Material Facts Obscured in Hansen's Modern Gauss-Markov Theorem

TL;DR

This paper critically examines Hansen's modern Gauss-Markov theorem, revealing that it obscures key facts about estimator bias and efficiency, potentially misleading students about the true nature of estimator optimality.

Contribution

It clarifies the limitations of the modern Gauss-Markov theorem and highlights the importance of bias-reducing techniques over traditional unbiasedness in estimator efficiency.

Findings

MGMT emphasizes OLS and efficiency bounds, overshadowing bias-reduction methods.

Most nonlinear estimators are biased due to the link between linearity and unbiasedness.

The theorem's scope is limited, extending the classical GMT only to a trivial set.

Abstract

We show that the abstract and conclusion of Hansen's {\it Econometrica} paper, \cite{Hansen22}, entitled a modern Gauss-Markov theorem (MGMT), obscures a material fact, which in turn can confuse students. The MGMT places ordinary least squares (OLS) back on a high pedestal by bringing in the Cramer-Rao efficiency bound. We explain why linearity and unbiasedness are linked, making most nonlinear estimators biased. Hence, MGMT extends the reach of the century-old GMT by a near-empty set. It misleads students because it misdirects attention back to the unbiased OLS from beneficial shrinkage and other tools, which reduce the mean squared error (MSE) by injecting bias.