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?