obtained the identification region for the marginal distributions under IV conditions. 5 The joint distribution and the DTE have not been investigated in these studies. The literature on identification of the joint distribution and the DTE is relatively small. ? established sharp bounds on the joint distribution and the DTE in semiparametric triangular systems using Fr´ echet-Hoeffding bounds and Makarov bounds, respectively. Their identification is for the entire population under the full support of IV. Also, ? point-identified the DTE based on a random coefficients specification for the selection equation. To do this, they also relied on the full support of the IV. ? studied identification of the joint distribution and the DTE in the extended Roy model, a particular case of triangular systems. 6 Although he point-identified the joint distribution and the DTE by taking advantage of the particular structure of the extended Roy model, his identification only applies to the group of compliers. ? , ? , and Aakvik et al. (2005) considered factor structures in outcome unobservables and assumed the presence of additional proxy variables to identify the joint distribution. ? considered Roy models with a binary outcome variable. They derived sharp bounds on the marginal distributions and the joint distribution of the potential outcomes. Although they did not assume the full support of IV and rank similarity, for the joint distribution bounds they focused on a one-factor structure, as proposed in ? . The main contribution of this paper is to partially identify the joint distribution and the DTE as well as marginal distributions for the entire population without the full support condition of IV and rank similarity. To avoid strong assumptions and impose plausible information on the model, I consider weak restrictions on dependence between unobservables and between potential outcomes. First, I obtain sharp bounds on the distributional parameters for the worst case, which only assumes the latent index model of ? . Next, I explore three different types of restrictions to tighten the worst bounds and investigate how each restriction contributes to improving the identification regions of these parameters. The first restriction that I consider is negative stochastic monotonicity (NSM) between ε$_{d}$ and U for d ∈ { 0 , 1 } . NSM means that ε$_{d}$ increases as U increases for d ∈ { 0 , 1 } . This assumption has been adopted in the literature including ? for its plausibility in practice. 7 The role of NSM in my paper $^{5}$The IV restrictions that he considers are (i) IV independence of each potential outcome, (ii) IV joint independence of the pair of potential outcomes, and (iii) LATE restrictions. $^{6}$The extended Roy model models individual self-selection based on the potential outcomes and observable char- acteristics without allowing for any additional selection unobservables. $^{7}$? also considered stochastic monotonicity to identify triangular systems with a multivalued discrete endogenous 4