How Robust Standard Errors Expose Methodological Problems They Do Not Fix, and What to Do About It
How Robust Standard Errors Expose Methodological Problems They Do Not Fix, and What to Do About It (DOI: https://doi.org/10.1093/pan/mpu015) was written by Gary King and Margaret E. Roberts in 2015. It was published in Political Analysis (vol. 23, no. 2).
The authors argue that heteroskedasticity consistent errors are inappropriate to use in many circumstances. Generally, if estimators under classical and robust standard errors diverge, there must be a specification error.
If an OLS model assuming homoskedastic errors is fit to data generated with heteroskedastic errors, the estimators are not biased. It is the classical standard errors that are affected by that assumption's failure, as the estimates are now inefficient. So if the estimates change with introduction of robust standard errors, there must be a specification error. (For example, an omitted variable leads to heteroskedastic-appearing errors. The variance that could have been explained devolves to the error term, and usually is not constant.)
More generally, classical and robust standard errors should be very similar. Their similarity demonstrates that the model is robust to heteroskedasticity.
Furthermore, heteroskedasticity consistent errors are asymptotically unbiased; they can be biased for small n.
The authors generalize the information matrix test using a bootstrapping method. They recommend this for evaluation of a model for detecting specification errors.