are obtained by Fr´ echet-Hoeffding bounds as follows: for any ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ $^{2}$, Since P$_{1}$ ( y$_{1}$ | 0 , z ) is only partially identified, sharp bounds on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) are ob- tained by taking the union over all possible values of P$_{1}$ ( y$_{1}$ | 0 , z ) . Therefore, sharp bounds on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$, D = 0 | z ) = P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) (1 − p ( z )) are derived as follows: Similarly, By (12), sharp bounds on P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) are obtained by taking the intersection of the bounds over all values of z ∈ Ξ , 36 F $^{L}$( y$_{0}$, y$_{1}$ ) = sup z ∈ Ξ $^{{}$max$^{{}$( P ( y$_{0}$ | 0 , z ) − 1) (1 − p ( z )) + L wst 10 ( y$_{1}$, z ) , 0 } + max $^{{}$L wst 01 ( y$_{0}$, z ) + ( P ( y$_{1}$ | 1 , z ) − 1) p ( z ) , 0 $^{}}}$, F U ( y$_{0}$, y$_{1}$ ) = inf z ∈ Ξ $^{{}$min$^{{}$P ( y$_{0}$ | 0 , z ) (1 − p ( z )) , U wst 10 ( y | z ) } + min $^{{}$U wst 01 ( y$_{0}$, z ) , P ( y$_{1}$ | 1 , z ) p ( z ) $^{}}}$. max $^{{}$L wst 01 ( y, z ) + ( P ( y$_{1}$ | 1 , z ) − 1) p ( z ) , 0 } ≤ P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$, D = 1 | z ) ≤ min $^{{}$U wst 01 ( y, z ) , P ( y$_{1}$ | 1 , z ) p ( z ) $^{}}$. max $^{{}$P ( y$_{0}$, 0 | z ) + L wst 10 ( y, z ) − (1 − p ( z )) , 0 } ≤ P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$, D = 0 | z ) ≤ min $^{{}$P ( y$_{0}$, 0 | z ) , U wst 10 ( y, z ) $^{}}$. max { P ( y$_{0}$ | 0 , z ) + P$_{1}$ ( y$_{1}$ | 0 , z ) − 1 , 0 } ≤ P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) ≤ min { P ( y$_{0}$ | 0 , z ) , P$_{1}$ ( y$_{1}$ | 0 , z ) } .