We know that the definition of stationarity (either weak or strong) of a random time series involves having the same joint distribution or statistic (like mean or variance) for "any" set of time points, extending to infinity.

Since we can only observe finite series in reality, how can any stationarity test work? Let me rephrase: it seems to me that many seemingly non-stationary time series we acquire may be generated by a doubly-stochastic time series where the parameters that describe the series are coming from a stationary random process that changes over an arbitrarily larger time-scale. Then how do the stationarity tests work from this perspective?

  • $\begingroup$ I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? $\endgroup$ – Richard Hardy Feb 12 '17 at 12:49

Precisely by your argument

the definition of stationarity (either weak or strong) of a random time series involves having the same joint distribution or statistic (like mean or variance) for "any" set of time points

we should have the same joint distributions for small sets of points, too. Since one counterexample is enough to reject a statement that applies to all possible examples, if we reject for small sets we no longer care about any other sets (not rejecting for large sets will not change the conclusion, so why even inspect that).

In your example of doubly-stochastic process, you seem to have a hierarchy. There is a "superprocess" generating the population parameters and a "subprocess" generating the series for the particular population parameters that come from the "superprocess". When testing for stationarity you should then specify whether you are interested in the "superprocess" or the "subprocess" and act accordingly. If this structure is not imposed, we treat the outcome of this hierarchical structure as coming out of one fixed process; if it is nonstationary we do not care whether it is due to the nonstationarity of the "superprocess" or the "subprocess", we only care about their combination (which is nonstationary).

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  • $\begingroup$ Your argument is sound. The problem is, we don't have the distributions, we just have a realization of some random process. We are not exactly checking the distributions/statistics point-by-point because we don't know them, yet you seem to suggest that. I also don't think the question of whether or not a certain time-series is the realization of a stationary random process should depend on an assumed model. To me, it seems that if we decide that the time series is non-stationary, we are also rejecting the possibility of a doubly-stochastic process, which doesn't seem quite right. $\endgroup$ – eymen Mar 22 '16 at 3:15
  • $\begingroup$ I agree that stationarity/nonstationarity does not depend on whether the DGP is hierarchical or not. But rejecting stationarity does not mean we are rejecting a doubly-stochastic process (!). If you are not willing to accept the hierarchy when testing even though it is present, then refer to my last sentence: nonstationarity will be detected regardless of whether the super- or the subprocess causes it. However, if we do allow for the hierarchy and reject stationarity of the outcome, then we know that at least one of the super- and subprocess is nonstationary. We do not know which, though. $\endgroup$ – Richard Hardy Mar 23 '16 at 18:18
  • $\begingroup$ ...(cont'd) And without further assumptions we cannot test which of them is nonstationary. Only by accepting extra assumptions regarding the hierarchical structure can we do that. $\endgroup$ – Richard Hardy Mar 23 '16 at 18:20

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