negative stochastic monotonicity (NSM) between each outcome unobservable and the selection unobservable, and show how identification regions improve under the additional restriction. Third, I consider conditional positive quadrant dependence (CPQD) as a restriction between two outcome unobservables ε$_{0}$ and ε$_{1}$ conditional on the selection unobservable U . I also explore identifying power of this restriction on each parameter, when it is imposed on top of M. 1 − M. 4. Lastly, I consider monotonicity between two potential outcomes as a different type of restriction. Henceforth, I call this monotone treatment response (MTR). I derive sharp bounds under MTR in addition to M. 1 − M. 4. First, I present the definition of NSM, CPQD, and MTR. I also illustrate them using a toy model and discuss the underlying intuition with economic examples. To better understand these restrictions, consider a particular case where ε$_{0}$ and ε$_{1}$ have a one-factor structure as follows: for d ∈ { 0 , 1 } where ( ν$_{0}$, ν$_{1}$ ) ⊥ U. Here U is the unobservable in the selection equation, while ν$_{0}$ and ν$_{1}$ represent treatment specific heterogeneity. 9 In this setting, NSM requires that ρ$_{0}$ and ρ$_{1}$ be nonpositive. Note that the direction of the sign of the monotonicity is not crucial because my identification strategy can be applied to negative stochastic monotonicity. Intuitively, NSM implies that as the level of U increases, both ε$_{0}$ and ε$_{1}$ decrease or stay constant. This condition is plausible in many empirical applications. In job training $^{9}$This one-factor structure has been discussed in the context of the effects of employment programs in the literature including ? and ? . 8 ε$_{d}$ = ρ$_{d}$U + ν$_{d}$, (2) P [ ε$_{0}$ ≤ e$_{0}$, ε$_{1}$ ≤ e$_{1}$ | U = u ] ≥ P [ ε$_{0}$ ≤ e$_{0}$ | U = u ] P [ ε$_{1}$ ≤ e$_{1}$ | U = u ] . NSM (Negative Stochastic Monotonicity) Both ε$_{0}$ and ε$_{1}$ are first order stochastically non- increasing in U. That is, P ( ε$_{d}$ ≤ e | U = u ) is nondecreasing in u ∈ (0 , 1) for any e ∈$_{R}$ and d ∈ { 0 , 1 } . CPQD (Conditional Positive Quadrant Dependence) ε$_{0}$ and ε$_{1}$ are positively quadrant de- pendent conditionally on U. That is, for ( ε$_{0}$, ε$_{1}$ ) ∈$_{R}$ ×$_{R}$ and u ∈ (0 , 1) ,