  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].
