# Using Weighted Least Squares with Robust Standard Errors

Weighted least squares (WLS) and robust standard errors are sometimes presented as alternative approaches for obtaining reliable standard errors of estimates of regression coefficients in the presence of heteroscedasticity. However, I notice that my software (gretl) offers robust standard errors as an option when using WLS.

A situation in which it seems this might be useful is where, in a regression of Y on X, there is a clear reason for heteroscedasticity, for example a scale effect such that larger values of Y are expected to be associated with larger variances. One might then use WLS, giving a higher weighting to observations with smaller Y (or, perhaps better, to observations with smaller E[Y | X], as inferred from an initial OLS regression). However, it might be found that the WLS residuals suggested some remaining heteroscedasticity that the weighting had not eliminated. This would suggest that the standard errors estimated by WLS might not be entirely reliable, and to address this one might opt for robust standard errors (rather than attempting to do so via some more complex weighting pattern).

Question: Assuming the number of observations is reasonably large (over 100, say), are there any pitfalls in using WLS with robust standard errors when estimating standard errors of regression coefficients?

• This answer math.stackexchange.com/questions/681332/robust-standard-errors/… , although dealing with robust standard errors in the OLS case, and not in the WLS case, it discusses why one should not use robust errors uncritically, and so perhaps it may be of some use to you. Mar 21 '14 at 2:30
• @AlecosPapadopoulos Thanks ... so applying this to WLS, it is important to test for any remaining heteroscedasticity, eg by examining the WLS residuals, rather than automatically opting for robust standard errors. Mar 21 '14 at 7:23

I'll define the efficiency of the estimate as the true asymptotic variance/covariance matrix of the coefficients. Of course, from Gauss-Markov we know that only when you select weights proportional to the inverse conditional variance of each observation will you achieve the best limiting unbiased limit.$^1$ So based on first order, asymptotic concerns, we may just take the best stab at estimating the weightings we can, then go ahead and just robust standard errors to guard against mistakes in the weights.
$^1$ Proof here, apparently originally due to Aitchen.