necessary to ensures sharpness of the bounds. It requires that when some elements of the IV are continuous, the propensity score function p ( · ) be continuous for the continuous elements of IV when the discrete elements of IV are held constant. See ? for details. Remark 2 ? showed that under M. 3 , the selection equation in the model (1) is equivalent to the assumptions in the LATE framework developed by ? : independence and monotonicity. The LATE independence condition assumes that Z ⊥ ( Y$_{0}$, Y$_{1}$, U ) and that the propensity score p ( z ) is a nonconstant function. The LATE monotonicity condition assumes that either D$_{z}$ ≥ D$_{z}$ ′ or D$_{z}$ ′ ≥ D$_{z}$ with probability one for ( z, z $^{′}$) ∈ Z × Z with z = z $^{′}$. Numerous examples fit into the model (1). I refer to the following three examples throughout the paper. Example 1 (The effect of job training programs on wages) Let Y be a wage and D be an indicator of enrollment for the program. Let Z be the random assignment for the training service when the program designs randomized offers in the early application process. Note that such a randomized assignment has been widely used as a valid instrument in the LATE framework, which is equivalent to the model (1) considered in this paper. Example 2 (College premium) Let Y be a wage and D be the college education indicator. The literature including ? has used the distance to college, local wage, local unemployment rate, and average tuition for public colleges in the county of residence as IV. Example 3 (The effect of smoking on infant birth weight) Let Y be an infant birth weight and D be a smoking indicator. In the empirical literature, state cigarette taxes, policy interventions including tax hikes, and randomized counselling have been used as IV. The objects of interest here are the marginal distribution functions of Y$_{0}$ and Y$_{1}$, F$_{0}$ ( y$_{0}$ ) and F$_{1}$ ( y$_{1}$ ), their joint distribution function F ( y$_{0}$, y$_{1}$ ), and the DTE F$_{Δ}$ ( δ ) = P ( Y$_{1}$ − Y$_{0}$ ≤ δ ) for fixed y$_{0}$ , y$_{1}$ , and δ in $_{R}$. I obtain sharp bounds on F$_{0}$ ( y$_{0}$ ) , F$_{1}$ ( y$_{1}$ ) , F ( y$_{0}$, y$_{1}$ ) , and F$_{Δ}$ ( δ ) under various weak restrictions. First, I derive worst case bounds making use of only M. 1 − M. 4 in the model (1). The conditions M. 1 − M. 4 are maintained throughout this study. Second, I impose 7 2.2 Objects of Interest and Assumptions