Bruce E. Hansen

A Modern Gauss-Markov Theorem
Econometrica, forthcoming.

December 2020
Revised: September, 2021


Abstract:

This paper presents finite sample efficiency bounds for the core econometric problem of estimation of linear regression coefficients. We show that the classical Gauss-Markov Theorem can be restated omitting the unnatural restriction to linear estimators, without adding any extra conditions. Our results are lower bounds on the variances of unbiased estimators. These lower bounds correspond to the variances of the the least squares estimator and the generalized least squares estimator, depending on the assumption on the error covariances. These results show that we can drop the label "linear estimator" from the pedagogy of the Gauss-Markov Theorem. Instead of referring to these estimators as BLUE, they can legitimately be called BUE (best unbiased estimators).

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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.