : \sigma_{\epsilon}^2>0$. Again, just for the sake of argument, let’s say we are comfortable assuming a 95% confidence level. Then, we can interpret the KPSS test as follows: