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Selection is allowed to be endogenous since the selection unobservable U is dependent on potential outcomes Y$_{0}$ and Y$_{1}$ for ρ ̸ = 0. I consider negative values of ρ to make the specification satisfy NSM discussed in Subsection 3.2. CPQD holds due to the common factor ε in Y$_{0}$ and Y$_{1}$ , which is independent of U . Lastly, MTR is obviously satisfied as P ( Y$_{1}$ ≥ Y$_{0}$ ) = 1 since η ≥ 0 with probability one. Also, to rule out the full support of the instrument, Z is assumed to be a uniformly distributed random variable on ( z, − z ) for z = 2 , 1 . 5 , 1 , . 5 . 

First, for ρ = − 0 . 75 and Z ∼ Unif (1 , − 1) , I obtain the sharp bounds on the marginal distri- butions of potential outcomes Y$_{0}$ and Y$_{1}$ as proposed in Section 3. Figure 5 shows the bounds on each potential outcome distribution as well as the true distribution. Solid curves represent the true marginal distributions of Y$_{0}$ and Y$_{1}$ and dash-dot curves, dotted curves, and dashed curves rep- resent their worst bounds, bounds under NSM, and bounds under MTR, respectively. Remember that bounds on marginal distributions under CPQD are identical to worst bounds. Figure 5 shows that NSM substantially improves the upper bound on F$_{0}$ and the lower bound on F$_{1}$ , compared to worst bounds. As shown in Lemma 2, NSM improves the upper bound on P ( Y$_{0}$ ≤ y, 1 | z ) and the lower bound on P ( Y$_{1}$ ≤ y, 0 | z ) for y ∈$_{R}$ , which are used in obtaining the upper bound on F$_{0}$ and the lower bound on F$_{1}$, respectively. On the other hand, MTR improves the lower bound on F$_{0}$ and the upper bound on F$_{1}$ . Note that in contrast to NSM, MTR improves the lower bound on P ( Y$_{0}$ ≤ y, 1 | z ) and the upper bound on P ( Y$_{1}$ ≤ y, 0 | z ) for all y ∈$_{R}$ , which are used in obtaining 

