With random regressors, least squares inference is robust to correlated errors with unknown correlation structure

TL;DR

This paper demonstrates that with random regressors, linear regression inference remains valid even when errors are correlated with unknown structure, challenging traditional assumptions and highlighting the robustness of randomized approaches.

Contribution

It proves the asymptotic normality of t statistics under unknown error correlation with random regressors, extending the applicability of linear regression inference.

Findings

Asymptotic normality of t statistics is established.

Error correlation can enhance test power under weak signals.

Linear regression inference is robust to unknown error correlation with random regressors.

Abstract

Linear regression is arguably the most widely used statistical method. With fixed regressors and correlated errors, the conventional wisdom is to modify the variance-covariance estimator to accommodate the known correlation structure of the errors. We depart from the literature by showing that with random regressors, linear regression inference is robust to correlated errors with unknown correlation structure. The existing theoretical analyses for linear regression are no longer valid because even the asymptotic normality of the least-squares coefficients breaks down in this regime. We first prove the asymptotic normality of the t statistics by establishing their Berry-Esseen bounds based on a novel probabilistic analysis of self-normalized statistics. We then study the local power of the corresponding t tests and show that, perhaps surprisingly, error correlation can even enhance power in the regime of weak signals. Overall, our results show that linear regression is applicable more broadly than the conventional theory suggests, and further demonstrate the value of randomization to ensure robustness of inference.