  The control function approach allows the researcher to identify various
causal effects of interest. While powerful, it requires a strong invertibility
assumption, which limits its applicability. This paper expands the scope of the
nonparametric control function approach by allowing the control function to be
set-valued and derive sharp bounds on structural parameters. The proposed
generalization accommodates a wide range of selection processes involving
discrete endogenous variables, random coefficients, treatment selections with
interference, and dynamic treatment selections.
