GaussMarkovi

GaussMarkovi is a term sometimes used in statistical writing to refer to the Gauss–Markov theorem, a foundational result in linear regression. In informal usage, GaussMarkovi points to the idea that, under specific assumptions, the ordinary least squares estimator is the Best Linear Unbiased Estimator (BLUE) of the regression coefficients.

The classical linear regression model is written as y = Xβ + ε, where X is an n-by-k design matrix of regressors with full column rank, β is a vector of coefficients, and ε is the error term. The Gauss–Markov conditions require E[ε|X] = 0 and Var(ε|X) = σ^2 I, implying homoscedasticity and no autocorrelation, with the regressors treated as fixed or non-stochastic. Under these conditions, the OLS estimator β̂ = (X′X)⁻¹X′y is unbiased and has the minimum variance among all linear unbiased estimators. The variance is Var(β̂) = σ^2 (X′X)⁻¹.

Extensions and caveats: If the Gauss–Markov assumptions are violated, OLS may no longer be BLUE, though it can remain unbiased. In the presence of heteroskedasticity or autocorrelation, generalized least squares (GLS) or feasible GLS, as well as robust standard errors, can provide more reliable inference.

Origin: The theorem is named after Carl Friedrich Gauss and Andrey Markov, reflecting historical contributions to the theory of linear models. In some texts, GaussMarkovi is used as a variant spelling or shorthand for the Gauss–Markov result.

See also: Gauss–Markov theorem, ordinary least squares, generalized least squares, BLUE, heteroskedasticity-robust methods.