Questions: Generalized Least Squares (GLS) for Non-Spherical Errors

Question 1

A researcher runs OLS on panel data where urban observations have much higher error variance than rural ones. The OLS estimates are unbiased. Why might she prefer GLS?

• A. OLS estimates are biased under heteroskedasticity, so GLS corrects this bias
• B. OLS is unbiased but inefficient — it weights all observations equally, while GLS down-weights noisy observations to achieve lower variance
• C. GLS guarantees unbiasedness in cases where OLS does not
• D. OLS cannot be computed when error variances differ across observations

Answer: (B)

This is the key: OLS remains unbiased under heteroskedasticity — the bias critique is wrong. The problem is inefficiency. OLS treats every observation as equally informative when computing the estimator, but observations with high error variance are noisy signals about the true relationship and should count for less. GLS formalizes this by down-weighting high-variance observations (pre-multiplying by Ω^{-1/2}), which restores spherical errors in the transformed regression and makes OLS on the transformed data BLUE — the Best Linear Unbiased Estimator.

Question 2

What happens to the GLS estimator formula β̂_GLS = (X'Ω⁻¹X)⁻¹X'Ω⁻¹y when the error covariance matrix is Ω = σ²I?

• A. GLS becomes infeasible because σ²I is not invertible
• B. GLS reduces to the standard OLS estimator (X'X)⁻¹X'y
• C. GLS produces different, more efficient estimates than OLS even under spherical errors
• D. GLS is only defined for non-spherical errors and cannot handle the Ω = σ²I case

Answer: (B)

Substituting Ω = σ²I into the GLS formula: (X'(σ²I)⁻¹X)⁻¹X'(σ²I)⁻¹y = (X'(1/σ²)X)⁻¹X'(1/σ²)y = σ²(X'X)⁻¹(1/σ²)X'y = (X'X)⁻¹X'y. The σ² cancels, and you recover standard OLS. This is the elegant consistency of GLS: it is a generalization of OLS that reduces to OLS when the errors are already spherical. When errors are not spherical, GLS 'transforms away' the problem by rescaling the data.

Question 3

When the error covariance matrix Ω is known, GLS is BLUE — but when Ω must be estimated from OLS residuals (Feasible GLS), the estimator is no longer exactly BLUE in finite samples.

• T. True
• F. False

Answer: True

True GLS with known Ω satisfies the Gauss-Markov conditions in the transformed regression, making it BLUE. But Ω is almost never known in practice. FGLS estimates Ω from OLS residuals (e.g., regressing squared residuals on covariates to estimate the heteroskedasticity pattern, or estimating the autocorrelation parameter ρ). This introduces first-step estimation error that propagates into the second step. In finite samples, FGLS is consistent and asymptotically efficient but not exactly BLUE — the small-sample properties depend on how well Ω is estimated.

Question 4

OLS is biased when regression errors are heteroskedastic.

• T. True
• F. False

Answer: False

Heteroskedasticity violates the Gauss-Markov assumption of homoskedasticity but does NOT cause bias. OLS is still unbiased (and consistent) under heteroskedasticity — the expected value of the OLS estimator still equals the true parameter. What heteroskedasticity causes is inefficiency: OLS is no longer BLUE because it ignores the information in the variance pattern. It also invalidates standard OLS standard errors and t-statistics (though these can be fixed with Huber-White robust standard errors without changing the point estimates).

Question 5

Why might a researcher prefer robust standard errors over Feasible GLS, even though FGLS can be more efficient?

Model answer: FGLS requires correctly specifying and estimating the error covariance structure Ω. If the model for Ω is misspecified — for example, if you assume a particular heteroskedasticity pattern that doesn't match the true one — FGLS estimates can be worse than OLS. Robust standard errors (Huber-White for heteroskedasticity, Newey-West for serial correlation) make no assumptions about the form of Ω: they leave the OLS point estimates unchanged and only correct the standard errors. In small samples or when the variance structure is uncertain, the weaker assumptions of robust standard errors make them safer and more credible.

This is a classic robustness-efficiency tradeoff. FGLS is potentially more efficient (lower variance) but relies on getting the structure of Ω right. Robust standard errors sacrifice some efficiency but are valid under much broader conditions. In practice, many applied econometricians default to OLS with robust standard errors and only use FGLS when the error structure is well-understood and the efficiency gain is substantial — for instance, in time-series models where AR(1) errors are theoretically motivated and the autocorrelation parameter is precisely estimated.