# 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. – tagoma Feb 4 '13 at 21:39

## 1 Answer

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.

• nice explanation! – Matt Feb 5 '13 at 3:45
• So, if I understand Ryogi correctly if the data is splitted in smaller parts, in general the test will reject the hyphothesis right? But in my case, just the opposite is the case. I have read in Forums, that Hyphothesis test are sensitve to small variations from the process and in large data sets they tend to reject the hypothesis. So I tested the following in R: " x<-rnorm(714000,mean=100,sd=9);kpss.test(x) KPSS Test for Level Stationarity data: x KPSS Level = 0.5604, Truncation lag parameter = 194, p-value = 0.02807 " as we can see, the kpss rejects the data to be stationarity. But if I spli – jcc Feb 5 '13 at 10:09
• On splitted data the tests will not reject the hypothesis, as long as the sample size is not too small. – Ryogi Feb 5 '13 at 13:46
• I guess 35400 for one sample part is not too small. Ryogi, do you know any reference where my problem is described in detail? Because I look already for one or two weeks and needed for my thesis...Thanks already for your help. – jcc Feb 5 '13 at 15:02
• All you need is to spend more time getting familiar with the definition of stationary process with some examples and counterexamples. It is standard material in any time series analysis book. – Ryogi Feb 5 '13 at 16:04