Abstract

The control function approach is a popular method for identifying causal parameters in binary response models with endogenous variables. We extend the control function approach so that it can admit discrete endogenous variables. We characterize models using a set-valued control function that contains unobservable control variables. Conditioning on the set-valued control function yields an incomplete threshold-crossing structure, which we exploit to obtain sharp bounds on structural parameters and their functionals, such as average treatment effects. We show that the approach can be combined flexibly with outcome equations that admit rich treatment effect heterogeneity and selection equations that allow rich compliance types and multiple instrumental variables. Numerical illustrations show that implied bounds can provide complementary identifying information to existing bounds.