Questions: Gauss-Markov Theorem and OLS Efficiency

Question 1

A researcher estimating a wage equation discovers the error variance is much larger for high-wage workers than for low-wage workers. Which consequence for OLS is most accurate?

• A. OLS estimates are biased because the zero-mean-error assumption is violated
• B. OLS estimates are unbiased but no longer have minimum variance among linear unbiased estimators; GLS weights observations by inverse error variance and would be more efficient
• C. OLS estimates are inconsistent and should be replaced with instrumental variables
• D. The results are unaffected because heteroskedasticity only matters when errors are also autocorrelated

Answer: (B)

Heteroskedasticity violates the homoskedasticity assumption (Var(uᵢ) = σ² for all i), but it does NOT violate exogeneity (Cov(X, u) = 0). Because exogeneity still holds, OLS remains unbiased — the expected value of the estimates still equals the true parameters. However, OLS is no longer efficient: it treats all observations equally, while GLS exploits the fact that low-variance observations contain more information per unit. GLS minimizes a weighted sum of squared residuals, giving higher weight to observations with lower error variance, and achieves strictly lower variance than OLS. Only a violation of exogeneity (option A/C logic) causes bias.

Question 2

An econometrician claims to have found a nonlinear unbiased estimator with strictly lower variance than OLS under all Gauss-Markov conditions. What does this imply about the Gauss-Markov theorem?

• A. It disproves the theorem — no such estimator should exist if the proof is correct
• B. It is entirely consistent with the theorem — Gauss-Markov only guarantees OLS is best among linear unbiased estimators, and a nonlinear estimator may achieve lower variance
• C. The estimator must be biased — the theorem guarantees OLS has minimum variance among all unbiased estimators, linear or not
• D. This is impossible; the theorem covers all estimators regardless of their functional form

Answer: (B)

The 'B' in BLUE stands for 'Best among linear unbiased estimators.' This is a crucial qualifier. The theorem makes no claim that OLS beats nonlinear estimators. In fact, nonlinear estimators can often achieve lower variance — for example, if errors are non-normal, maximum likelihood estimation (a nonlinear procedure) can be strictly more efficient than OLS. The Gauss-Markov theorem is a restricted optimality result: OLS wins the race within the class of linear unbiased estimators. Students often over-read BLUE as claiming universal optimality.

Question 3

Under the Gauss-Markov assumptions, OLS is the most efficient estimator among most unbiased estimators — linear or nonlinear.

• T. True
• F. False

Answer: False

Gauss-Markov guarantees OLS is Best Linear Unbiased — minimum variance within the class of linear unbiased estimators only. Nonlinear unbiased estimators (such as maximum likelihood under non-normal errors) can achieve lower variance than OLS while remaining unbiased. The restriction to linear estimators is essential to the theorem's claim; without it the result does not hold.

Question 4

If the exogeneity assumption fails — for instance, because a relevant variable is omitted that is correlated with an included regressor — OLS is still unbiased but loses its efficiency advantage over GLS.

• T. True
• F. False

Answer: False

Endogeneity (Cov(X, u) ≠ 0) causes OLS to be biased, not merely inefficient. This is a qualitatively more serious problem than heteroskedasticity or autocorrelation, which cause inefficiency while preserving unbiasedness. When exogeneity fails, E[β̂_OLS] ≠ β — the estimator is systematically wrong in expectation, and no amount of additional data will fix this. This is why instrumental variables are needed to address endogeneity: GLS addresses inefficiency but cannot fix bias caused by correlation between regressors and errors.

Question 5

The Gauss-Markov theorem says OLS is 'best' — but best in what restricted sense, and what are the boundaries of that claim?

Model answer: OLS is Best Linear Unbiased (BLUE): it has the lowest variance among all estimators that are (1) linear functions of the observed outcome Y, and (2) unbiased (expected value equals the true parameter). The claim is bounded in two important ways. First, it applies only within the linear class — nonlinear estimators like maximum likelihood can achieve lower variance. Second, all five Gauss-Markov assumptions must hold: if homoskedasticity fails, GLS (a different linear unbiased estimator) beats OLS on variance; if exogeneity fails, OLS is not even unbiased, let alone efficient, and the theorem no longer applies at all.

The key is understanding BLUE as a restricted optimality claim, not an absolute one. Students often treat 'best' as universal superiority, missing that (a) the comparison class is limited to linear estimators, and (b) the result is conditional on all five assumptions holding. Each assumption plays a specific role in the proof — linearity defines the class, unbiasedness requires exogeneity, and efficiency requires homoskedasticity and no autocorrelation.