# ADF-Test indicates stationarity for a non-stationary time series

I have a minor issue and am not sure what to do. The link below leads to an image of two time series I plotted, the upper being the original, the bottom one obtained by taking the first differences.

While neither of both is stationary, at least that's what I think (correct me if I'm wrong), performing an ADF test to test for stationarity only indicates non-stationarity for the original time series, but not for the first differences one.

Below you can find the ADF-Test outputs for each of the series:

Results of Dickey-Fuller Test (Original Time Series):
Test Statistic                 -0.991032
p-value                         0.756521
Lags Used                      7.000000
Number of Observations Used    48.000000
Critical Value (1%)            -3.574589
Critical Value (5%)            -2.923954
Critical Value (10%)           -2.600039

Results of Dickey-Fuller Test (First Differences):
Test Statistic                 -3.316947
p-value                         0.014138
#Lags Used                      6.000000
Number of Observations Used    49.000000
Critical Value (1%)            -3.571472
Critical Value (5%)            -2.922629
Critical Value (10%)           -2.599336


Am I missing something here? Why does the second test indicate stationarity at the 5% level?

Thanks a lot!

Cheers, IG

Original time series and first differences

• What do you think about my answer? If it is clear, you may accept it by clicking on the tick mark to the left. Otherwise you may ask for further clarification. This is how Cross Validated works. May 5 '20 at 12:33

## 1 Answer

The ADF test is not a test of nonstationarity in general, but of a very specific kind of nonstationarity, namely, presence of a unit root. Thus it cannot indicate stationarity in general, only lack of a unit root.

Judging from the graph, the second series clearly does not have a unit root, and the test statistics shows that. The first series does not look entirely like one with a unit root either, but at least it has some features of it, and the test statistic apparently picks those features up.