Some time ago I wrote about a paper by Arthur Lewbel in the Journal of Business & Economic Statistics in which he develops a method to do two-stage least squares regressions without actually having an exclusion restrictions in the model. The approach relies on higher moment restrictions in the error matrix and works well for linear or partly linear models. Back then, I expressed concerns that the estimator does not seem to work when an endogenous regressor is binary though; at least not in the simulations I have carried out.
After a bit of email back-and-forth we were able to settle the debate now. Arthur has written a note in which he shows that it’s indeed possible to fulfill all assumptions for his method to work in the binary case. The drawback, however, is that you have to accept very strong distributional assumptions on the error structure—much stronger than in the linear case. It might be hard to justify these assumptions on the grounds of economic theory. Moreover, even if all requirements are fulfilled exactly with simulated data, the small-sample properties of the estimator seem to be quite bad. Anyhow, having these limitations in mind, it’s good to have these tools available in applied work. Nevertheless, you should probably consider taking a continuous measure instead of a binary, if you have the option available.