An essential feature common to all empirical social research is variability across units of analysis. Individuals differ not only in background characteristics, but also in how they respond to a particular treatment, intervention, or stimulation. Moreover, individuals may self-select into treatment on the basis of their anticipated treatment effects. To study heterogeneous treatment effects in the presence of self-selection, Heckman and Vytlacil (1999, 2001a, 2005, 2007b) have developed a structural approach that builds on the marginal treatment effect (MTE). In this paper, we extend the MTE-based approach through a redefinition of MTE. Specifically, we redefine MTE as the expected treatment effect conditional on the propensity score (instead of all observed covariates) as well as a latent variable representing unobserved resistance to treatment. The redefined MTE improves upon the original MTE in a number of aspects. First, while it is conditional on a unidimensional summary of covariates, it is sufficient to capture all of the treatment effect heterogeneity that is consequential for selection bias. Second, the new MTE is a bivariate function, and thus is easier to visualize than the original MTE. Third, as with the original MTE, the new MTE can also be used as a building block for evaluating standard causal estimands such as ATE and TT. However, the weights associated with the new MTE are simpler, more intuitive, and easier to compute. Finally, the redefined MTE immediately reveals treatment effect heterogeneity among individuals who are at the margin of treatment. As a result, it can be used to evaluate a wide range of policy changes with little analytical twist, and to design policy interventions that optimize the marginal benefits of treatment
