Robust standard error in generalized least squares regression

Question

Suppose we have a correlated outcome $\mathbf{y}$ and a bunch of predictors $\mathbf{X}$. For some reason, we know the variance/covariance matrix of the error term $(\epsilon)$, say $\mathbf{V}$.

In this scenario, it is reasonable to utilize the Generalized Least Squares.

Through Cholesky decomposition, we can calculate $\mathbf{P'}\mathbf{P}=\mathbf{V}$. It follows that we can use Ordinary Least Squares for the model $\mathbf{z}=\mathbf{Q\beta}+\mathbf{f}$, where $\mathbf{z}=\mathbf{P^{-1}y}$, $\mathbf{Q}=\mathbf{P^{-1}X}$ and $\mathbf{f}=\mathbf{P^{-1}\epsilon}$.

What would be in this case, if any, the advantage(s) introduced by using a robust variance estimator for the model $\mathbf{z}=\mathbf{Q\beta}+\mathbf{f}$? Would it relax the assumption that the matrix $\mathbf{V}$ we used is actually the right variance/covariance matrix for $\epsilon$? Does it even make sense using a robust variance estimator in this scenario, since we (pretend to) know $\mathbf{V}$?

Answer 1

Pretending to know the true variance is always a funny thing to see. I would never trust that sort of an assumption, unless it comes from a carefully designed experiment where an artificial correlation structure was imposed (although frankly I cannot really imagine a practical set up that would lead to this).

From the GEE perspective, you should be able to get more efficient estimates when your assumption about $\bf V$ is correct, as compared to OLS, but you would still want to use the sandwich variance estimator in the (highly unlikely) case that you are mistaken about the covariances. Usually, robustness to model assumptions is considered a greater issue than efficiency, unless you have really tiny sample sizes, and any 20% efficiency gain is a big deal. So I would run this with as the GLS (or feasible GLS, if you only know the structure of $\bf V$, but not the specific parameter values), but still correct for clustering using the sandwich variance estimator.