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. 

## 2.2 Objects of Interest and Assumptions 

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 

