throughout this paper and present concrete examples of support restrictions. I review the existing method of identifying the DTE given marginal distributions without support restrictions to demonstrate its limits in the presence of support restrictions. I also briefly discuss the optimal transportation approach to describe the key idea of my identification strategy. Section 3 formally characterizes the identification region of the DTE under general support restrictions and derives informative bounds for economic examples from the characterization. Section 4 provides numerical examples to assess the informativeness of my new bounds and analyzes sources of identification gains. Section 5 illustrates the usefulness of these bounds by applying DTE bounds derived in Section 3 to an empirical analysis of the impact distribution of smoking on infant birth weight. Section 6 concludes and discusses interesting extensions. In this section, I present the potential outcomes setup that this study is based on, the notation, and the assumptions used throughout this study. I demonstrate that the bounds on the DTE established without support restrictions are not the best possible bounds in the presence of support restrictions. Then I propose a new method to derive sharp bounds on the DTE based on the optimal transportation framework. The setup that I consider is as follows: the econometrician observes a realized outcome variable Y and a treatment participation indicator D for each individual , where D = 1 indicates treatment participation while D = 0 nonparticipation. An observed outcome Y can be written as Y = DY$_{1}$ + (1 − D ) Y$_{0}$ . Only Y$_{1}$ is observed for the individual who takes the treatment while only Y$_{0}$ is observed for the individual who does not take the treatment, where Y$_{0}$ and Y$_{1}$ are the potential outcome without and with treatment, respectively. Treatment effects Δ are defined as Δ = Y$_{1}$ − Y$_{0}$ the difference of potential outcomes. The objective of this study is to identify the distribution function of treatment effects F$_{Δ}$ ( δ ) = Pr ( Y$_{1}$ − Y$_{0}$ ≤ δ ) from observed pairs ( Y, D ) for fixed δ ∈$_{R}$ . To avoid notational confusion, I differentiate between the distribution and the distribution function . Let µ$_{0}$ , µ$_{1}$ and π denote marginal distributions of Y$_{0}$ and Y$_{1}$ , and their joint distribution, respectively. That is, for any measurable set A$_{d}$ in $_{R}$, µ$_{d}$ ( A$_{d}$ ) = Pr { Y$_{d}$ ∈ A$_{d}$ } for d ∈ { 0 , 1 } and π ( A ) = Pr { ( Y$_{0}$, Y$_{1}$ ) ∈ A } for any measurable set A in R $^{2}$. In addition, let F$_{0}$, F$_{1}$ and F denote marginal distribution functions of Y$_{0}$ and Y$_{1}$, and their joint distribution function, respectively. That is, F$_{d}$ ( y$_{d}$ ) = µ$_{d}$ (( −∞ , y$_{d}$ ]) and 5 2.1 Basic Setup 2 Basic Setup, DTE Bounds and Optimal Transportation Ap- proach