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

Question

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

Answer 1

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.