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Let $Y=X\beta +\epsilon$ with a fixed full column rank design matrix, $\mathbb{E}\epsilon =0$ and $\mathrm{Cov}(\epsilon )={\sigma }^{2}I$. Then ordinary least squares is a best linear unbiased estimator: every response-linear unbiased estimator has at least as much variance in every coefficient direction. A normal distribution is unnecessary.^[ref]

For a proof put $A=(X^{\mathsf{T}}X{)}^{-1}{X}^{\mathsf{T}}$. A competing estimator $CY$ is unbiased for every $\beta$ exactly when $CX=I$. Write $C=A+D$; then $DX=0$, so the cross terms vanish andThis positive semidefinite matrix inequality proves the simultaneous directional comparison. The scope is linear unbiased estimators, not all conceivable estimators: a biased ridge regression estimate may improve mean squared error for some signals. With correlated or unequal-variance errors, the corresponding known-covariance result uses generalized least squares.