You can get robust standard errors by using the Sandwich estimator. The optimization problem for the wOLS problem is: $$ \max_{\beta} -\frac{1}{2} \sum_{i=1}^{n} w_i(y_i - X_i\beta)^{2}. $$ The maximizer of this objective function is the MLE for the model defined by: \begin{equation} y_i \sim N(X_i\beta, w_i^{-1}) \end{equation}

Taking a the first derivative, we get an estimating equation, the solution of which is the MLE for $\beta$: $$ \sum_{i=1}^{n} w_i X_i(y_i - X_i\beta) = 0 $$

The information is given by: $$ A(\beta) = X^{T}WX. $$

If one believes the normal heteroskedastic model then the MLE is an efficient estimator of $\beta$ and it has a covariance matrix $A(\beta)^{-1}$. Otherwise, the MLE is consistent (though not efficient) and a consistent estimator for the variance is given by: $$ A(\beta)^{-1} B(\beta) A(\beta)^{-1} \rightarrow^P Cov(\hat\beta) $$ with: $$ B(\beta) = \sum_{i=1}^{n} w_i^{2}(y_i - X_i\beta)^{2}X_i^{T}X_i, $$ this is the sandwich estimator. A good reference for the derivation of the sandwich variance estimate is Van Der Vaart's Asymptotic Statistics http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521784506.

You can get robust standard errors by using the Sandwich estimator. The optimization problem for the wOLS problem is: $$ \max_{\beta} -\frac{1}{2} \sum_{i=1}^{n} w_i(y_i - X_i\beta)^{2}. $$ The maximizer of this objective function is the MLE for the model defined by: \begin{equation} y_i \sim N(X_i\beta, w_i^{-1}) \end{equation}

Taking a the first derivative, we get an estimating equation, the solution of which is the MLE for $\beta$: $$ \sum_{i=1}^{n} w_i X_i(y_i - X_i\beta) = 0 $$

The information is given by: $$ A(\beta) = X^{T}WX. $$

If one believes the normal heteroskedastic model then the MLE is an efficient estimator of $\beta$ and it has a covariance matrix $A(\beta)^{-1}$. Otherwise, the MLE is consistent (though not efficient) and a consistent estimator for the variance is given by: $$ A(\beta)^{-1} B(\beta) A(\beta)^{-1} \rightarrow^P Cov(\hat\beta) $$ with: $$ B(\beta) = \sum_{i=1}^{n} w_i^{2}(y_i - X_i\beta)^{2}X_i^{T}X_i, $$ this is the sandwich estimator. A good reference for the derivation of the sandwich variance estimate is Van Der Vaart's Asymptotic Statistics | Cambridge Series in Statistical and Probabilistic Mathematics.