Sharp bounds on the MTE with sample selection

I propose a Generalized Roy Model with sample selection that can be used to analyze treatment effects in a variety of empirical problems. First, I decompose, under a monotonicity assumption on the sample selection indicator, the MTR function for the observed outcome when treated as a weighted average of (i) the MTR on the outcome of interest for the always-observed sub-population and (ii) the MTE on the observed outcome for the observed-only-when-treated sub-population, and show that such decomposition can provide point-wise sharp bounds on the MTE of interest. I, then, show how to point-identify these bounds when the support of the propensity score is continuous. After that, I show how to (partially) identify the MTE of interest when the support of the propensity score is discrete.