The DTE in the extended Roy model is written as follows: F$_{Δ}$ ( δ ) = E [Pr ( Y$_{1}$ − Y$_{0}$ ≤ δ | X )] = E [Pr ( Y$_{1}$ − Y$_{0}$ ≤ δ | X, z )] = E [ F$_{Δ}$ ( δ | 1 , X, z )] p ( z ) + E [ F$_{Δ}$ ( δ | 0 , X, z )] (1 − p ( z )) , where p ( z ) = Pr ( D = 1 | Z = z ), F$_{Δ}$ ( δ | d, , X, z ) = Pr ( Y$_{1}$ − Y$_{0}$ ≤ δ | D = d, X, Z = z ) for d ∈ { 0 , 1 } . French and Taber (2011) listed sufficient conditions under which the marginal distributions of potential outcomes are point-identified in the generalized Roy model. 11 Those assumptions also apply to the extended Roy model since it is a special case of the generalized Roy model. Under their conditions, conditional marginal distributions of Y$_{0}$ and Y$_{1}$ on the treated ( D = 1) and untreated ( D = 0) are also all point-identified. Note that given Z = z , the treated and untreated groups correspond to the regions { Y$_{1}$ − Y$_{0}$ ≥ m$_{C}$ ( z ) } and { Y$_{1}$ − Y$_{0}$ < m$_{C}$ ( z ) } respectively. Let F$_{d}$$_{1}$ ( y | d$_{2}$, z ) = Pr ( Y$_{d}$$_{1}$ ≤ y | D = d$_{2}$, Z = z ) . Bounds on the DTE are obtained based on the identified marginal distributions on the treated and untreated as follows: for d ∈ { 0 , 1 } , where with $^{11}$See Assumption 4.1-4.6 in French and Taber (2011). These assumptions include some high level conditions such as the full support of both instruments and of exclusive covariates for each sector. If those conditions are not satisfied, the marginal distributions may only be partially identified. 27 a$_{k}$ ≤ a$_{k}$$_{+1}$ ≤ a$_{k}$ + δ − m$_{C}$ ( z ) , F L Δ ( δ | 1 , z ) =        sup { a$_{k}$ } ∞ k = −∞ ∞ ∑ k = −∞ max$^{}$     F$_{1}$ ( a$_{k}$$_{+1}$ + m$_{C}$ ( z ) | 1 , z ) − F$_{0}$ ( a$_{k}$ | 1 , z ) , 0      , for δ ≥ m$_{C}$ ( z ) , 0 , for δ < m$_{C}$ ( z ) , F L Δ ( δ | d, z ) ≤ F$_{Δ}$ ( δ | d, z ) ≤ F U Δ ( δ | d, z ) ,