I apologize in advance if this proves to be a nonsensical question but it is something I have been struggling with.

Is there a way to prove/discern that covariance stationarity is not an attribute of the sample at hand and is an attribute of the underlying process?


The motivation of the question comes from unit root testing; when we perform unit root testing on a series, we may accept the null ie that the series in question, has a unit root on one sample, but reject the null on a different (sub-)sample.


Covariance stationarity is described as constancy over time of the first two moments of $X_t$. This is clearly a property of the underlying process: the moments are a process property, not a sample property.

  • $\begingroup$ Since samples have moments too, this reasoning would seem not to answer the question. $\endgroup$ – whuber Jan 12 '18 at 15:22
  • $\begingroup$ Let me add, then: "Clearly, the constancy implied by stationarity rules out that the moments are sample moments, for randomness would prevent these from being constant." $\endgroup$ – F. Tusell Jan 12 '18 at 16:04
  • $\begingroup$ That helps. I'm not sure it addresses the spirit of the question, though. The question seems to concern making inferences about a process from a sample. It appears to ask, "how confidently may we conclude that the process is stationary based on properties of a sample?" To me, that looks like a version of the basic question, "how can we test any statistical hypothesis based on a sample?" From this point of view, observing that the sample can vary because it's random is correct but not terribly useful. $\endgroup$ – whuber Jan 12 '18 at 16:16
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    $\begingroup$ May be you are right. I thought I understood the question on my first reading. I am no longer sure, so I will refrain from making additional comments that rather than being of help may obfuscate the matter. $\endgroup$ – F. Tusell Jan 12 '18 at 16:42

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