# Testing for stationarity in large sample sizes

I keep struggling with testing 9 samples if they are stationary. Each of these samples is a real valued time series with 714.000 values. If I use the KPSS test with the each compleete sample set, the hypothesis is rejected. But, if I split each sample in 20 parts of equal size, then test all these sample parts with KPSS in all most all cases KPSS accepts the sample parts to be stationary.

But, I cannot find any good explanation for this behaviour. Can you give me any explanation for it and maybe a reference?

• Hasn't that just to do with the very process of churning out smaller series from largers series? I mean I am not surprised that happens. Feb 4, 2013 at 21:39

Here is a possible explanation. Consider $X_t \sim N(0,1)$ and $Y_t \sim N(1,1)$. Then $(X_t)_0^n$ and $(Y_t)_0^n$ are realizations from stationary time series and I would expect the null hypothesis of stationarity not to be rejected (compatibly with the size of your test). Instead, the sample $(Z_t)_1^{2n} = (X_1, \dots, X_n, Y_1, \dots, Y_n)$ is drawn from a non-stationary process (the mean is not constant) and a test with enough power will in general reject the null of stationarity.