equation. Since Y is an realized outcome as a result of selection D, Y can be written as Y = D × Y$_{1}$ + (1 − D ) × Y$_{0}$ , where Y$_{0}$ and Y$_{1}$ are potential outcomes for the treatment status 0 and 1, respectively. Let Z denote a scalar or vector-valued IV that is excluded from the outcome equation and Z denote the support of Z . For each z ∈ Z , let D$_{z}$ be the potential treatment participation when Z = z . Note that I allow the distribution of outcome unobservables to vary with the selection D . Also, I do not impose an additively separable structure on the unobservable in the outcome equation. In the selection equation, p ( Z ) − U can be interpreted as the net utility from treatment participation. 8 Note that selection on unobservables arises from dependence between ε$_{D}$ and U. Remark 1 Without loss of generality, I assume that U ∼ Unif (0 , 1) for normalization. Then p ( z ) = P [ D = 1 | Z = z ] is interpreted as a propensity score. Throughout this study, I impose the following assumptions on the model (1). M.1 (Monotonicity) m ( d, ε$_{d}$ ) is strictly increasing in a scalar unobservable ε$_{d}$ for each d ∈ { 0 , 1 } . M.2 (Continuity) For d ∈ { 0 , 1 } , the distribution function of ε$_{d}$ is absolutely continuous with respect to the Lebesgue measure on$_{R}$. M.3 (Exogeneity) Z ⊥ ( ε$_{0}$, ε$_{1}$, U ). M.4 (Propensity Score) The function p ( · ) is a nonconstant and continuous function for the continuous element in Z . M. 1 and M. 2 ensure the continuous distribution of Y$_{d}$ and invertibility of the function m ( d, ε$_{d}$ ) in the second argument, which is a standard assumption in the literature on nonparametric models with a nonseparable error. M. 3 is an instrument exogeneity condition. That is, the instrument Z exogenously affects treatment selection and it affects the outcome only through the treatment status. Furthermore, Z does not affect dependence among unobservables ε$_{0}$, ε$_{1}$ , and U . M. 4 is $^{8}$? showed that selection equation in the model (1) is equivalent to the most general form of the latent index selection model D = 1 [ s ( Z, V ) ≥ 0] where s is unknown function and V is a (possibly) vector-valued unobservable under monotonicity of the selection in the instruments. Technically, the condition means that for any z and z ′ in Z , if s ( z, v$_{0}$ ) > s ( z $^{′}$, v$_{0}$ ) for some v$_{0}$ ∈ V , s ( z, v ) > s ( z $^{′}$, v ) for almost every value of v ∈ V where V is the support of V. Intuitively, this implies that the sign of the change in net utility caused by the instruments does not depend on the value of the unobservable V . 6