Identifying Marginal Treatment Effects in the Presence of Sample Selection

This article presents identification results for the marginal treatment effect (MTE) when there is sample selection. We show that the MTE is partially identified for individuals who are always observed regardless of treatment, and derive uniformly sharp bounds on this parameter under four increasingly restrictive sets of assumptions. The first result imposes standard MTE assumptions with an unrestricted sample selection mechanism. The second set of conditions imposes monotonicity of the sample selection variable with respect to the treatment, considerably shrinking the identified set. Third, we incorporate a stochastic dominance assumption which tightens the lower bound for the MTE. Finally, we provide a set of conditions that allows point identification for completeness. Our analysis extends to discrete instruments and distributional MTE. All the results rely on a mixture reformulation of the problem where the mixture weights are identified. This extends Lee (2009) trimming procedure to the MTE context. We propose nonparametric estimators for the bounds derived, provide a numerical example and simulations that corroborate the bounds feasibility and usefulness as an empirical tool. We use data made available by Deb, Munkin, and Trivedi (2006) to illustrate the identifying power of our assumptions in a real-world data generating process