and ˆ S ∗ n = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ ) g ( X$_{i}$, θ$_{0}$ ) V$_{i}$ + o$_{p}$ (1) . Thus, by Lemmas A.6 and A.7, uniformly in u , leading to the multiplier bootstrapped version of R p n $_{0}$( u ) in (2.14). The rest of the proof then follows from the multiplier central limit theorem applied to process R p ∗ n $_{0}$( u ); see van der Vaart and Wellner (1996, Theorem 2.9.2, p.179), and the continuous mapping theorem. □ ˆ R p ∗ n ( u ) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ )(1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } − G $^{′}$( u, θ$_{0}$ )Δ − $^{1}$( θ$_{0}$ ) g ( X$_{i}$, θ$_{0}$ )) V$_{i}$ + o$_{p}$ (1) = 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( θ$_{0}$ ) P 1 { q ( X$_{i}$, θ$_{0}$ ) ≤ u } V$_{i}$ + o$_{p}$ (1) := R p ∗ n $_{0}$( u ) + o$_{p}$ (1) , Appendix B: Simulation results with trimming In Section 4 we found that the inverse probability weighting type ATE estimators and tradi- tional balancing tests can perform poorly when estimated propensity scores are relatively close to zero or one. In this section, we conduct the same Monte Carlo study as in Section 4 but we trim observations with estimated propensity score outside [0 . 05 , 0 . 95] when considering ̂ ATE$_{n}$ in (4.3) and the classical balancing tests; for the other test statistics, we do not use any trimming. Table B.1 and Table B.2 present the results under designs DGP 1- DGP 5, and DGP 6- DGP 10, respectively. As discussed in the main text, when the propensity score is correctly specified ( DGP 1 and DGP 6), the IPW estimator ̂ ATE$_{n}$ has very attractive finite sample properties, with little to no bias, coverage probability close to its nominal level, and 95% confidence interval length shrinking with sample size n . Here it is important to emphasize that in DGP 1, in contrast with the “untrimmed” ATE estimator in Table 1, the coverage probability of the “trimmed” ATE estimator under is close to the nominal level. However, when the propensity score is misspecified, bias does not vanish as sample size increases, and coverage probability can be substantially lower than its nominal level. In terms of size, we note that under DGP 1 balancing tests seem to perform relatively well once we trim observations with “extreme” propensity score estimates, though with n = 1 , 000, the Wald test seems to be over-rejecting. On the other hand, when the dimension of the 41