Endogeneity-Robust OLS Estimation (?)

Imagine White-style robust OLS inference, but with robustness to endogeneity as opposed to heteroskedasticity/autocorrelation (or maybe even robustness to all three).  It sounds too good to be true.  Actually, it sounds impossible, and even if somehow possible, it would of course require not only post-OLS tweaking of standard errors but also post-OLS tweaking of coefficient estimates.  But there are actually some emerging results – embryonic and requiring strict conditions -- but results nonetheless.  Quite intriguing.  Check out Jan Kiviet’s latest paper, "When is it Really Justifiable to Ignore Explanatory Variable Endogeneity in a Regression Model?"