The result that, in the classical linear regression model with errors having zero mean, being uncorrelated and homoscedastic, the ordinary least squares (OLS) estimator of the regression coefficients is the best linear unbiased estimator (BLUE), i.e., it has the smallest variance among all linear unbiased estimators.

Principle

Under the Gauss–Markov conditions the estimation problem is a projection in a Hilbert space of observables onto the column space of regressors; linear unbiasedness plus orthogonality to the residual space yields minimal variance among linear estimators.

Demonstration

In a simple linear regression with intercept and i.i.d. errors of common variance, the OLS slope and intercept are linear functions of the observed responses; any other linear unbiased estimator can be shown via variance algebra to have equal or larger covariance matrix, demonstrating the BLUE property.

Misapplication

Assuming the theorem guarantees optimality in a non-linear class (it constrains to linear estimators) or that normality is required; applying OLS as BLUE when errors are heteroscedastic, autocorrelated, or when regressors are endogenous invalidates the conclusion.

Consequence

Justifies use of OLS for estimation and motivates standard error formulas and inference under the stated assumptions; indicates when alternative estimators (weighted least squares, GLS, instrumental variables) are needed if assumptions fail.

Reversal

If homoscedasticity or independence is violated, the OLS estimator is no longer guaranteed to minimize variance among linear unbiased estimators; generalized least squares (GLS) or other estimators may dominate, reversing the BLUE assertion.

Boundary

Applies only to linear estimators under the classical linear model with finite second moments and the specified error conditions; it does not address efficiency among nonlinear or biased estimators, nor does it provide finite-sample optimality outside the linear class.

Semantic Tension

Often compared with the Cramér–Rao notion of efficiency which addresses the lower bound for all unbiased estimators under parametric models; Gauss–Markov restricts attention to linear estimators and does not require distributional assumptions like normality.

Synthesis

Gauss–Markov states that, within the linear unbiased class under zero-mean, uncorrelated, homoscedastic errors, OLS is variance-minimizing; it is a projection-based, assumption-dependent optimality statement that guides estimator choice and signals when to adopt GLS or IV methods if assumptions break.