Many of us are familiar with the connection between spurious correlation and its relationship to cointegration. Granger explains in his seminal 1974 paper "Spurious Regressions in Econometrics" how hypothesis testing on two time series that are I(1) can lead to significance tests that are very biased according to experimental results (resulting in far more "significant" correlations than there should be). Later, Philips (1986) showed with functional limit theory exactly why these I(1) time series led to biased results: because in fact there is no convergent asyptotic behavior for the correlation coefficient for these series (among other relevant statistics).

At this point, there seems to be a gap in my understanding. There seems to be dozens of articles online confidently proclaiming that you can "get around" the spurious correlation problem if the two time series in question are not only I(1), but also cointegrated. The explanation seems to be that since the error term is I(0), the usual assumptions of hypothesis testing for correlation are met. Of course, this makes intuitive sense, since the original assumption for the correlation significance test is that the residuals are iid normal. But saying the residuals are I(0) is not the same as saying they are iid normal.

I am wondering if there is a reference that explains this relationship in more detail? Who was it that showed that regular hypothesis testing is valid when the two series are cointegrated? And is this an "if and only if", or are there other circumstances other than co-integration under which regular OLS correlation testing is still valid?

  • $\begingroup$ What you should be looking at IMHO is the consistency property of the OLS estimator, which merely requires the usual Gauss-Markov assumptions: exogeneity, homoskedasticity and absence of serial correlation in the linear regression residuals (hence no "normality" nor "independent" assumption contrary to population belief). Cointegration guarantees I(0) residuals. If residuals are not I(0), hence weakly stationary, they cannot possibly verify the GM conditions, hence OLS is not consistent and your R2 can be therefore be quite misleading. $\endgroup$ – Quantuple Dec 15 '16 at 8:16
  • 1
    $\begingroup$ Hi Paul! I've taken the liberty to add links to the papers you mentioned in the question. $\endgroup$ – olaker Dec 15 '16 at 10:02
  • 4
    $\begingroup$ I'm voting to close this question as off-topic because it is purely about statistics rather than finance, and as such belongs on Cross Validated. $\endgroup$ – Richard Hardy Dec 15 '16 at 16:25
  • $\begingroup$ @Quantuple, thank you for the detailed comment. I think that what I am looking for is probably related to consistency. Though it sounds from what you are saying that the argument is an "only if", i.e. the OLS estimator cannot be consistent if the residuals are not I(0). What I am even more interested is what we can say about the OLS estimator if the residuals are I(0). $\endgroup$ – Paul Dec 15 '16 at 16:42
  • $\begingroup$ It has been posted at stats which indeed seems to fit better. $\endgroup$ – Bob Jansen Dec 15 '16 at 19:28