and these bounds are sharp. Now, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ are obtained by plugging the results in Lemma 2 into the counterfactual probabilities. Note that under NSM, sharp bounds on the joint distribution and sharp bounds on the DTE are still obtained from Fr´echet-Hoeffding bounds and Makarov bounds. To illustrate this, consider the case where ρ$_{0}$ = ρ$_{1}$ = 0 in the example (2). 13 This case satisfies NSM and NSM does not impose any restriction on the dependence between ν$_{0}$ and ν$_{1}$ . Therefore, sharp bounds on the joint distribution and the DTE are obtained by the same token as in Subsection 3.1. 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 and NSM are provided in Corollary 1 in Appendix. Unlike NSM, CPQD has no additional identifying power for the joint distribution and the DTE. In this subsection, I impose weak positive dependence between ε$_{0}$ and ε$_{1}$ conditional on U by considering CPQD as follows: for any ( e$_{0}$, e$_{1}$ ) ∈$_{R}$ $^{2}$, $^{13}$Note that NSM restricts the sign of ρ$_{d}$ as nonnegative for d ∈ { 0 , 1 } . 3.3 Conditional Positive Quadrant Dependence where P$_{0}$ ( y, 1 | z ) ∈ $^{[}$L wst 01 ( y, z ) , U sm 01 ( y, z ) $^{]}$, P$_{1}$ ( y, 0 | z ) ∈ $^{[}$L wst 10 ( y, z ) , U sm 10 ( y, z ) $^{]}$, L sm 10 ( y, z ) =      ( lim p ( z ) → p P ( y, 1 | z ) − P ( y, 1 | z ) p − p ( z ) ) (1 − p ( z )) , for any z ∈ Z \ p − $^{1}$( p ) , 0 , for z ∈ p − $^{1}$( p ) , , U sm 01 ( y, z ) =      ( P ( y, 0 | z ) − lim p ( z ) → p P ( y, 0 | z ) p ( z ) − p ) p ( z ) , for any z ∈ Z \ p − 1 $^{(}$p$^{)}$, p ( z ) , for z ∈ p − 1 $^{(}$p$^{)}$, , P [ ε$_{0}$ ≤ e$_{0}$ | u ] P [ ε$_{1}$ ≤ e$_{1}$ | u ] ≤ P [ ε$_{0}$ ≤ e$_{0}$, ε$_{1}$ ≤ e$_{1}$ | u ] . (6) 16 Lemma 2 Under M. 1 − M. 4 and NSM, P$_{0}$ ( y, 1 | z ) and P$_{1}$ ( y, 0 | z ) are bounded as follows: