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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?

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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.

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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.

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  • $\begingroup$ 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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