Augmented Dickey Fuller


No cake for spunky
The ADF has three different tests. The first is random walk, the second random walk with drift, and the third a deterministic trend. For all three tests, If p > alpha non-Stationarity is indicated. The problem I have is that the first test indicates non-Stationarity and the 2nd and third tests indicate Stationarity.

One advice I found is that if any of the three tests show Stationarity you should conclude Stationarity. Another writer stated something pretty similar.. "In interpreting the output, one starts from the most general model of a constant (single mean) and trend and continues to the most specific (zero mean and no trend) stopping when one can reject the null hypothesis of a unit root [non-Stationarity] using the results of t and F type test...

by that definition my data is stationary because the single mean does strongly reject the null [my concern is that the zero mean test does not].

I also ran the Phillip Pheron test which has the same null but a different method. I get essentially identical results, the single mean and trend test strongly rejects the null, the zero mean does not.

Finally I ran the KPSS where the null is that the series is stationary (the fact that the null is the opposite of the other two tests has led some to recommend running them together to see if they agree). Sadly there are multiple suggestions to determine the kernel used in this test. Using one of them I rejected the null of Stationarity, in one I would not have rejected it, and in the third it was really close .p=.0482:)

The author comments...."Unlike the ADF and PP tests, the KPSS test is a test of stationarity with the null being that the series is stationary (i.e. I(0)). To this extent the KPSS might serve as a complement to unit root tests where the null hypothesis – and thus the “benefit of the doubt” – is that the series is I(1). A rejection of the null hypothesis of stationarity in the KPSS test would then tend to corroborate a failure to reject the null hypothesis of a unit root in a ADF or PP test. However, power concerns means that one needs to be cautious in this interpretation."

So in the end I am unsure if my data is stationary or not :p Maybe its confused to...


No cake for spunky
One thing I did not know is KPSS and ADF apparently have different definitions of stationarity. ADF has linear or difference stationarity and KPSS has trend stationarity. So for ARIMA, in deciding which to use which test would you chose ADF or KPSS in regards to deciding if you need to difference.