GaussMarkovteoremet

Gauss-Markovteoremet, commonly known as the Gauss–Markov theorem, is a fundamental result in linear regression theory. It characterizes the efficiency of the ordinary least squares estimator within a broad class of estimators.

In its standard form, the theorem applies to the linear model y = Xβ + ε, where y is an n-dimensional vector of observations, X is an n-by-k matrix with full column rank, β is a k-dimensional vector of coefficients, and ε is an error vector with E[ε] = 0 and Var(ε) = σ²I_n (i.e., errors are uncorrelated with equal variance). No normality assumption is required.

The theorem states that, among all linear unbiased estimators of β, the ordinary least squares (OLS) estimator β̂ = (X′X)⁻¹X′y has the minimum variance. In other words, OLS is the Best Linear Unbiased Estimator (BLUE) of the regression coefficients under these conditions. The result relies on unbiasedness and linearity with respect to the observed data.

If the homoskedasticity or absence of autocorrelation assumptions are violated (for example, with heteroskedastic or correlated errors), the OLS estimator is generally no longer BLUE. In such cases, generalized least squares (GLS) can achieve the BLUE property when the error covariance structure is known (or approximated) by a suitable weighting matrix.

The Gauss-Markov theorem does not claim optimality among all unbiased estimators, only among the class of linear unbiased estimators for the coefficients.