I recently came across the Dickey-Fuller test for existence of a unit root in an AR(1) series, definition on Wikipedia. If a unit root exists, the series is not stationary. Fine by me.

Now looking at some applications and interpretations of the Dickey-Fuller test, apparently people say that if the null hypothesis is rejected, there is evidence that the process is stationary. More so, this "logic" is apparently still applied in case time series are obviously seasonal or other time-dependent things go on. I realise that there also is an augmented Dickey-Fuller test that allows to detect unit roots for some more sophisticated models, but anyway...

The thing that bothers me is the following. Stationarity is a standard model assumption in time series analysis. It's quite restrictive in my view, any time-dependent pattern is not allowed. Normally when testing model assumptions (e.g., normality, independence...), the restrictive model assumption is the null hypothesis and the data can reject it or not, but we will never have evidence in favour of the model assumption, as this is an idealisation, will not hold precisely, and we can be happy enough if it's just not obviously incompatible with the data.

For the Dickey-Fuller test it's apparently the opposite. Stationarity is the alternative, rejecting the unit root amounts to rejecting non-stationarity, or, in other words, to observe more or less strong evidence for stationarity. This seems to be a misinterpretation to me, because there are lots and lots of possibilities to have non-stationary series that do not fulfill the Dickey-Fuller unit root model (seasonal series to start with), and may therefore lead to rejection of the unit root model. So this doesn't seem to provide positive evidence in favour of stationarity at all; the only thing is that one specific form of non-stationarity is ruled out.

Am I misunderstanding something, or is it indeed the case that rejection of a unit root is pervasively misinterpreted?


You are not misunderstanding anything as far as I can tell. You are presenting a solid story with logical supporting arguments. I could stop here and I think your question would be answered, but let me add some other comments that might be relevant.

The (augmented) Dickey-Fuller test is based on an autoregressive model for the time series of interest. It is testing presence of a unit root against a specific alternative, a stationary process. The universe of cases explicitly considered is restricted to these two (both being autoregressive processes). They are used for deriving the distribution of the test statistic under $H_0$ and probably for examining the test's power against the specific alternative. (It has been a while since I read the original paper, so I do not remember the details; feel free to correct me.)

In reality, not all time series are autoregressive with constant parameters and all the other nice features we tend to assume about them. You could very well argue none of the real-world time series follow any of the relatively simple models that we use. So the test should be understood as a simplification.

Moreover, perfect stationarity is not necessary in practice. Approximate stationarity is good enough to get approximately correct results from models and tests that rely on the assumption of stationarity. Even though we know that all models are wrong, we still find some of them useful. I suggest using and interpreting the (A)DF test in this perspective.

Responding to comments:

arguably inferring any specific alternative from rejection of the $H_0$ is even worse than inferring the $H_0$ from non-rejection

You can test the assumptions of the test. If there are no violations, a rejection of $H_0$ typically points to a specific alternative by construction of the test statistic. If so, inferring $H_1$ need not be so problematic, since the data contains something characteristic of $H_1$; otherwise the test statistic would not pick it up.

at least the $H_0$ (in many tests) has a point shape, so something specific is tested, and we are entitled to say "data are compatible with the $H_0$". Can anything like this be stated for the stationary DF-alternative?

$H_0$ as considered in the DF test contains the single undersirable parameter value (the root being equal to unity); $H_1$ contains all the desirable alternatives (the root being less than unity; keep in mind that values in the negative territory are usually irrelevant in practice). The possibility of an explosive process (the root above unity) is ruled out a priori. (Though there are versions of the test which have explosive process as the alternative, ruling out lower-than-unit roots a priori.) Thus it is quite satisfactory to have the test set up as is, targeting the single undersirable outcome and (hopefully) rejecting it with a high degree of confidence.

In other words, the DF test targets a specific violation of stationarity, one that is probably more pernicious than others when it comes to messing up estimator's properties and inference. I think one incurs smallers losses by neglecting, say, a shift in variance than a unit root. The former only makes estimators inefficient, while the latter makes unconditional moments undefined/infinite etc. So a test that allows testing the $H_0$ of a unit root and (hopefully) rejecting it at a low significance level makes sense.

  • $\begingroup$ Fair enough, I don't disagree with this. Obviously if we'd demand our models to hold precisely, we couldn't do anything in statistics. However I have two issues. 1) Despite being able to use the test in this way, we shouldn't talk as if we had proved something which in fact we haven't proved. I have seen intelligent non-statisticians in despair about statistics because they couldn't understand some things that statisticians said in this manner that to them obviously couldn't be true (and in fact weren't). (2 in another comment.) $\endgroup$ – Lewian May 17 '20 at 23:20
  • $\begingroup$ 2) It seems to me that rejecting the DF test leaves quite some possibility for the series to be dangerously non-stationary in the sense of having the potential to seriously mislead inference from a stationarity assumption. (No examples here, I just don't see that very much is ruled out by the test really.) $\endgroup$ – Lewian May 17 '20 at 23:22
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    $\begingroup$ @Lewian, as to point shapes and such: in the DF test, a specific violation of stationarity is being tested, one that is probably more pernicious than others when it comes to messing up estimator's properties and inference. I think you incur less losses by neglecting, say, a shift in variance than a unit root. So it makes sense to try rejecting $H_0$ of a unit at a low significance level; at least for me this would give some confidence in the results of the analysis that follows. $\endgroup$ – Richard Hardy May 19 '20 at 5:51
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    $\begingroup$ @adrCoder, wrong. Lack of stationarity can arise from a variety of reasons, only one of them being a unit root. The same holds for testing. $\endgroup$ – Richard Hardy Dec 2 '20 at 13:18
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    $\begingroup$ @adrCoder, if you followed my comment to your question and looked at the tag descriptions, you should be able to see a difference. A simple example: a series without a unit root but with a deterministic trend or unconditional heteroskedasticity is nonstationary. $\endgroup$ – Richard Hardy Dec 2 '20 at 13:53

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