M.5 The propensity score p ( z ) is bounded away from 0 and 1. Under M. 5, Chebyshev’s integral inequality yields the lower bound as follows: The inequality in the third line of (7) follows from CPQD and the inequality in the fourth line of (7) is due to Chebyshev’s integral inequality. Consequently, I obtain the following: Similarly, the lower bound on P ( Y$_{0}$ ≤ y$_{0}$, Y ≤ y$_{1}$ | 1 , z ) is obtained as follows: Interestingly, the DTE is still bounded by Makarov bounds under CPQD although the lower bound on the joint distribution improves. The rigorous proof is provided in Appendix. Here I discuss the reason intuitively using a graphical illustration. As shown in Figure 1, the DTE is a probability corresponding to the region below the straight line y$_{1}$ = y$_{0}$ + δ and the Makarov lower bound is obtained from the rectangle { Y$_{0}$ ≥ y − δ, Y$_{1}$ ≤ y } below the straight line Y$_{1}$ = Y$_{0}$ + δ for y ∈ R that maximizes the Fr´ echet-Hoeffding lower bound. Since the Fr´ echet-Hoeffding lower bound on P ( Y$_{0}$ ≥ y − δ, Y$_{1}$ ≤ y ) for each y ∈$_{R}$ is achieved when the joint distribution of Y$_{0}$ and 18 P ( Y$_{0}$ ≤ y$_{0}$, Y ≤ y$_{1}$ | 1 , z ) ≥ P$_{0}$ ( y$_{0}$ | 1 , z ) P ( y$_{1}$ | 1 , z ) (9) ≥ L wst 01 ( y$_{0}$, z ) P ( y$_{1}$ | 1 , z ) p ( z ) . P ( Y ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) ≥ P ( y$_{0}$ | 0 , z ) P$_{1}$ ( y$_{1}$ | 0 , z ) (8) ≥ P ( y$_{0}$ | 0 , z ) L wst 10 ( y$_{1}$, z ) 1 − p ( z ) . P ( ε$_{0}$ ≤ e$_{0}$, ε$_{1}$ ≤ e$_{1}$ | U > p ( z )) (7) = 1 1 − p ( z ) 1 ∫ p ( z ) P [ ε$_{0}$ ≤ e$_{0}$, ε$_{1}$ ≤ e$_{1}$ | u ] du ≥ 1 1 − p ( z ) 1 ∫ p ( z ) P [ ε$_{0}$ ≤ e$_{0}$ | u ] P [ ε$_{1}$ ≤ e$_{1}$ | u ] du ≥ ( 1 1 − p ( z )$^{)}$$^{2}$ 1 ∫ p ( z ) P [ ε$_{0}$ ≤ e$_{0}$ | u ] du 1 ∫ p ( z ) P [ ε$_{1}$ ≤ e$_{1}$ | u ] du.