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Gauss–Markov theorem
Theorem related to ordinary least squares
Gauss–Markov theorem
In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that, under certain conditions, the ordinary least squares (OLS) estimator for linear regression has the lowest variance among all linear unbiased estimators. The theorem guarantees that OLS is the most precise way to estimate regression coefficients within that class of estimation methods. The conditions are that the model errors are each mean zero, have finite equal variances, and are all mutually uncorrelated. The errors do not need to be normal, nor do they need to be independent and identically distributed. This theorem holds only for unbiased estimators, since there exist biased estimators with lower variance. See, for example, ridge regression or the James–Stein estimator (which is a nonlinear biased estimator). The theorem is named after Carl Friedrich Gauss and Andrey Markov, although Gauss's work significantly predates Markov's. Gauss derived the result under the stronger assumptions of...
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