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I did a quick Dickey-Fuller test, to test whether my time series is stationary, and the result gives a P value of 9.6*10^-15, or in other words stationary. How is this possible, my data clearly doesn't seem sttionary. The mean seems to be constant(to an extent), no seasonality issue but it's the variance?

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  • $\begingroup$ you say "not augmented" but use the augmented DF test, what did you mean by this? $\endgroup$
    – jros
    Feb 16, 2022 at 12:57
  • $\begingroup$ Yes sorry I just realized, I did do the augmented DF test, but still, how does the test say that it's stationary, although it clearly isn't? $\endgroup$
    – sir.edward
    Feb 16, 2022 at 13:01
  • $\begingroup$ @jros, if you look at my question from yesterday, asking whether my time series is stationary or not, for that time series I did a ADF test as well, and that one was considered non-stationary, but from what I see, that one would seem more stationary than this one. $\endgroup$
    – sir.edward
    Feb 16, 2022 at 13:03
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    $\begingroup$ Related: stats.stackexchange.com/questions/564467 $\endgroup$ Feb 16, 2022 at 13:43

2 Answers 2

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The ADF test has a null hypothesis of presence of a unit root. Clearly, your series does not appear to have a unit root, and the ADF correctly rejects $H_0$. However, absence of a unit root does not imply stationarity. Consequently, rejection of $H_0$ with an ADF test does not imply stationarity. How so?

Absence of a unit root is a necessary but not a sufficient condition for stationarity. Time-varying variance is one characteristic of nonstationarity, and it applies to your case. The ADF test is not designed to capture that, however, so no wonder the results are what they are.

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    $\begingroup$ ah yes that makes sense, stationary requires the absence of unit root, but the absence of unit root is not necessarily stationary. $\endgroup$
    – sir.edward
    Feb 16, 2022 at 13:59
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I would agree you have visual evidence the series as non-stationary variance, and would suggest the unit root test that drives the ADF test does not detect this. While the ADF is great for testing trend stationarity, you can justify using your own discretion here on whether the variance is stationary.

There is also the possibility your version of the ADF test doesnt detect variance stationarity. From the ADF Wikipedia page:

The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend-stationarity.

There is a good Stack Overflow: Cross Validated post about the versions of the ADF test

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  • $\begingroup$ There is also the possibility your version of the ADF test doesnt detect variance stationarity. Take a look at my answer for why ADF is not even supposed to do that. Are you aware of a version of ADF test that does? $\endgroup$ Feb 16, 2022 at 14:23
  • $\begingroup$ @RichardHardy I understand you're saying the unit root test is not designed to evaluate stationarity of variance, but can you educate me on what the "versions" of the ADF test that are capable of doing so are doing? Or are you refuting that there are versions of the ADF capable of doing so? $\endgroup$
    – jros
    Feb 16, 2022 at 15:10
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    $\begingroup$ I have never seen such a version, but your answer seems to suggest it exists. This is why I posed the question. $\endgroup$ Feb 16, 2022 at 15:42

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