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In this post mpiktas showed that the sample correlation measure for two random walks (possible correlated) is a random variable and does not estimate the theoretical correlation. When trying to find a cointegrating relationship between two non-stationary variables using Engle-Granger we estimate the cointegrating vector through OLS, which implicitly involves using the sample correlation.

Given that the sample correlation does not give a consistent estimate (at least for the random walk case) for correlation, should we really use OLS to carry out the estimation or is there some other technicality that validates the usage of OLS?

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It is not the correlation coefficient that is the takeaway from the Engle-Granger procedure; it is rather the regression coefficient. The OLS estimate of the regression coefficient is superconsistent in the case of a bivariate regression of two cointegrated series, so the Engle-Granger procedure is legitimate.
(Superconsistent is even better than consistent as the estimator converges faster than usual.)

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  • $\begingroup$ Thanks for the quick reply. I guess my point was on the fact that the slope coefficient is just the correlation times the ratio of standard deviations of Y and X respectively, given that the sample correlation is not consistent in the non-stationary case it is interesting that the slope coefficient is. $\endgroup$
    – barbarossa
    Commented Jun 9, 2015 at 16:24
  • $\begingroup$ Yes, that is a valid point; my answer does not address it. I do not know an intuitive explanation, while the formal derivation needs some time to get comfortable with. Let's hope someone else will pitch in with a more detailed response. $\endgroup$
    – Richard Hardy
    Commented Jun 9, 2015 at 17:09
  • $\begingroup$ Well but is mpiktas result for random walks or cointegrated processes? $\endgroup$
    – Matifou
    Commented Jun 12, 2015 at 5:58
  • $\begingroup$ His result is for random walks generated by correlated processes, which is where I got confused. But from more thinking and some simulation I resolved my issue. Essentially, when estimation Engle-Granger beta, correlation is typically not important, rather the ratio of variancies (correlation typically tends to one due to I(1) processes). I guess mpiktas's point comes in when we test for cointegration using Engle-Granger and the null of no cointegration is valid; to my knowledge Engle-Granger with Dickey-Fuller does not have a very good power. $\endgroup$
    – barbarossa
    Commented Jun 17, 2015 at 15:39

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