a$_{k}$ ≤ a$_{k}$$_{+1}$ ≤ 1 S$_{1}$$^{{}$ ( a$_{k}$ + δ ) + 1 S$_{0}$ w$^{}}$ , S$_{1}$ = t$_{1}$ − t$_{W}$ t$_{0}$ − t$_{W}$ , S$_{0}$ = t$_{0}$ − t$_{W}$ t$_{1}$ − t$_{0}$ . with Proof. See Appendix A. 3.2.3 Roy Model Establishing sharp DTE bounds under support restrictions allows us to derive sharp DTE bounds in the Roy model. In the Roy model, each agent selects into treatment when the net benefit from doing so is positive. The Roy model is often divided into three versions according to the form of its selection equation: the original Roy model, the extended Roy model, and the generalized Roy model. Most of the recent literature considers the extended or generalized Roy model that accounts for nonpecuniary costs of selection. Consider the generalized Roy model in Heckman et al. (2011) and French and Taber (2011): where X is a vector of observed covariates while ( U$_{1}$, U$_{0}$ ) are unobserved gains in the equation of potential outcomes. In the selection equation, Z is a vector of observed cost shifters while U$_{C}$ is an unobserved scalar cost. The main assumption in this model is As two special cases of the generalized Roy model, the original Roy model assumes that µ$_{C}$ ( Z ) = U$_{C}$ = 0 and the extended Roy model assumes that each agent’s cost is deterministic with U$_{C}$ = 0. My result provides DTE bounds in the extended Roy model: 26 Y = m ( D, X ) + U$_{D}$, D = 1 { Y$_{1}$ − Y$_{0}$ ≥ m$_{C}$ ( Z ) } . ( U$_{1}$, U$_{0}$, U$_{c}$ ) ⊥ ( X, Z ) . Y = µ ( D, X ) + U$_{D}$, D = 1 { Y$_{1}$ − Y$_{0}$ ≥ m$_{C}$ ( Z ) + U$_{C}$ } ,