It has been suggested by Angrist and Pischke that Robust (i.e. robust to heteroskedasticity or unequal variances) Standard Errors are reported as a matter of course rather than testing for it. Two questions:
Using robust standard errors has become common practice in economics. Robust standard errors are typically larger than non-robust (standard?) standard errors, so the practice can be viewed as an effort to be conservative.
In large samples (e.g., if you are working with Census data with millions of observations or data sets with "just" thousands of observations), heteroskedasticity tests will almost surely turn up positive, so this approach is appropriate.
Another means for combating heteroskedasticity is weighted least squares, but this approach has become looked down upon because it changes the estimates for parameters, unlike the use of robust standard errors. If your weights are incorrect, your estimates are biased. If your weights are right, however, you get smaller ("more efficient") standard errors than OLS with robust standard errors.
Robust standard errors provide unbiased standard errors estimates under heteroscedasticity. There exists several statistical text books that provide a large and lengthy discussion on robust standard errors. The following site provides a somewhat comprehensive summary on robust standard errors:
https://economictheoryblog.com/2016/08/07/robust-standard-errors/
Coming back to your questions. Using robust standard errors is not without caveats. According to Woolridge (2009 edition, page 268) using robust standard errors, the t-statistics obtained only have distributions which are similar to the exact t-distributions if the sample size is large. If the sample size is small, the t-stats obtained using robust regression might have distributions that are not close to the t distribution. This could throw off inference. Furthermore, in case of homoscedasticity, robust standard errors are still unbiased. However, they are not efficient. That is, conventional standard errors are more precise than robust standard errors. Finally, using robust standard errors is common practice in many academic fields.
In Introductory Econometrics (Woolridge, 2009 edition page 268) this question is addressed. Woolridge says that when using robust standard errors, the t-statistics obtained only have distributions which are similar to the exact t-distributions if the sample size is large. If the sample size is small, the t-stats obtained using robust regression might have distributions that are not close to the t distribution and this could throw off inference.
There are a lot of reasons to avoid using robust standard errors. Technically what happens is, that the variances get weighted by weights that you can not prove in reality. Thus robustness is just a cosmetic tool. In general you should thin about changing the model.
There are a lot of implications to deal with heterogeneity in a better way than just to paint over the problem that occurs from your data. Take it as a sign to switch the model. The question is close related to the question how to deal with outliers. Some people just delete them to get better results, it's nearly the same when using robust standard errors, just in another context.
I thought that the White Standard Error and the Standard Error computed in the "normal" way (eg, Hessian and/or OPG in the case of maximum likelihood) were asymptotically equivalent in the case of homoskedasticity?
Only if there is heteroskedasticity will the "normal" standard error be inappropriate, which means that the White Standard Error is appropriate with or without heteroskedasticity, that is, even when your model is homoskedastic.
I can't really talk about 2, but I don't see the why one wouldn't want to calculate the White SE and include in the results.
I have a textbook entitled Introduction to Econometrics, 3rd ed. by Stock and Watson that reads, "if the errors are heteroskedastic, then the t-statistic computed using the homoskedasticity-only standard error does not have a standard normal distribution, even in large samples." I believe you cannot do proper inference/hypothesis testing without being able to assume your t-statistic is distributed as standard normal. I have a LOT of respect for Wooldridge (in fact, my graduate-level class also used his book) so I believe what he says about the t-stats using robust SEs require large samples to be appropriate is definitely correct, but I think we often have to deal with the large-sample requirement, and we accept that. However, the fact that using non-robust SEs won't give a t-stat with the proper standard normal distribution even if you DO have a large sample creates a much bigger challenge to overcome.
