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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?

Richard Hardy
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Christian Hennig
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1 Answers1

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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.

Richard Hardy
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  • 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.) – Christian Hennig May 17 '20 at 23:20
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    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.) – Christian Hennig May 17 '20 at 23:22
  • @Lewian, these points are similar to what you already have included in the OP. And as I said, I agree with it/them! – Richard Hardy May 18 '20 at 05:33
  • But can you shed any light on why the DF test is used the opposite way than other model misspecification tests, in which the model of interest (which we may want to assume) is the H0? Of course a test can neither prove the H0 nor any specific alternative, but arguably inferring any specific alternative from rejection of the H0 is even worse than inferring the H0 from non-rejection; at least the H0 (in many tests) has a point shape, so something specific is tested, and we are entitled to say "data are compatible with the H0". Can anything like this be stated for the stationary DF-alternative? – Christian Hennig May 18 '20 at 20:54
  • @Lewian, I don't know why the $H_0$ and $H_1$ are "reversed" in the DF test. It might be due to mathematical convenience. I would say, however, that for me the "reversal" makes sense. I would feel more comfortable after having rejected the $H_0$ of a unit root at 5% level than not having rejected an AR process without a unit root at 5% level; there is in a sense more "power" in the first scenario. – Richard Hardy May 19 '20 at 05:47
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    @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. – Richard Hardy May 19 '20 at 05:51
  • Your answer is clear and I agree with it, however it didn't tell me much I wouldn't have known before. Particularly there is no reference to the reversal of standard goodness-of-fit logic in it. I will upvote and/or accept it if you include the stuff you responded to me on May 19, expanding a bit on why the unit root is "probably more pernicious" than other violations of stationarity. – Christian Hennig May 24 '20 at 14:56
  • The only way to assess stationarity is through a unit root test, right? In other words, the existence of a unit root is the same as non stationarity, and vice versa. Right or wrong? – adrCoder Dec 02 '20 at 12:47
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    @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. – Richard Hardy Dec 02 '20 at 13:18
  • @RichardHardy can you please tell me or give me to some references on how you can have non stationarity without a unit root? Because I thought these two were the same thing. – adrCoder Dec 02 '20 at 13:28
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    @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. – Richard Hardy Dec 02 '20 at 13:53
  • @adrCoder: see here an example of non-stationarity without unit root: https://stats.stackexchange.com/q/541428/198058 – ColorStatistics Sep 15 '21 at 11:34
  • @RichardHardy: Wouldn't it be better to say "It is testing presence of a unit root against a specific alternative, an I(0) process." instead of "It is testing presence of a unit root against a specific alternative, a stationary process." – ColorStatistics Sep 15 '21 at 11:46
  • @ColorStatistics, thanks for a good comment. Yes, technically that would be the more precise term. In practice, the difference between the two is rarely of concern. – Richard Hardy Sep 16 '21 at 08:14
  • @RichardHardy: This is something I am trying to understand. What are the consequences of fitting an ARIMA model to an I(0) with non-constant variance? Let's take this one https://stats.stackexchange.com/q/541428/198058 It is I(0) but not not stationary. Can I fit an ARIMA model to it although its variance increases over time? What about an I(0) series that has ARCH-looking nonconstant variance? Can I fit an ARIMA model to that one as well? Let me know if you prefer I post this as a new question. Or if you'd rather point me to a book, I am happy to look wherever this may be well explained. – ColorStatistics Sep 16 '21 at 20:05
  • @ColorStatistics, I think you could post this as a new question. – Richard Hardy Sep 17 '21 at 06:27
  • @RichardHardy: I just did https://stats.stackexchange.com/q/545016/198058 Thank you in advance, Richard. – ColorStatistics Sep 17 '21 at 14:36