Question about OLS and BLUE in the presence of hetereoscadasticity and robust standard errors My understanding that if errors are non-spherical, OLS is no longer the minimum variance linear unbiased estimator (assume the error terms are fully independent of all covariaties- so unbiasedness holds).
So we can use robust standard errors to still obtain consistent estimation of standard errors when they are not homoscedastic. Now, if we use robust standard errors and that's it- have we restored 'BLUE"? i.e. does this just give us consistent estimation of the standard errors, or does it also restore the 'Best', minimum variance property?
 A: 
Now, if we use robust standard errors and that's it- have we restored
'BLUE"? i.e. does this just give us consistent estimation of the
standard errors, or does it also restore the 'Best', minimum variance
property?

It just gives you consistent estimation of the standard errors. It doesn't change the point estimates, so it can't possibly make them any more BLUE.
A: First choice in practice, I would (by examining regression residues, for example) postulate a crude relationship to account for the change in volatility as a function of a model's variable(s). This is, in effect, an attempt to restore BLUE properties.
Common models in practice include, for example, weighted least-squares, which follows from noting that error magnitudes may historically, for select datasets, move roughly in accord with the size of a variable.
Interestingly, if you believe that underlying fundamentals (like in risk levels) have changed, you could quite originally postulate (based on sound arguments) that a regime change in volatility, starting at a particular time point, has occurred (so a set of more recent observations could be down-weighted).
Note: applying an appropriate data transformation (see discussion here), at times, may obviate the need for such a volatility model correction.
If the above provides marginal results, consider employing robust regression as an option.
