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