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Suppose we're doing univariate linear regression between X and Y. Let's say X are daily observations, and Y reflects how some variable changes 1 year into the future. So Y observations will be overlapping and autocorrelated. The daily observations in X could be autocorrelated as well.

How do we measure $R^2$ and variance of $\beta$ in this situation? How do we know if the relationship between X and Y is statistically significant?

I have looked at a number of papers on this:

  • "Dividend Yields and Expected Stock Returns: Alternative Procedures for Inference and Measurement", Hodrick R., 1992
  • "Tests of Financial Models in the Presence of Overlapping Observations", Richardson M. and Smith T., 1991
  • "Improved Inference and Estimation in Regression with Overlapping Observations", Britten-Jones M. and Neuberger A., 2011

But it is still not clear to me what I should be doing in practice in these situations. Is there any tutorial that clearly summarizes how estimates should be corrected in the presence of overlapping observations, and what the assumptions behind each procedure are, stating which procedure tends to be the most conservative?

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  • $\begingroup$ What's wrong with the usual definitions? $\endgroup$
    – Dave
    Nov 19, 2021 at 16:40

1 Answer 1

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Not a full answer unfortunately, but I am thinking in the context of linear regression with just one predictor, you can use the correction outlined in:

"Long-run predictability tests are even worse than you thought" https://onlinelibrary.wiley.com/doi/full/10.1002/jae.2930

This gives you an adjustment for the variance of the coefficient of the predictor, and therefore an adjusted t-stat.

I think (but am not certain) that since in OLS with size sample n and t-stat t for the coefficient you have:

R^2=[1+(n-1)/(t^2)]^(-1)

The adjusted t-stat, gives you and adjusted R^2 as well.

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