# Stationarity of time series data

The following are plots of

1. the original data
2. first order difference values of original data and
3. the first order difference of the log transformed data.

Can someone please tell me which of 2 and 3 can be considered stationary or can you suggest a remedy? Raw data can be downloaded here.

• have you tried to apply statistical tests for unit root like, e.g., the Dickey-Fuller test? – utobi Jun 26 '15 at 16:33
• For the first order difference, p value<0.01, DF stat= -5.7411 and for log(first order difference), pvalue<0.01, DF stat= -5.9478. Which sounds better? But the variance in both does not remain constant? – user80685 Jun 26 '15 at 17:29
• DF may not be enough; consider other unit root tests (there are plenty available in R) and apply them also to the second order difference as the series may be integrated of order > 1. As about the heteroschedasticity that's another matter. For this you may consider fitting GARCH-type models and looking at their residuals. – utobi Jun 26 '15 at 19:47

## 1 Answer

A good way of checking if data is stationary or not is using Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test. Actually the KPSS tests for the null hypothesis that x is level or trend stationary.

see this paper: D. Kwiatkowski, P. C. B. Phillips, P. Schmidt, and Y. Shin (1992): Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root. Journal of Econometrics 54, 159–178.

in R you need to have package tseries and you can use kpss.test() function:

kpss.test(x, null = c("Level", "Trend"), lshort = TRUE)