Lemma 1 (?) Under M. 1 − M. 4 , for any z ∈ Z , P$_{0}$ ( y, 1 | z ) and P$_{1}$ ( y, 0 | z ) are bounded as follows: where and these bounds are sharp. Proof. The proof is in Appendix. Remark 3 If p = 0 , then P$_{0}$ ( y, 1 | z ) is point-identified as L wst 01 ( y, z ) = U wst 01 ( y, z ) . On the other hand, if p = 1 , then P$_{1}$ ( y, 0 | z ) is point-identified as L wst 10 ( y, z ) = U wst 10 ( y, z ) . Therefore, when the instruments shift the propensity score from 0 to 1 , both counterfactual probabilities are point- identified, and thus both marginal distributions of potential outcomes are point-identified. This full support condition implies that treatment participation is completely determined by instruments in the limits, and unobservables do not exert any influence on treatment selection in the limits of the propensity score. Therefore, the distributions of potential outcomes are point-identified as they are point-identified in the absence of selection on unobservables. Note that under M. 1 − M. 4, the model (1) does not impose any restriction on dependence between Y$_{0}$ and Y$_{1}$ . Hence, Fr´ echet-Hoeffding bounds and Makarov bounds can be employed to establish sharp bounds on the joint distribution and the DTE, respectively. Specifically, for any 13 L wst 01 ( y, z ) = lim p ( z ) → p P ( y | 0 , z ) $^{(}$1 − p ) − P ( y | 0 , z ) (1 − p ( z )) , U wst 01 ( y, z ) = lim p ( z ) → p P ( y | 0 , z ) $^{(}$1 − p ) − P ( y | 0 , z ) (1 − p ( z )) + p, L wst 10 ( y, z ) = lim p ( z ) → p P ( y | 1 , z ) p − P ( y | 1 , z ) p ( z ) , U wst 10 ( y, z ) = lim p ( z ) → p P ( y | 1 , z ) p − P ( y | 1 , z ) p ( z ) + 1 − p, P$_{0}$ ( y, 1 | z ) ∈ $^{[}$L wst 01 ( y, z ) , U wst 01 ( y, z ) $^{]}$, P$_{1}$ ( y, 0 | z ) ∈ $^{[}$L wst 10 ( y, z ) , U wst 10 ( y, z ) $^{]}$,