The Gauss Markov Theorem: A Pedagogical Note

Abstract

When stating the Gauss-Markov theorem, undergraduate econometric textbooks generally imply that the ordinary least squares (OLS) estimator has minimum variance. However, the proof of the Gauss Markov theorem indicates that the weights produced by the OLS estimator–not the formula per se–produce the unique minimum–variance estimator. An example demonstrates that other linear unbiased estimators can yield the same variance as the OLS estimator; however, the weights from such formulas are identical to the OLS weights. To avoid this pedagogical confusion, the statement of the theorem should emphasize the weights rather than the estimator itself.