What does the Gauss-Markov theorem tell us about least squares estimators?

• A. If the classical assumptions hold, the least squares estimator is the best linear unbiased estimator (BLUE).
• B. Least squares estimators are always biased.
• C. Least squares estimators are always efficient.
• D. Least squares estimators do not depend on variance.

Answer: (A)

The Gauss-Markov theorem says that in a linear regression model where the errors have zero mean, are uncorrelated, and have constant variance (plus the design matrix has full rank), the ordinary least squares estimator is the best linear unbiased estimator. In other words, among all estimators that are linear in the observations and unbiased, none has a smaller variance than the OLS estimator. Mathematically, E(β̂) = β and Var(β̂) = σ^2 (X'X)^{-1}, and no other linear unbiased estimator can beat this variance.

This is why the answer is the most appropriate: it captures the essential guarantee of OLS under the standard assumptions. If those assumptions fail—such as heteroskedasticity or autocorrelation—the BLUE property no longer holds, and alternative methods (like GLS or robust standard errors) are used. The other statements contradict the theorem: least squares estimators are not always biased, they are not guaranteed to be efficient among all estimators (only among linear unbiased ones under the assumptions), and their variance does depend on the error variance and the design.