What Estimators Are Unbiased For Linear Models?

TL;DR

This paper investigates which estimators are unbiased for linear models, reviewing historical work, unifying previous results, and establishing new theorems that generalize existing findings under various conditions.

Contribution

It introduces a unified framework for unbiased estimators in linear models and presents new representation theorems that extend prior results to broader settings.

Findings

Established new representation theorems for unbiased estimators.

Unified and extended previous results on unbiased estimators.

Generalized claims under various restrictions on linear models.

Abstract

The recent thought-provoking paper by Hansen [2022, Econometrica] proved that the Gauss-Markov theorem continues to hold without the requirement that competing estimators are linear in the vector of outcomes. Despite the elegant proof, it was shown by the authors and other researchers that the main result in the earlier version of Hansen's paper does not extend the classic Gauss-Markov theorem because no nonlinear unbiased estimator exists under his conditions. To address the issue, Hansen [2022] added statements in the latest version with new conditions under which nonlinear unbiased estimators exist. Motivated by the lively discussion, we study a fundamental problem: what estimators are unbiased for a given class of linear models? We first review a line of highly relevant work dating back to the 1960s, which, unfortunately, have not drawn enough attention. Then, we introduce notation that allows us to restate and unify results from earlier work and Hansen [2022]. The new framework also allows us to highlight differences among previous conclusions. Lastly, we establish new representation theorems for unbiased estimators under different restrictions on the linear model, allowing the coefficients and covariance matrix to take only a finite number of values, the higher moments of the estimator and the dependent variable to exist, and the error distribution to be discrete, absolutely continuous, or dominated by another probability measure. Our results substantially generalize the claims of parallel commentaries on Hansen [2022] and a remarkable result by Koopmann [1982].