report these results in Table B.1 in the Appendix B. Finally, note that when the propensity score is correctly specified, the bias of the ̂ ATE$_{n}$ estimator in (4.3) is small, the length of the 95% confidence interval reduces as sample size increases, but the coverage probability is smaller than its nominal value even when n = 1 , 000. However, as we show in Table B.1 in the Appendix B, such undercoverage disappears when we trim observations with extreme estimated propensity scores. Note that when the propensity score is misspecified in DGP 2 − DGP 5, the ATE estimator (4.3) can be severely biased, and its confidence interval can be “too small” when the propen- sity score is misspecified, leading to severe undercoverage $^{5}$. Thus, detecting propensity score misspecifications can prevent misleading inference about ATE . Our proposed KS$_{n}$ and CvM$_{n}$ tests perform admirably well in such a task, particularly in moderate sample sizes. In these scenarios, CvM$_{n}$ performs slightly better than KS$_{n}$ . Looking at the results from Shaikh et al. (2009)’s test, we note that bandwidth choices can play an important role, and the choice of the “best” bandwidth h$_{n}$ via c varies across DGPs. Perhaps, what is more important to emphasize in terms of power is that in all alternative hypotheses and sample sizes analyzed, our projec- tion based tests have higher power than Shaikh et al. (2009)’s test, regardless of the bandwidth choice. Our proposed tests also dominate the balancing tests, as balancing tests have little to no power in all DGPs considered but DGP 2, even with n = 1 , 000. Finally, the results in Table 1 show that projection-based tests perform either better or as well as the other ICM tests in the DGPs considered. Among the considered DGPs, DGP 3 is the only one where some existing specification testing has higher power than our proposed procedure. For this particular DGP, ICM type tests KS trad n and CvM trad n based on (4.1) have higher power than our proposed tests when sample size n is large. Given the discussion in Section 3.2 and the fact that none of the ICM tests are strictly better than the others uniformly over the space of alternatives, such a finding does not come with a surprise. In this simulation, we push forward the dimensionality of the covariates to see how our proposed tests and the other alternative tests perform in scenarios with 10 continuous covariates. To 4.2 Simulation 2 5 In DGP 3, the bias, confidence interval length, and coverage of the ATE estimator are good. However, it is important to have in mind that we consider only one particular DGP for the ATE , and that such “robust” results may not translate to other DGPs. 21