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I'm not sure, but I think I've read somewhere that because the Classical Linear Regression model assumes to have a random sample, when researchers think they might not be in presence of a sample with that property, they try to use some randomization technique to make sure the usual theory may be applied.

In Hayashi's Econometrics, chapter 2, he develops the OLS and studies its large-sample properties in a generalization of iid sample. He assumes that the sample is ergodic, and stationary, and that the regressors multiplied by the error terms (the same that define the orthogonality conditions) also follow a martingale difference sequence.

My question is, given a sample, is there a way to know if it satisfies ergodicity, stationarity, and martingale diff. seq.? Also, even if the sample does not satisfy it, is there a way to make sure that we are able to obtain such a sample, when randomization techniques are not possible to apply?

Any help would be appreciated.

P.S.: This question is also posted in CrossValidated, but with a different intention. When posting it here, I'm looking more for a perspective of applied econometricians, not so much for formal mathematical explanation of the possible existing methods. But of course, if you're willing to formalize and going into depth I would also be very thankful.

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Ergodic and strict stationarity are the essentially the weakest assumptions for which you have a LLN, i.e. can do large sample estimation. Given the very liberal way applied econometricians use laws of large numbers, ergodicity and strict stationarity is almost always assumed. If that is in question, you're simply not in business for consistent estimation.

Similarly, if one would like to have a CLT, i.e. do large sample inference, MDS is a general weak assumption. (There are other CLT's around but more technical. The MDS CLT and Lindenberg CLT are the two immediate generalizations of the classical CLT.)

The MDS assumption can be tested. An MDS is just a discrete-time martingale with mean zero. In particular, its increments are uncorrelated. (The martingale assumption is an intermediate condition between serially uncorrelated and independence.) So a rejection by a serial correlation test would also reject the MDS assumption.

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  • $\begingroup$ Is there a way to test precisely for MDS, instead of serial correlation? Also, how can I make a sample to have exactly MDS? $\endgroup$
    – An old man in the sea.
    Mar 25, 2015 at 10:54
  • $\begingroup$ Yes, there are formal test for the joint hypothesis of stationarity and ergodicity, and test for the martingale assumption. As for simulating a MDS, I suppose one can always cook up Markov chains where the conditional means are zero. $\endgroup$
    – Michael
    Mar 25, 2015 at 11:25
  • $\begingroup$ Could you talk about those tests in your answer? Also Instead of simulating a MDS, I was thinking something similar as with experiment design, where we design a experiment to make sure that we get a random sample. Is there a way to design some experiment so that we can make sure that the sample satisfies MDS? $\endgroup$
    – An old man in the sea.
    Mar 26, 2015 at 0:23
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I have to disagree on some points in @Michale answer.

For a Weak Law of Large numbers to hold, stationarity, either strict or "weak"(alternatively called "2nd-order" or "covariance-stationarity"), is not a necessary condition.

There are simple versions of LLN where each random variable in the sequence examined may have a different mean and/or variance. This rules out stationarity, but the probability limit may exist, subject to a condition on the variance ("Markov's condition").

Of course, the probability limit being an average of the different expected values, it may not have a useful interpretation -but this is not a "technical" problem as regards the validity of the LLN, but rather whether we can interpret, in a useful way, the averaging over a sequence of random variables that each has different moments.

As a compromise, we assume that the mean is the same, but the variance may differ. This imposes only "mean-stationarity", but not covariance-stationarity, and certainly not strict stationarity (strict stationarity requires that the joint distribution of any subsequence is identical). Again, we make this assumption not in order for the LLN to hold, but in order to be able to interpret usefully the results of the LLN: after all, to pool and examine data together, they must have something in common, otherwise, what's the point?

But ergodicity is required in order for the sample averages to converge to the theoretical quantity, this is what Kinchine's Law is all about.

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