substantial identification gains from the distance between my new lower bound and the Makarov lower bound. The most remarkable improvement arises around q = 0 . 5 and the refinement gets smaller as q approaches 0 and 1 , in turn as δ approaches 0 and 2000. This can be intuitively understood through Figure 7(c). As δ gets closer to 2000 , the number of triangles, which is one source of identification gains, decreases to one in the bounded support of each potential outcome. This causes the new lower bound to converge to the Makarov lower bound as δ approaches 2000. Also, as δ converges to 0 , the identification gain generated by each triangle, which is written as max { F$_{1}$ ( y ) − F$_{0}$ ( y − δ ) , 0 } , converges to 0 under MTR, which implies F$_{1}$ ( y ) ≤ F$_{0}$ ( y ) for each y ∈$_{R}$ . The quantiles of smoking effects can be obtained by inverting these DTE bounds. Specifically, the upper and lower bounds on the quantile of treatment effects are obtained by inverting the lower bound and upper bound on the DTE, respectively. Note that quantiles of smoking effects show q -quantiles of the difference ( Y$_{1}$ − Y$_{0}$ ), while QTE gives the difference between the q -quantiles of Y$_{1}$ and those of Y$_{0}$ . These two parameters typically have different values. Fan and Park (2009) pointed out that QTE is identical to the quantile of treatment effects under strong conditions. 21 The bounds on the quantile of treatment effects are reported in Table 7 with comparison to QTE, already reported in Table 5. In the entire sample, my new bounds on the quantiles of the treatment effect show 33% - 45% refinement for q = 0 . 15, 0 . 25, 0 . 5, 0 . 75 compared to Makarov bounds. For the entire sample, my new bounds yield [0 , 457] grams for the median of the benefit of smoking cessation on infant birth weight, while Makarov bounds yield [0 , 843] grams. Compared to Makarov bounds, my new bounds are more informative and show that (457 , 843] should be excluded from the identification region for the median of the effect. $^{21}$Specifically, QTE = the quantile of treatment effects when (i) two potential outcomes are perfectly positively dependent Y$_{1}$ = F − 1 1 ( F$_{0}$ ( Y$_{0}$ )) AND (ii) F − 1 1 ( q ) − F − 1 0 ( q ) is nondecreasing in q . Figure 17: Bounds on the effect of smoking on birth weight for the entire sample 43