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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 

<img src='content_image/313593.jpg'>

where A$_{δ}$ = $^{{}${ a$_{k}$ } ∞ k = $_{−∞}$; 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for every integer k $^{}}$. 

<img src='content_image/313594.jpg'>

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. 

## 4 Discussion 

## 4.1 Testable Implications 

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 

