Counterfactual Mapping and Individual Treatment Effects in Nonseparable Models with Discrete Endogeneity

Author(s): Quong Vuong and Haiqing Xu
Date: August 2014
Type: CRATE Working Papers, CRATE-2014-4
doi: download pdf

Abstract

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, and the instrumental variable can take binary values. We distinguish the cases where the model includes or does not include a selection equation for the binary endogenous regressor. First, we establish point identification of the structural function when it is continuous and strictly monotone in the latent variable. The key to our results is the identification of a so-called “counter- factual mapping” that links each outcome with its counterfactual. This then identifies every individual treatment effect. Second, we characterize all the testable restrictions on observables imposed by the model with or without the selection equation. Lastly, 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.