content_image/308367.jpg content_image/308366.jpg
z ∈ Z , 

<img src='content_image/308367.jpg'>

The first equality follows from M. 3. Now Fr´ echet-Hoeffding bounds can be established on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) and P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 1 , z ) based on point-identified P ( y$_{0}$ | 0 , z ) and partially identified P$_{1}$ ( y$_{1}$ | 0 , z ) , and partially identified P$_{0}$ ( y$_{0}$ | 1 , z ) and point-identified P ( y$_{1}$ | 1 , z ) , respectively. 

Note that when marginal distributions are partially identified, sharp bounds on the joint dis- tribution are obtained by taking the union of Fr´ echet-Hoeffding bounds over all possible pairs of marginal distributions. Similarly, the DTE can be written as 

<img src='content_image/308366.jpg'>

and Makarov bounds can be applied to P ( Y$_{1}$ − Y$_{0}$ ≤ δ | 0 , z ) and P ( Y$_{1}$ − Y$_{0}$ ≤ δ | 1 , z ) based on point- identified P ( y$_{0}$ | 0 , z ) and partially identified P$_{1}$ ( y$_{1}$ | 0 , z ) , and partially identified P$_{0}$ ( y$_{0}$ | 1 , z ) and point-identified P ( y$_{1}$ | 1 , z ) , respectively. 

The specific forms of sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$, their joint distribu- tion, and the DTE under M. 1 − M. 4 are provided in Theorem 1 in Appendix. 

## 3.2 Negative Stochastic Monotonicity 

In this subsection, I additionally impose NSM on dependence between ε$_{0}$ and U and between ε$_{1}$ and U . I show that NSM has additional identifying power for marginal distributions, but not on the joint distribution nor on the DTE. 

First, I use NSM to tighten the bounds on counterfactual probabilities P$_{1}$ ( y, 0 | z ) and P$_{0}$ ( y, 1 | z ). Consider a counterfactual distribution P$_{1}$ ( y | 0 , z ) = P $^{(}$ε$_{1}$ ≤ m − $^{1}$(1 , y ) | p ( z ) < U $^{)}$. If p ( z ) < p, 

