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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.

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I would still like others to maybe contribute if they have something substantive to say, but I think this personally cleared the issue up for me:

http://www.nber.org/WNE/lect_6_controlfuncs.pdf

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.

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Petrin and Train (2009) is one example of endogeneity being handled with a control function. E.g. a WP here: http://eml.berkeley.edu/~train/petrintrain.pdf

I have always found the distinction a bit tricky. I guess in CF you typically invoke more specific assumptions about how two error terms are related. In IV, you rely on an instrument. The classical example covered by Imbens and Wooldridge as well is the Heckman selection model: here, under normality, you get the inverse mill's ratio. Therefore, you can estimate the model even in absence of an instrument, just relying on functional form only (although it is not advisable).

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