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 we derive sharp bounds on this parameter under four sets of assumptions. The first identification result combines the standard MTE assumptions without any restrictions to the sample selection mechanism. The second result 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. We therefore extend the Lee (2009) trimming procedure to the MTE context. We propose some preliminary estimators for the bounds derived, provide a numerical example and simulations that corroborate the bounds feasibility and usefulness as an empirical tool. In future drafts, we plan to highlight the practical relevance of the results by analyzing the impacts of managed health care options on health outcomes and expenditures, following Deb, Munkin, and Trivedi (2006).