Autocorrelated errors break your standard errors, not your coefficients

Question

You regress a return series on a signal, and the residuals are serially correlated: today's error is correlated with yesterday's (common with overlapping windows or persistent variables).

Does autocorrelation bias your OLS coefficients? If not, what does it break, and how do you fix it?

Accepted Answer

Coefficients are fine. OLS remains unbiased and consistent as long as \mathbb{E}[\varepsilon \mid X] = 0. Serial correlation is a property of how errors relate to each other, not of whether they relate to the regressors, so it leaves \hat\beta unbiased. What you lose is efficiency (OLS is no longer the minimum-variance linear estimator) and, more importantly, valid inference. Standard errors are wrong. The classical variance \widehat{\operatorname{Var}}(\hat\beta) = \sigma^2 (X'X)^{-1} assumes \operatorname{Var}(\varepsilon \mid X) = \sigma^2 I, an identity matrix with zeros off the diagonal.…

Autocorrelated errors break your standard errors, not your coefficients

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