can be written as follows: for each z ∈ Z and any y ∈$_{R}$ , While the probability P ( y, 1 | z ) is observed, the counterfactual probability P$_{1}$ ( y, 0 | z ) is never ob- served. Let p = sup z ∈Z p ( z ) , p = inf z $_{∈Z}$p ( z ). Note that p and p are well defined under M. 4. For z ∈ Z such that p ( z ) < p, the counterfactual probability P$_{1}$ ( y, 0 | z ) can be decomposed as follows: The second equality follows from M. 3. Note that P ( Y$_{1}$ ≤ y, p ( z ) < U ≤ p ) is point-identified as follows: However, P ( Y$_{1}$ ≤ y, p < U ) is never observed. Note that for lim p ( z ) → p P$_{1}$ ( y | 0 , z ) can be any value between 0 and 1. Therefore, I can derive bounds on P ( Y$_{1}$ ≤ y, p < U ) by plugging 0 and 1 into the counterfactual distribution P ( y | 0 , z ). Similarly, the other counterfac- tual probability P$_{0}$ ( y, 1 | z ) can be partially identified. 12 P ( Y$_{1}$ ≤ y, p < U ) = lim p ( z ) → p P$_{1}$ ( y | 0 , z ) (1 − p ) , P ( Y$_{1}$ ≤ y, p ( z ) < U ≤ p ) = P ( Y$_{1}$ ≤ y, U ≤ p ) − P ( Y$_{1}$ ≤ y, U ≤ p ( z )) = lim p ( z ) → p P ( y | 1 , z ) p − P ( y | 1 , z ) p ( z ) . P$_{1}$ ( y, 0 | z ) (4) = P ( Y$_{1}$ ≤ y, p ( z ) < U | z ) = P ( Y$_{1}$ ≤ y, p ( z ) < U ) = P ( Y$_{1}$ ≤ y, p ( z ) < U ≤ p ) + P ( Y$_{1}$ ≤ y, p < U ) , F$_{1}$ ( y ) = P ( Y$_{1}$ ≤ y | Z = z ) (3) = P ( y, 1 | z ) + P$_{1}$ ( y, 0 | z ) .