any y ∈$_{R}$ , This yields the following testable form of functional inequalities: Next, MTR has two testable implications. First, MTR implies stochastic dominance. In our model, marginal distributions are partially identified for the entire population. Therefore, it can be tested by applying econometric techniques for testing stochastic dominance for partially identified marginal distributions as proposed in the literature including Jun et al. (2013). Also, the sharp lower bound on the DTE under MTR can be greater than the upper bound and furthermore the lower bound could be even above 1, when MTR is violated for the true joint distribution of Y$_{0}$ and Y$_{1}$ . In Section 3, I explored the identifying power of NSM, CPQD, and MTR, separately. In this subsection, I briefly discuss how sharp bounds are constructed when some of these conditions are combined. Establishing sharp bounds under NSM and CPQD and sharp bounds under NSM and MTR is straightforward from the results in Subsection 3.2 - Subsection 3.4. First, under NSM and CPQD, bounds on marginal distributions and bounds on the DTE are identical to those under NSM only, since CPQD has no identifying power on the marginal distributions and the DTE. The bounds on the joint distribution under NSM and CPQD can be established by plugging the bounds on the counterfactual probabilities P$_{0}$ ( y$_{0}$, 1 | z ) and P$_{1}$ ( y$_{1}$, 0 | z ) under NSM into the upper bound 24 4.2 NSM+CPQD and NSM+MTR P ( y | 1 , z ) ≤ P $^{(}$y | 1 , z ′ $^{)}$, (10) P ( y | 0 , z ) ≤ P $^{(}$y | 0 , z ′ $^{)}$. P $^{(}$ε$_{1}$ ≤ m − $^{1}$(1 , y ) | U ≤ p ( z ) ) ≤ P $^{(}$ε$_{1}$ ≤ m − $^{1}$(1 , y ) | U ≤ p $^{(}$z ′ $^{))}$, P $^{(}$ε$_{0}$ ≤ m − $^{1}$(0 , y ) | U > p ( z ) ) ≤ P $^{(}$ε$_{0}$ ≤ m − $^{1}$(0 , y ) | U > p $^{(}$z ′ $^{))}$.