Identification of the Marginal Treatment Effect with Multivalued Treatments

Heckman et al. (2008) examine the identification of the marginal treatment effect (MTE) with multivalued treatments by extending the local instrumental variable (LIV) approach of Heckman and Vytlacil (1999). Lee and Salani\'e (2018) study the identification of conditional expectations given unobserved heterogeneity; in Section 5.2 of their paper, they analyze the identification of MTE under the same selection mechanism as in Heckman et al. (2008). We note that the construction of their model in Section 5.2 in Lee and Salani\'e (2018) is incomplete, and we establish sufficient conditions for the identification of MTE with an improved model. While we reduce the unordered multiple-choice model to the binary treatment setting as in Heckman et al. (2008), we can identify the MTE defined as a natural extension of the MTE using the binary treatment defined in Heckman and Vytlacil (2005). Further, our results can help identify other parameters such as the marginal distribution of potential outcomes.