Counterfactual Mapping and Individual Treatment Effects in Nonseparable Models With Binary Endogeneity

This paper establishes nonparametric identification of individual treatment effects in a nonseparable model with a binary endogenous regressor. The outcome variable may be continuous, discrete, or a mixture of both, while the instrumental variable can take binary values. First, we study the case where the model includes a selection equation for the binary endogenous regressor. We establish point identification of the individual treatment effects and the structural function when the latter is continuous and strictly monotone in the latent variable. The key to our results is the identification of a so-called counterfactual mapping that links each outcome of the dependent variable with its counterfactual. Second, we extend our identification argument when there is no selection equation. Last, we generalize our identification results to the case where the outcome variable has a probability mass in its distribution such as when the outcome variable is censored or binary.