equation. This structure allows for general unobserved heterogeneity in potential outcomes and selection on unobservables. The error term in the outcome equation represents unobserved factors causing heterogeneity in potential outcomes among observationally equivalent individuals. 1 The selection model with a latent index crossing a threshold has been widely used to model selection into programs. In the model, the latent index p ( Z ) − U is interpreted as the net expected utility from participating in the program. ? showed that the model is equivalent to the local average treatment effect (LATE) framework developed by ? . 2 In the literature, the identification method has relied on either the full support of IV or rank similarity to consider the entire population. The full support condition requires IV to change the probability of receiving the treatment from zero to one. 3 As discussed in ? , and ? , however, the applicability of the identification results is very limited because such instruments are difficult to find in practice. Rank similarity assumes that the distribution ε$_{d}$ conditional on U does not depend on d for d ∈ { 0 , 1 } . As a relaxed version of rank invariance, it allows for a random variation between ranks with and without treatment. 4 However, rank similarity is invalid when individuals select treatment status based on their potential outcomes, as in the Roy model. The literature on identification in triangular systems has stressed marginal distributions more than the joint distribution or the DTE. ? point-identified marginal distributions relying on the full support condition. ? showed that the marginal distributions are point-identified for the entire population under rank similarity. Without these conditions, most of the literature has focused on local identification for compliers , to circumvent complications in considering the whole population. ? , and ? showed that under LATE assumptions presented by ? , marginal distributions of potential outcomes are point-identified for compliers who change their selection in a certain direction accord- ing to the change in the value of IV. ? contrasts with other work in the sense that his identification is for the entire population without relying on the full support of IV and rank similarity. He $^{1}$Since it determines the relative ranking of such individuals in the distribution of potential outcomes, it is also referred to as the rank variable in the literature. See ? . $^{2}$The LATE framework consists of two main assumptions: independence and monotonicity. The former assumes that the instrument is jointly independent of potential outcomes and potential selection at each value of the in- strument, while the latter assumes that the instrument affects the selection decision in the same direction for every individual. Since the contribution of ? , the selection structure has been widely recognized as the model which is not only motivated by economic theory but also as weak as LATE assumptions. $^{3}$This type of identification is also referred to as identification at infinity. $^{4}$In this sense, rank similarity is also called expectational rank invariance. See ? . ? , ? , ? , and ? made use of rank similarity to identify average treatment effects for models with a binary outcome variable. Note that these results are readily extended to identification of marginal distributions for continuous outcome variables. 3