GaussMarkovTheorem

Gauss-Markov theorem is a fundamental result in statistics and econometrics. It states that, in the classical linear regression model with certain assumptions, the ordinary least squares (OLS) estimator of the coefficients is the Best Linear Unbiased Estimator (BLUE).

Consider the model y = Xβ + ε, where y is the n-by-1 vector of observed responses, X is the n-by-k matrix of regressors with full column rank, β is the k-by-1 coefficient vector, and ε is the n-by-1 error vector. The assumptions include E[ε|X] = 0 and Var(ε|X) = σ^2 I_n, i.e., the errors are uncorrelated and homoscedastic (constant variance), with regressors treated as fixed (or strictly exogenous).

Under these conditions, the OLS estimator β̂ = (X'X)^{-1} X'y is unbiased, and its variance is Var(β̂) = σ^2 (X'X)^{-1}. The Gauss-Markov theorem asserts that among all estimators that are linear in the observations and unbiased for β, β̂ has the smallest variance for every component of β.

Implications and extensions: The theorem provides a justification for using OLS in ordinary least squares regression when the CLRM assumptions hold. If errors are heteroskedastic or autocorrelated, the BLUE property may fail, and OLS can be inefficient. In such cases, generalized least squares (GLS) or robust standard errors are used to obtain efficient or valid inference.

History and naming: The result is associated with Carl Friedrich Gauss and Andrey Markov, and it is a core component of the theory of linear models in econometrics and statistics.