Correlated Errors

What is the effect of positively correlated errors in OLS on confidence intervals and hypothesis testing? What about negative correlation?

Answer

If the errors are positively correlated but we still use the usual OLS formulas (which assume independence), then the variance of the estimators is underestimated.

Confidence intervals: CIs have less coverage in reality.
Hypothesis testing: Standard errors are too small, so the reported $$t$$ -statistics are larger (lower p-valies), meaning we reject $$H_0$$ more often than we should (false confidence).

For negatively correlated errors (doesn't really happen in finance), the opposite happens: the variance of the estimators is overestimated, CIs have more coverage in reality, and tests reject too rarely.

Note that (assuming $\mathbb{E}[\varepsilon\mid X]=0$ ), both positive and negative error correlation do not affect bias.