for d ∈ { 0 , 1 } , where β ∼ χ $^{2}$( k$_{1}$ ) , ε ∼ N (0 , k$_{2}$ ), and β ⊥ ε. Obviously, treatment effects Δ = β ∼ χ $^{2}$( k$_{1}$ ) satisfy MTR and marginal distribution functions F$_{0}$ and F$_{1}$ are given as where G ( · ; k$_{1}$ ) is the distribution function of a χ $^{2}$( k$_{1}$ ) and Φ ( · ) are the standard normal probability density function and its distribution function, respectively. Recall that the sharp upper bound under MTR is identical to the Makarov upper boun, and the sharp lower bound on the DTE under MTR is given as follows: for δ ≥ 0, where A$_{δ}$ = $^{{}${ a$_{k}$ } ∞ k = $_{−∞}$; 0 ≤ a$_{k}$$_{+1}$ − a$_{k}$ ≤ δ for each integer k $^{}}$. The lower bound requires computing the optimal sequence of a$_{k}$ . The specific computation procedure is described in Appendix B. Figure 12 shows the true DTE as well as Makarov bounds and the improved lower bound under MTR for k$_{1}$ = 1 , 5 , 10 and k$_{2}$ = 1 , 10 , 40 . To see the effect of marginal distributions for the fixed true DTE Δ ∼ χ $^{2}$( k$_{1}$ ) , I focus on how the DTE bounds change for different values of k$_{2}$ and fixed k$_{1}$ . Figure 12 shows that Makarov bounds and my new lower bound become less informative as k$_{2}$ increases. My data generating process assumes Y$_{1}$ − Y$_{0}$ ∼ χ $^{2}$( k$_{1}$ ), Y$_{0}$ ∼ N (0 , k$_{2}$ ) and Y$_{1}$ − Y$_{0}$ ⊥ Y$_{0}$. When the true DTE is fixed with a given value of k$_{1}$, both Makarov bounds and my new bounds move further away from the true DTE as the randomness in the potential outcomes Y$_{0}$ and Y$_{1}$ increases with higher k$_{2}$ . If k$_{2}$ = 0 as an extreme case, in which Y$_{0}$ has a degenerate distribution, obviously Makarov bounds as well as my new bounds point-identify the DTE. Interestingly, as k$_{2}$ increases, my new lower bound moves further away from the true DTE much more slowly than the Makarov lower bound. Therefore, the information gain from MTR, which is represented by the distance between my new lower bound and the Makarov lower bound, increases as k$_{2}$ increases. This shows that under MTR, my new lower bound gets additional information from the larger variation of marginal distributions. To develop intuition, recall Figure 7(c). Under MTR, the larger variation in marginal distributions 29 sup { a$_{k}$ } ∞ k = $_{−∞}$∈A$_{δ}$ ∞ ∑ k = −∞ max { F$_{1}$ ( a$_{k}$$_{+1}$ ) − F$_{0}$ ( a$_{k}$ ) , 0 } , (13) F$_{1}$ ( y ) = ∞ ∫ −∞ G ( y − x ; k$_{1}$ ) φ$^{(}$ x √ k$_{2}$$^{)}$ dx, F$_{0}$ ( y ) = Φ$^{(}$ y √ k$_{2}$$^{)}$ , Y$_{d}$ = βd + ε,