Gauss–Markov theorem

In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors)[1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance (variance of the estimator across samples) within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero.[2] The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance). The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator (which also drops linearity), ridge regression, or simply any degenerate estimator.