Robust standard error in generalized least squares regression

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 $\mathbf{V}$?

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.
• One last thing: unfortunately, the sample sizes I'm working with are usually ridiculously small: 4-7 points, as those are the number of dose-specific relative risks typically reported in epidemiological papers (and I have to fit a trend passing through those data). Thus, I am not only worried about the SE, but also about the $\beta{}$ coefficient for the trend. This is why I would exclude OLS+robust SE as a possible solution to my problem. – boscovich Sep 29 '12 at 12:46