Checking for normality with robust errors I am running a linear regression (just a single IV) and have selected the robust error option (vce robust) in Stata due to heteroscedasticity (and because it is sometimes recommended to do so anyway). However, try as I might, I cannot find any advice on whether I should be testing for normality after I have selected this robust option or whether running the robust option negates the need to do so. Any help on whether normality should be tested with this option checked would be greatly appreciated. 
BASED ON ANSWERS: My main focus is on understanding the regression model, so I will be looking at the (slope) coefficient and its 95% CI as well as statistical significance (I have a continuous IV). In another linear regression I had hoped to make predictions (with CI and, hopefully, PI) also.  I was OK with checking the assumptions of a regression analysis until I reached the option to use robust standard errors. From the answers received am I correct in saying that asymptotic normality is needed, but not readily/easily tested for (and is rarely tested in practice)? So I could run the regression with robust errors and not test for normality. I assume that other assumptions (e.g., unusual points) still hold. I checked Stata and it does seem that it predicts when robust errors are used. Is it correct to use these predictions?
 A: To make things simple, suppose you have 3 observations. Robust standard errors allow for a variance-covariance matrix of the errors to look like this:
$$\Sigma =
\begin{bmatrix}
    \sigma_{1}  & 0 & 0\\
    0 & \sigma_{2} & 0 \\
    0 & 0 & \sigma_{3}
\end{bmatrix}
$$
The diagonal terms are the variances of the errors for each of the 3 observations. The covariance terms are all zero because we still assume that the errors are uncorrelated across observations. If you want to relax that, you will need cluster-robust errors.
Ordinary, non-robust errors assume that $\sigma_1=\sigma_2=\sigma_2=\sigma$: all observations have the same (unknown) error variance. Neither the robust nor the non-robust VCE make any assumptions about the distribution of the error, such as normality. They only make assumptions about the variance being the same (homoskedasticity) or different across observations (heteroskedasticity).
Thus, depending on what you are doing, you may still need to test for normality of the errors. Alternatively, you may be able to bootstrap or rely on asymptotics for inference.  
A: If your goal is to use a fitted model to do prediction, then you cannot use robust standard errors because you need to make the assumption of normality as well as homoscedasticity. This is not how most people use linear models. Most use linear models to test for statistical significance of model parameters.
In this case, robust standard errors give you a general test of significance that depends neither on normality nor homoscedasticity. Furthermore, it has very comparable power to the classical (non-robust) standard errors when those assumptions happen to be met. For that reason, I agree with Stata and would advocate using robust standard errors for most analyses.  You would only not use robust standard errors if your $n<40$ because it behaves poorly in small samples.
It is important to note that even with classic standard errors, there is no normality assumption. I honestly don't know where this assumption came about, though I see it frequently mentioned in introductory texts. Even Gauss, in his seminal paper on the least squares estimation, specifically noted that normality is not a necessary assumption. 
It's the Lindeberg Feller central limit theorem ensures that, under a minimal number of assumptions the normal approximation to the sampling distribution of model parameters is very good. Regardless of whether robust or non-robust standard errors are used.
Nonetheless the answer to your question is that no, normality assumptions are not important with robust standard errors.
A: To answer your question: Should I test for normality, when I have used robust standard errors in my analysis?
The answer is: no.
Why? The assumption of normality gives OLS finite sample propperties when you also assume the Gauss-Markov (GM) assumptions to be true. When you use robust standard errors, you do not assume GM because homoscedlasticity is one of these assumptions.
Buttom line, the robust errors only have asymptotic justification and adding normality in this case gives you exactly nothing. OLS is already asymptotically normal regardless of the underlying distribution. 
EDIT: Note that when you add normality to the GM assumptions you are in the classical linear model setup. In this case OLS has i) finite sample properties and is the minimum variance unbiased estimator (it reaches the Cramer Rao lower bound). That is, you can fint no other (unbiased) estimator that gives you a better result.
