# Difference Between IV and Control Function in a Non-Linear Model

I've noticed that when addressing the problem of linear endogeneity that the different techniques result in the same estimator for the potentially endogenous variable. Specifically, the control function approach ends up with the same estimator as instrumental variable approaches.

Considering that, I've heard it mentioned that in non-linear models these approach do not result in the same estimator. Can anyone explain or potentially provide me with a good reference with the derivations? If so, that would be very much appreciated.

I quite like the Imbens and Wooldridge notes, but they're not very general.

So, it appears in the non-linear setting the CF approach imposes a linear conditional expectation on the endogenous variable, i.e. $E[u_{2}|z,y_2]$ has a linear conditional expectation. Note: I have listed $u_2$ as the error term in the regression of the endogenous variable, $y_2$ on the exogenous variables $z$. Apparently this assumption is more stringent than simply relying on linear projections as IV does.