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The time series is used in a regression (OLS) and then the diagnostics are been run.

Or does stationarity imply homoskedasticity in all cases?

I get heteroskedasticity through a breusch pagan test but I get stationarity from a Unit Root (Dickey Fuller Test). Is this possible? Are these two mutually exclusive?

adrCoder
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  • This question would make some sense if the word "regression" didn't appear in the title. But with it there, we need to know something about how you conceive of regression as applying to concepts of stationarity and heteroscedasticity in time series. – whuber Dec 01 '20 at 17:32
  • You run a regression and then do the diagnostics. Which part of my question is not clear? – adrCoder Dec 01 '20 at 17:34
  • Running a regression and doing diagnostics ordinarily have nothing to do with time series; and the concept of "stationarity" simply doesn't apply to regression. I'm guessing you might be trying to ask something about the Durbin-Watson test, but that's purely a guess. – whuber Dec 01 '20 at 17:35
  • Yes I get heteroskedasticity through a Durbin Watson test but I get stationarity from a Unit Root (Dickey Fuller Test) – adrCoder Dec 01 '20 at 17:37
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    The Durbin-Watson test has nothing to do with heteroskedasticity. It is for first-order autocorrelation. – Richard Hardy Dec 01 '20 at 18:16
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    I meant the breusch pagan test. apologies. – adrCoder Dec 01 '20 at 19:42

1 Answers1

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Answering to the general question in the title of this post:

Assuming that by "stationarity" the OP means "second-order", "covariance-", "weak-" stationarity, i.e. the stationarity concept that relates to a constant mean and variance throughout a stochastic process, then the answer depends on whether the heteroskedasticity is conditional or unconditional.

If it is unconditional, it is incompatible with covariance-stationarity, because it negates it directly.

If it is conditional on certain co-variates, then, if these covariates are themselves stationary of the appropriate order (depending on how they enter the heteroskedastic function), the unconditional variance will be constant, and so conditional heteroskedasticity given such covariates is compatible with second-order stationarity.

But the above are relations at population-level.

Statistical tests are based on finite samples, and so the phenomenon where two statistical tests may provide conflicting indications is nothing new.

Alecos Papadopoulos
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  • so the breusch pagan can give heteroskedasticity but the augmented dickey fuller stationarity. it is possible, right? what type of heteroskedasticity and stationarity do these tests show from the ones you mention? – adrCoder Dec 01 '20 at 19:43
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    @adrCoder Yes it is. – Alecos Papadopoulos Dec 01 '20 at 19:44
  • excellent answer thank you. if you could tell me what types of stationarity and heteroskedasticity these tests get from the ones you mentioned, I would be grateful. – adrCoder Dec 01 '20 at 19:50
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    @adrCoder The Breusch-Pagan test test for heteroskedasticity that depends on a selection of variables determined by the researcher. So it is always "conditional". Dickey Fuller tests for the existence of a unit root. If it exists, it makes the unconditional variance non-stationary. – Alecos Papadopoulos Dec 01 '20 at 20:00
  • thank you for clarifying this – adrCoder Dec 01 '20 at 20:15
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    A naive question: how can heteroskedasticity be unconditional? It seems that the very definition requires conditioning. E.g. our [tag definition](https://stats.stackexchange.com/tags/heteroscedasticity/info) says *Heteroscedasticity refers to the property of a random process that has non-constant variance along some continuum* where it seems the conditioning is done w.r.t. this continuum. – Richard Hardy Dec 01 '20 at 20:19
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    @RichardHardy The "continuum" part I don't really understand. "Heteroskedasticity" is a descriptive term, describing a situation where in a collection of a random variables, collected together for any reason, these variables have different variance. – Alecos Papadopoulos Dec 02 '20 at 02:55
  • @AlecosPapadopoulos do you have some references you can share where it is claimed that the results of the tests could be different due to the small sample size? – adrCoder Dec 02 '20 at 09:01
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    @adrCoder Nothing comes to mind, but this is a general and known issue: with finite samples, anything goes, because we never really know how actually well a sample represents the population (even if it is an i.i.d. sample). – Alecos Papadopoulos Dec 02 '20 at 10:15
  • @AlecosPapadopoulos what type of stationarity does the Augmented Dickey Fuller test for? Weak? Second Order? First Order? – adrCoder Dec 02 '20 at 10:21
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    @adrCoder This has already been answered in a previous comment. Please pay attention. – Alecos Papadopoulos Dec 02 '20 at 12:20
  • Ok I read your previous comment. So my understanding is that I might get an indication of stationarity wrt to the unconditional variance (Augmented Dickey Fuller), but the conditional heteroskedasticity (Breusch Pagan) may indicate that the conditional variance is not constant across time. Right? – adrCoder Dec 02 '20 at 15:19
  • @AlecosPapadopoulos, I posted [a question](https://stats.stackexchange.com/questions/499018) asking for clarification of this mysterious *continuuum* and [another one](https://stats.stackexchange.com/questions/499046) that you have already answered briefly in your comment. Perhaps you will be interested. – Richard Hardy Dec 02 '20 at 19:14
  • @AlecosPapadopoulos: +1 Excellent answer. I just posted a question that builds heavily on your answer. I'd love to hear your take on it. Do you have a reference fore "if all series are stationary then even if there conditional heteroskedasticity there is no unconditional heteroskedasticity"? – ColorStatistics Sep 24 '21 at 19:13