nominal value. When one trims observations with extreme estimated propensity score, these properties further improve; see Table B.2 in the Appendix B. Note that when the propensity score is misspecified, the ATE estimator (4.3) can be severely biased, and inference can be unreliable. Thus, tests with higher power to detect such misspecifi- cations can prevent one to make misleading conclusions about the effectiveness of a given policy. What is clear from Table 2 is that, regardless of the sample size and bandwidth considered, Shaikh et al. (2009)’s test seems to have little to no power to detect the alternatives described in DGP 7, DGP 9, and DGP 10. For DGP 8, the maximum power for their test is approximately 55% when n = 1 , 000 and c = 0 . 1 . However, with c = 0 . 01 , the power of Shaikh et al. (2009)’s test reduces to approximately 30%, highlighting again how important (and non-trivial) is to “appropriately” choose the bandwidth. In sharp contrast with Shaikh et al. (2009)’s test, note that for moderately sized samples, our proposed KS$_{n}$ and CvM$_{n}$ tests have non-trivial power to detect all the alternatives. Our projection-based tests seem to dominate the other tests in the scenarios considered. Note that the traditional balancing tests have no power to detect the alternatives considered. Finally, ICM tests based on the traditional empirical process (4.1) have substantially less power than our proposed tests, reflecting the cost of the “curse of di- mensionality”. The power gains from using the projection-based process (2.7) instead of the unprojected process (2.10) can also be noted. Overall, the simulation results highlight that the proposed projection-based tests perform favorably compared to other alternative testing procedures in terms of size and power. Impor- tantly, the simulations illustrate that the gains in power can be credited to each distinguished feature of our tests, that is, ( i ) the avoidance of smoothing parameters by using the ICM approach, ( ii ) the dimension-reduction coming from considering 1 { q ( X,θ$_{0}$ ) ≤ u } instead of 1 { X ≤ u } , and ( iii ) the use of orthogonal projections. Given these attractive features, we believe that our tests can be of great use in practice. In this section, we provide an empirical illustration of our testing procedure. We revisit the analysis of Frankel and Rose (2005) and Millimet and Tchernis (2009), and study the effect of trade on the environment. More specifically, following Millimet and Tchernis (2009), we assess the effect of a country being a member of the General Agreement on Tariffs and Trade (GATT) 5 Empirical Illustration 24