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I know that GLS estimators only have exact distributions asymptotically, so the efficiency gains in finite samples are not all that clear. But otherwise, I'm struggling on how to attack this discussion.

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The answer to the question in the title is "Not really".
We have a linear regression model (matrix notation) $y = X\beta + u$, where $\operatorname {Var}(u) = \sigma^2V$, with $V$ unknown. Then the Feasible Generalized Least Squares estimator (FGLS) is

$$\hat \beta_{FGLS} = \left(X'\hat V^{-1}X\right)^{-1}X'\hat V^{-1}y$$

What are the finite-sample properties of this estimator? To quote Hayashi (2000), p.59

"If $V$ is estimated from the sample, $\hat V$ becomes a random variable, which affects the distribution of the GLS estimator. Very little is known about the finite-sample properties of the FGLS estimator".

This has not changed much in the intervening years, although for example, Ullah, A., & Huang, X. (2006). Finite Sample Properties of FGLS Estimator for Random-Effects Model under Non-Normality. ch. 3 in Contributions to Economic Analysis, 274, 67-89., provide (approximate) results for the Bias and MSE of the FGLS estimator in the context of panel-data under normality and non-normality of the errors (under normality FGLS is approximately unbiased).

Asymptotically, with only heteroskedasticity present, a simple version of FGLS, the Weighted Least Squares Estimator (WLS) has been proven to be more efficient than OLS, even when the $V$ is estimated from the sample, but under the assumption that the functional form of heteroskedasticity is correctly specified -if it is not, then the finite-sample reality may favor the OLS estimator because it estimates fewer population parameters.

So, as is usual the case, no clear-cut rule of thumb can be offered.

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