# How to determine if GLS improves on OLS?

I have a multiple regression model, which I can estimate either with OLS or GLS. The weights for the GLS are estimated exogenously (the dataset for the weights is different from the dataset for the model). I’m trying to determine if one estimation technique is “better” than the other by looking at the coefficients, t-stats, f-stat, $R^2$, … . I guess it’s a question about the weights in general. If I have multiple sets of weights (including uniform), how can I choose the set that will give the “best” estimation?

The real difference between OLS and GLS is the assumptions made about the error term of the model. In OLS we (at least in CLM setup) assume that $Var(u)=\sigma^2 I$, where I is the identity matrix – such that there are no off diagonal elements different from zero. With GLS this is no longer the case (it could be, but then GLS = OLS). With GLS we assume that $Var(u) = \sigma^2 \Sigma$, where $\Sigma$ is the variance-covariance matrix.

Many text books introduce GLS with WLS, which is the GLS function that eliminates heteroskedasticity (or tries to). This means that the usual t/F statistics can be valid for the GLS estimation, but not for the OLS. This is less troublesome today, since you can just compute robust variance estimates and base you inference on that – same as you normally would.

This implies that difference between OLS and GLS is in the variance of the estimates. And the real reason, to choose, GLS over OLS is indeed to gain asymptotic efficiency (smaller variance for n $\rightarrow \infty$. It is important to know that the OLS estimates can be unbiased, even if the underlying (true) data generating process actually follows the GLS model. If GLS is unbiased then so is OLS (and vice versa).

You can very easily proof this, but basically the assumptions for consistency/unbiasedness do not rely on the variance of the estimates at all.
A more subtle point is that, unless you know the actual GLS function, it is not unbiased but only consistent.

I would therefore argue that choosing between OLS and GLS based on estimates and $R^2$ is the wrong way to think about it. The estimates of both OLS and GLS should be close to one another, if not numerically then in the size of the “impact”. If they are not, then it would most likely indicate that you have a function form misspecification(s), of that you have left out variables.

I don’t know whether or not, excluding the GLS weights covariates is justified in your case – but perhaps it worth trying to include them in the OLS estimation and see what happens? It might make the reader less “suspicious” about your conclusion (but this is pure speculation on my part).