What Estimators Are Unbiased For Linear Models?

Understand

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.