Robust Variance Estimation in Linear Regression: A Projection-Geometry Perspective

2026-09-01

AI Summary

本文针对线性回归中因投影非局部导致的方差估计不准确问题，提出一种基于投影几何框架的新方法，结合组内和跨组残差矩进行稳健方差估计。

Abstract

Inference in linear regression commonly treats OLS residuals as proxies for unobserved errors. This approximation can fail when the regression projection is nonlocal relative to the error-dependence structure. Residualization then shifts covariance information across observations and clusters, while conventional heteroskedasticity-consistent (HC) and cluster-robust variance estimators (CRVE) retain only diagonal or within-cluster residual moments and may therefore understate sampling uncertainty. This paper develops a projection-geometry framework for robust variance estimation. The variance of the OLS estimator is represented exactly as a Riesz functional of latent covariance blocks, and observable residual moments are linked to the target through a linear operator determined by the full regression projection. This formulation reduces variance estimation to a linear inverse problem. I propose a Riesz variance estimator that combines within- and cross-cluster residual moments. Conventional HC and CRVE emerge as restricted approximations whose validity depends on negligible projection spillovers. The estimator remains well defined when cluster-specific leverage matrices are singular and is computed by an iterative algorithm that avoids explicit matrix inversion. Simulations show substantial undercoverage by conventional methods under projection spillovers, whereas the proposed estimator restores near-nominal coverage. In an application to colonial governor promotions, the correction changes the significance of four of five reported coefficients.

Problem

• linear regression
• OLS residuals
• error-dependence structure
• heteroskedasticity-consistent
• cluster-robust variance estimators

Innovation

• projection-geometry
• Riesz functional
• linear inverse problem
• Riesz variance estimator
• residual moments