Narrowing Bounds on QTE

bounds for non- Hispanic males are between 3.6 to 13.4 percent excluding the 0.05 quantile. 18

F. Narrowing Bounds on QTE

EE α Using a Covariate To narrow the trimming bounds on QTE EE α we follow the procedure outlined in Section IVB employing earnings in the year prior to randomization as a covariate and breaking up the sample into three groups, as in Section VIC. To estimate the bounds on F Y i 1|EE y m in Equation 22, we use 300 values of y that span the support of the out- come. The results are presented in Figures 3 and 4 for bounds under Assumptions A and B, and Assumptions A, B, and D, respectively. The main insights can be summarized as follows. First, reductions in the width of the estimated bounds on QTE EE α are observed in Figures 3 and 4 relative to those in Figures 1 and 2. Although most of the reductions are less than 20 percent, they range from 0 no reduction to 100 percent point identifi cation. Comparing the results in Figures 1 and 3, the reductions in estimated bounds’ width across quantiles for the analyzed groups were, on average, 6 percent for the full sample, 9 percent for non- Hispanics, 14 percent for whites, 2 percent for blacks and non- Hispanic females, and 16 percent for non- Hispanic males. Comparing the results in Figures 2 and 4 that employ stochastic dominance, the reductions in width are more modest as only about a quarter of the estimated bounds’ width were reduced recall that only the lower bounds are subject to trimming and thus to reductions. On average, the reductions in width across quantiles were 9 percent for the full sample, 14 percent for non- Hispanics, 10 percent for whites, 2 percent for blacks, 4 percent for non- Hispanic females, and 11 percent for non- Hispanic males. Second, for this empirical application, the reductions in the width of the estimated bounds do not change the qualitative results that were discussed in previous sections. Third, looking at the IM confi dence intervals of the estimated bounds that employ X to narrow them, it is evident that in our application the procedure results in wider IM confi dence intervals. This is likely due to the re- quired nonparametric estimation of the trimming bounds in Equation 22, which has to be performed for each of the three groups based on X.

G. Estimated Bounds for Hispanics