Instrumental variables vs control function: Which approach and why to handle endogeneity?

Question

I'm curious if anyone here can summarize the differences between the I.V. and control function approach to handling endogeneity. I think endogeneity is usually addressed using the 2SLS or IV approach and, having briefly studied the control function approach, am unsure what the CF approach offers that 2SLS and IV approaches do not offer.

Can anyone help me clarify here? When would I use a CF approach rather than 2SLS, what does this approach offer that the other two do not etc.

I am a newbie with metrics so any and all input is appreciated.

Answer 1

The Control Function Approach (proposed by Heckman-1979 if I remember correctly) , is similar to an IV strategy in spirit, except that it can be used in a non-linear setting, such as Probit/Logit models etc. For instance, if you have endogeneity in a logit model, you cannot just run a 2SLS procedure as the first step is a linear projection of your x on your z, and the second step is nonlinear. There are two ways of handling endogeneity in nonlinear models: control function approach and a special regressor approach.