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Lemma 2 Under M. 1 − M. 4 and NSM, P$_{0}$ ( y, 1 | z ) and P$_{1}$ ( y, 0 | z ) are bounded as follows: 

where 

<img src='content_image/322696.jpg'>

<img src='content_image/322697.jpg'>

and these bounds are sharp. 

Now, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ are obtained by plugging the results in Lemma 2 into the counterfactual probabilities. 

Note that under NSM, sharp bounds on the joint distribution and sharp bounds on the DTE are still obtained from Fr´echet-Hoeffding bounds and Makarov bounds. To illustrate this, consider the case where ρ$_{0}$ = ρ$_{1}$ = 0 in the example (2). 13 This case satisfies NSM and NSM does not impose any restriction on the dependence between ν$_{0}$ and ν$_{1}$ . Therefore, sharp bounds on the joint distribution and the DTE are obtained by the same token as in Subsection 3.1. 

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 NSM are provided in Corollary 1 in Appendix. 

## 3.3 Conditional Positive Quadrant Dependence 

Unlike NSM, CPQD has no additional identifying power for the joint distribution and the DTE. In this subsection, I impose weak positive dependence between ε$_{0}$ and ε$_{1}$ conditional on U by considering CPQD as follows: for any ( e$_{0}$, e$_{1}$ ) ∈$_{R}$ $^{2}$, 

<img src='content_image/322698.jpg'>

