In this paper we propose a new unifying approach to (partially) identify potential outcome distributions in a non-separable triangular model with a binary endogenous variable and a binary instrument. Our identification strategy provides a testable condition under which the objects of interest are point identified. When point identification is not achieved, we provide sharp bounds on the potential outcome distributions and the difference of marginal distributions. . 1 In the presence of the

more »

... nous covariate X, this assumption is strengthened. X is left out to simplify the notation, and its addition will be discussed in Section 3. 1 and the difference in marginal distributions of the whole population. Identification of these quantities are especially important for the analyses of heterogeneous treatment effects. As mentioned by Imbens and Rubin (1997) and Angrist and Pischke (2008) , these distributions are useful for policy makers who want to take into account differences in the dispersion of earnings when contemplating the merits of one program or treatment versus another. Our main contribution is to provide a new unifying approach to (partially) identify potential outcome distributions in this setup. When g is strictly monotone in U (Y is therefore continuously distributed), bounds collapse to a point and the point identification result achieved by Vuong and Xu (2014) is recovered. When the outcome variable Y is binary, bounds are exactly as those found by Shaikh and Vytlacil (2011). This identification strategy allows a testable condition to be derived under which objects of interest are point identified. This testable condition reveals that identification can be achieved, even if g is weakly monotone in U, which encompasses the cases where Y is either censored, truncated, discrete, or mixed continuous-discrete outcomes. A model similar to Equation (1) has also been studied by Vytlacil and Yildiz (2007). The identification strategy in their study requires the existence of an additional exogenous covariate and provides rank conditions based on exogenous covariates, under which it is possible to identify the average effect. Recently, Vuong and Xu (2014) generalized Vytlacil and Yildiz (2007)'s rank condition to point identify the quantile functions. Both papers do not discuss partial identification when the proposed rank condition fails to hold. Our paper complements this research by providing sharp bounds on potential outcome distributions whenever rank condition fails to hold. When the rank condition holds our bounds coincide with the identification results of these studies. The rest of the paper is organized as follows. Section 2 presents the identification strategy. Section 3 generalizes the method to the case where additional exogenous covariates are available. Proofs are collected in Appendix. IDENTIFICATION STRATEGY We make the following assumptions: