  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 three 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 treatment, considerably shrinking the identified set. Finally, we
incorporate a stochastic dominance assumption which tightens the lower bound
for the MTE. Our analysis extends to discrete instruments. The results rely on
a mixture reformulation of the problem where the mixture weights are
identified, extending Lee's (2009) trimming procedure to the MTE context. We
propose estimators for the bounds derived and use data made available by Deb,
Munking and Trivedi (2006) to empirically illustrate the usefulness of our
approach.
