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

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Linear regression is a fundamental statistical tool.
• Conventional methods assume fixed regressors and uncorrelated errors, requiring adjustments for known error correlation.
• Existing theory for linear regression breaks down with random regressors and correlated errors.

Purpose of the Study:

• To demonstrate the robustness of linear regression inference with random regressors and unknown correlated errors.
• To challenge existing theoretical limitations in linear regression analysis.
• To explore the impact of error correlation on statistical power.

Main Methods:

• Proving asymptotic normality of t-statistics using novel probabilistic analysis of self-normalized statistics.
• Establishing Berry-Esseen bounds for t-statistics.
• Analyzing the local power of t-tests under weak signal conditions.

Main Results:

• Linear regression inference is robust to unknown correlated errors when regressors are random.
• Asymptotic normality of least squares coefficients does not hold in this regime.
• Error correlation can surprisingly enhance statistical power in the presence of weak signals.

Conclusions:

• Linear regression is applicable in a broader range of scenarios than previously understood.
• Randomization is a valuable technique for ensuring the robustness of statistical inference.
• The findings necessitate a re-evaluation of standard linear regression theoretical assumptions.