The Phillips-Perron Test

Before fitting many time-series models, or before trusting a regression between two price series as meaningful rather than spurious, a researcher needs to check whether the series has a unit root — a form of non-stationarity where shocks never fully decay, so the series wanders rather than reverting to a stable mean. The classic Dickey-Fuller test checks this by regressing a series' change on its own lagged level and testing whether that lag's coefficient is significantly negative, but the test's textbook critical values assume the regression's errors are uncorrelated and have constant variance — an assumption financial return series routinely violate.

The Phillips-Perron (PP) test runs the same basic regression but corrects the test statistic itself, rather than adding more lagged terms to the regression as the augmented Dickey-Fuller test does. It uses a nonparametric adjustment based on the estimated long-run variance of the residuals, which absorbs the effect of autocorrelation and heteroskedasticity directly into the correction, leaving the regression itself simple.

In practice, researchers often run both PP and the augmented Dickey-Fuller test side by side, since they can occasionally disagree on borderline series and agreement between the two gives more confidence in a stationarity conclusion.