It is worth noting that my new bounds on the quantile of the effects of smoking are much tighter for SCCG women, compared to the entire sample and other subsamples. For q ≤ 0 . 5, the refinement rate ranges from 51% to 64% compared to Makarov bounds. For SCCG women, my new sharp bounds on the median are [0 , 299] grams, while Makarov bounds on the median are [0 , 764] grams. The higher identification gains result from relatively heavier potential nonsmokers’ infant birth weight, which leads to the shorter distance between two potential outcomes distributions as reported in Table 5. Note that the shorter distance between marginal distributions of potential outcomes improves both my new lower bound and the Makarov lower bound. 22 Although QTE is placed within the identification region for q = 0 . 15 to 0 . 85 and for all groups, at q = 0 . 15, QTE is very close to the upper bound on the quantile of smoking effects for SCCG and age 26-35 subgroups. Furthermore, at q = 0 . 10, QTE is placed outside of the improved identification region for SCCG group and age 26-35. This implies that QTE is not identical to the quantile of treatment effects in my example and so one should not interpret the value of QTE as a quantile of smoking effects. Table 7: QTE and bounds on the quantiles of smoking effects $^{22}$To develop intuition, recall Figure 7(c). The size of the lower bound on each triangle’s probability is related to the distance between marginal distribution functions of Y$_{0}$ and Y$_{1}$ . To see this, consider two marginal distribution functions F A 1 and F B 1 of Y$_{1}$ with F A 1 ( y ) ≤ F B 1 ( y ) for all y ∈$_{R}$ and fix the marginal distribution F$_{0}$ of Y$_{0}$ where ( Y$_{0}$, Y$_{1}$ ) satisfies MTR. Since MTR implies stochastic dominance of Y$_{1}$ over Y$_{0}$ for each y ∈$_{R}$ , F A 1 ( y ) < F B 1 ( y ) ≤ F$_{0}$ ( y ) . Thus, Since the probability lower bound on the triangle is written as max { F$_{1}$ ( y ) − F$_{0}$ ( y − δ ) } for some y ∈$_{R}$ , the above inequality shows that the closer marginal distributions F$_{0}$ and F$_{1}$ generates higher probability lower bound on each triangle. max $^{{}$F A ( y ) − F$_{0}$ ( y − δ ) , 0 $^{}}$< max $^{{}$F B ( y ) − F$_{0}$ ( y − δ ) , 0 $^{}}$. 44 Dep. var.= Birth weight (grams) Q$_{0}$$_{.}$$_{15}$ Q$_{0}$$_{.}$$_{25}$ Q$_{0}$$_{.}$$_{5}$ Q$_{0}$$_{.}$$_{75}$ Q$_{0}$$_{.}$$_{85}$ Entire Sample QTE 195 214 234 259 292 Makarov [0,405] [0,524] [0,843] [0,1317] [80,1634] New [0,265] [0,304] [0,457] [0,882] [80,1204] White QTE 204 212 212 227 255 Makarov [0,383] [0,505] [0,833] [0,1274] [65,1588] New [0,265] [0,308] [0,450] [0,891] [65,1239] SCCG QTE 109 165 187 244 194 Makarov [0,311] [0,428] [0,764] [0,1183] [69,1453] New [0,114] [0,193] [0,299] [0,579] [69,792] Age 26-35 QTE 233 180 179 262 283 Makarov [0,336] [0,458] [0,807] [0,1324] [79,1621] New [0,239] [0,276] [0,406] [0,746] [79,1204]