Can robust standard errors be less than those from normal OLS? I'm reading about Robust Standard Error Estimators for Panel Models from the developer of plm R package (Millo, 2017: 21). But my question is not about software.
In the example I see that some heteroscedasticity robust standard errors are in fact less than those in benchmark OLS. I thought that we normaly expected robust errors to be at least as large as their OLS counterparts.
Below you see the comparison of different HC procedures, for the same model with coefficients arranged by columns.

My question concerns this situation. Is it fine in general and how to interpret this fact if, i.e. in real model, such a change in s.e. changes the significance of a coefficient?

If the response is, conditionally on the covariates, Gaussian, then we know that OLS are best linear unbiased estimators by Gauss-Markov theorem; no other estimator can have lower variance than OLS.
If the Gaussian assumption is not met we have no such guarantee anymore. Indeed, since the influence function (IF) of the OLS estimator is unbounded and the variance of the OLS estimator is proportional to the IF, then the variance of the estimator is unbounded as well. In practice, this means that even a single aberrant observation can drive away the OLS estimate along with its variance.
On the other hand, robust estimators (e.g. Huber (1964) DOI: 10.1214/aoms/1177703732, Hampel et al. (1984) DOI:10.1002/9781118186435, etc.) typically have a bounded IF, thus they deliver bounded estimates with bounded variance.
The point of this is that a "bad" observation will have only a limited impact on the robust estimate and its variance, whereas it can have a large effect on an estimate obtained from an estimator with unbounded IF such as OLS.
