In Chapter 1, I obtained sharp bounds on the DTE when marginal distributions are fixed and MTR is imposed. Compared to Figure 1, Figure 4 shows that under MTR the lower bound on the DTE improves by allowing more mass to be added between Y$_{1}$ = Y$_{0}$ + δ and Y$_{1}$ = Y$_{0}$ . Lemma 5 presents sharp bounds on the DTE under MTR and fixed marginals F$_{0}$ an F$_{1}$ as follows: Lemma 5 ( ? ) Under MTR, sharp bounds on the DTE are given as follows: for fixed marginals F$_{0}$ an F$_{1}$ and any δ ∈$_{R}$ , where where A$_{δ}$ = $^{{}${ a$_{k}$ } ∞ k = $_{−∞}$; 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for every integer k $^{}}$. From Lemmas 3, 4, and 5, it is straightforward to derive sharp bounds on the joint distribution and the DTE under M. 1 − M. 4 and MTR. The specific forms of sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$, their joint distribu- tion, and the DTE under M. 1 − M. 4 and MTR are provided in Theorem 3 in Appendix. I here show that NSM and MTR yield testable implications. Note that NSM implies the following: for any ( z $^{′}$, z ) ∈ Z × Z such that p ( z $^{′}$) ≥ p ( z ) , and for 4 Discussion 4.1 Testable Implications 23 F U Δ ( δ ) =      1 + inf y ∈$_{R}$ { min ( F$_{1}$ ( y ) − F$_{0}$ ( y − δ )) , 0 } , for δ ≥ 0 , 0 , for δ < 0 . , F L $_{Δ}$( δ ) =      sup { a$_{k}$ } ∞ k = $_{−∞}$∈A$_{δ}$ ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } , for δ ≥ 0 , 0 , for δ < 0 , F L $_{Δ}$( δ ) ≤ F$_{Δ}$ ( δ ) ≤ F U Δ ( δ ) ,