F ( y$_{0}$, y$_{1}$ ) = π (( −∞ , y$_{0}$ ] × ( −∞ , y$_{1}$ ]) for any y$_{d}$ ∈$_{R}$ and d ∈ { 0 , 1 } . Let Y$_{0}$ and Y$_{1}$ denote the support of Y$_{0}$ and Y$_{1}$ , respectively. In this paper, the identification region of F$_{Δ}$ ( δ ) is obtained for fixed marginal distributions. When marginal distributions are only partially identified, DTE bounds are obtained by taking the union of the bounds over all possible pairs of marginal distributions. Marginal distributions of potential outcomes are point-identified in randomized experiments or under selection on observables. Furthermore, previous stud- ies have shown that even if the selection is endogenous, marginal distributions of potential outcomes are point or partially identified under relatively weak conditions. Imbens and Rubin (1997) and Abadie (2002) showed that marginal distributions for compliers are point-identified under the local average treatment ef- fects (LATE) assumptions, and Kitagawa (2009) obtained the identification region of marginal distributions under IV conditions. 3 I impose the following assumption on the fixed marginal distribution functions throughout this paper: Assumption 1 The marginal distribution functions F$_{0}$ and F$_{1}$ are both absolutely continuous with respect to the Lebesgue measure on$_{R}$. In this paper, I obtain sharp bounds on the DTE. Sharp bounds are defined as the best possible bounds on the collection of DTE values that are compatible with the observations ( Y, D ) and given restrictions. Let F L $_{Δ}$( δ ) and F U $_{Δ}$( δ ) denote the lower and upper bounds on the DTE F$_{Δ}$ ( δ ): If there exists an underlying joint distribution function F that has fixed marginal distribution functions F$_{0}$ and F$_{1}$ and generates F$_{Δ}$ ( δ ) = F L $_{Δ}$( δ ) for fixed δ ∈$_{R}$ , then F L $_{Δ}$( δ ) is called the sharp lower bound. The sharp upper bound can be also defined in the same way. Note that throughout this study, sharp bounds indicate pointwise sharp bounds in the sense that the underlying joint distribution function F achieving sharp bounds is allowed to vary with the value of δ. 4 To identify the DTE, I consider support restrictions, which can be written as F L $_{Δ}$( δ ) ≤ F$_{Δ}$ ( δ ) ≤ F U $_{Δ}$( δ ) . Pr (( Y$_{0}$, Y$_{1}$ ) ∈ C ) = 1 , $^{3}$Note that the conditions considered in these studies do not restrict dependence between two potential outcomes. $^{4}$If the underlying joint distribution function F does not depend on δ , then the sharp bounds are called uniformly sharp bounds. Uniformly sharp bounds are outside of the scope of this paper. For more details on uniform sharpness, see Firpo and Ridder (2008). 6