Set-Valued Control Functions
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.
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