Do I get the heteroskedasticity-robust standard errors from my OLS or WLS regression? I have a multiple regression linear model which I ran a simple OLS test on.
I then performed the White test and found that it was heteroskedastic.
Then I performed a Weighted Least Squares regression on the model, to account for the heteroskedasticity.
Now I need to use heteroskedasticity-robust standard errors (HRSE) for my inference, but I'm not sure whether to get the SEs from my OLS regression or my WLS one.
I'm asking because when I run a coefficient test on the WLS regression with or without HRSE, I get the same inference results; but I'm not sure if it makes sense to get HRSEs from a WLS regression, since WLS already accounts for heteroskedasticity.
Thanks in advance for any help!
Do I get the HRSEs from my OLS, or WLS regression?
 A: If your OLS model is heteroskedastic, you can either use heteroskedasticity-robust standard errors for the OLS model (such as Huber-White standard errors) or use a WLS model instead of your OLS model. 
Both are methods for correcting for the violation of the homoskedasticity assumption in the OLS model. 
A: 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.
A: You should run OLS and do the heteroskedasticity robust standard errors. Unless you have a very clear evidence that your variability changes by a factor of maybe five between the extremes in your sample, you should stick to OLS because doing it right with

*

*Test for heteroskedasticity and

*If rejected, run a conditional WLS model

-- that process produces crazy distributions and an extreme difficulties in controlling for the type I error. This used to be a popular niche theme in theoretical econometrics in 1990s (https://www.tandfonline.com/doi/abs/10.1080/07474939708800376), but it seems to have been forgotten since.
