subsample size b out of n observations s.t. j$_{1}$ = j$_{2}$ = . . . = j$_{b}$, b < n and j = 1 , ...,$^{}$   n b   $_{}$. Then the subsampling bias-adjusted estimator ˜ F NL Δ ( δ ) is In this paper, I have proposed a novel approach to identifying the DTE under general support restrictions on the potential outcomes. My approach involves formulating the problem as an optimal transportation linear program and embedding support restrictions into the cost function with an infinite Lagrange multiplier by taking advantage of their linearity in the entire joint distribution. I have developed the dual formulation for { 0 , 1 , ∞} -valued costs to overcome the technical challenges associated with optimization over the space of joint distributions. This contrasts sharply with the existing copula approach, which requires one to find out the joint distributions achieving sharp bounds given restrictions. I have characterized the identification region under general support restrictions and derived sharp bounds on the DTE for economic examples. My identification result has been applied to the empirical analysis of the distribution of smoking effects on infant birth weight. I have proposed an estimation procedure for the bounds. The empirical results have shown that MTR has a substantial power to identify the distribution of smoking effects when the marginal distributions of the potential outcomes are given. In some cases, information concerning the relationship between potential outcomes cannot be represented by support restrictions. Moreover, it is also sometimes the case that the joint distribution function itself is of interest. In a companion paper, I propose a method to identify the DTE and the joint distribution when weak stochastic dependence restrictions among unobservables are imposed in triangular systems, which consist of an outcome equation and a selection equation. 47 6 Conclusion where q$_{n}$ =    n b   $_{}$. ˜ F NL Δ ( δ ) = ̂ F NL Δ ( δ ) − 1 q$_{n}$ q$_{n}$ ∑ j =1 { ̂ F NL Δ ,n,b,j ( δ ) − ̂ F NL Δ ( δ ) } = 2$_{̂}$ F NL Δ ( δ ) − 1 q$_{n}$ q$_{n}$ ∑ j =1 ̂ F NL Δ ,n,b,j ( δ ) ,