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 observable 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 observable 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 for the always-observed sub-population. Moreover, I impose an extra mean dominance assumption and tighten the previous bounds. I, then, show how to point-identify those 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. At the end, I estimate bounds on the MTE of the Job Corps Training Program on hourly wages for the always-employed sub-population and find that it is decreasing with the likelihood of attending the program for the Non-Hispanic group. For example, I find that the ATT is between $.38 and $1.17 while the ATU is between $.73 and $3.14.