Questions: Feasible GLS (FGLS) with Estimated Covariance Structure

Question 1

A researcher has 40 observations with severe heteroskedasticity. She specifies an FGLS model assuming variance is proportional to x², estimates Ω̂ from OLS residuals, then applies GLS using Ω̂. But the true variance structure is actually proportional to x³. What is the most likely outcome?

• A. FGLS will still be efficient because any correction for heteroskedasticity improves on OLS
• B. FGLS may perform worse than both OLS and correct GLS due to misspecification of the covariance form
• C. FGLS will be unbiased but inefficient, with performance identical to OLS
• D. FGLS will be exactly as efficient as OLS because the sample size is too small for GLS improvements

Answer: (B)

This is the central danger of FGLS: if you specify the wrong covariance model, Ω̂ is systematically wrong, and the FGLS transformation distorts the data in the wrong way. The resulting estimator can have worse properties than plain OLS, which makes no transformation at all. With only 40 observations, the first-stage covariance estimation is also imprecise. The practical lesson: FGLS requires both a well-motivated covariance form AND a large enough sample for the first-stage estimation to be reliable.

Question 2

What does FGLS estimate in its first step, and why is this step necessary?

• A. The regression coefficients β, which are then used to construct Ω̂
• B. The error covariance structure Ω from OLS residuals, because true GLS requires knowing Ω a priori
• C. The instrumental variables needed to address endogeneity in the transformed model
• D. The optimal bandwidth for kernel-based heteroskedasticity correction

Answer: (B)

True GLS requires knowing the exact error covariance matrix Ω — which in practice you almost never have. FGLS makes GLS feasible by estimating Ω from the data. In step 1, you run OLS and collect residuals, then use those residuals to estimate the covariance structure (regressing squared residuals on regressors for heteroskedasticity, estimating ρ from an AR model for serial correlation, etc.). Only then can you perform step 2: apply GLS using Ω̂ in place of Ω. The necessity of step 1 is precisely what distinguishes FGLS from GLS.

Question 3

In large samples, FGLS is asymptotically equivalent to true GLS — both achieve the same efficiency gains over OLS.

• T. True
• F. False

Answer: True

As sample size grows, the first-stage estimate Ω̂ converges to the true Ω, so the FGLS transformation converges to the true GLS transformation. In the limit, the two estimators are asymptotically equivalent: both achieve the Gauss-Markov lower bound under the correctly specified covariance model. This is why the FGLS tradeoff depends heavily on sample size — in small samples, the estimation error in Ω̂ can dominate, but in large samples it becomes negligible.

Question 4

FGLS is generally more efficient than OLS because it corrects for non-spherical errors, so it should be the default estimator whenever heteroskedasticity or autocorrelation is suspected.

• T. True
• F. False

Answer: False

This is the most dangerous misconception about FGLS. Efficiency gains require that (a) the covariance structure is correctly specified and (b) the sample is large enough for step-1 estimation to be precise. If either condition fails, FGLS can have *higher* mean squared error than OLS. In small samples with misspecified covariance structure, the two-step estimation introduces noise that can more than offset the efficiency gains. This is why practitioners often prefer heteroskedasticity-robust standard errors for moderate samples — they require no assumption about the form of heteroskedasticity.

Question 5

Why do practitioners often prefer heteroskedasticity-robust standard errors over FGLS when facing heteroskedasticity, even though FGLS explicitly models and corrects for the heteroskedasticity?

Model answer: Robust standard errors correct the *inference* (standard errors and test statistics) without modifying the OLS point estimates, and they require no assumption about the *form* of heteroskedasticity — only its existence. FGLS requires specifying a parametric model for how variance depends on covariates, estimating that model from residuals, and transforming the data accordingly. If the specified form is wrong, FGLS introduces systematic bias in the transformation. Robust standard errors sacrifice the efficiency gains from GLS but avoid the risk of misspecification-induced distortions. FGLS is most valuable when the covariance structure is well-motivated theoretically and the sample is large enough for precise first-stage estimation.

The key asymmetry: robust standard errors are conservative (sacrifice efficiency) but robust to misspecification; FGLS gains efficiency conditional on correct specification but can fail badly when misspecified. In practice, the form of heteroskedasticity is rarely known with certainty, making the robustness of the simpler approach more attractive unless there is a strong theoretical prior about the variance structure.