covariates is relatively high as in DGP 6, these balancing tests tend to be conservative even with trimming, reflecting the “curse of dimensionality”. In terms of power, it is clear that balancing tests are dominated by our projection-based tests in all designs. In particular, as shown in Table B.2 balancing tests can have zero to no power when the dimension of covariates is high. The comparison between our tests and other specification tests is exactly as discussed in the main text and is therefore omitted. We would like to thank Han Hong and Elie Tamer (Editors), an Associate Editor, and four anonymous referees for comments that greatly improved this paper. We also thank Daniel Millimet for sharing with us the data we use in the empirical application, Tymon SGLYPH<suppress>loczy´ nski, Tatsushi Oka, and several seminar and conference participants for their useful comments. Pedro H. C. Sant’Anna acknowledge financial support from the Spanish Plan Nacional de I+D+I (Grant No. ECO2014-55858-P). Xiaojun Song acknowledge financial support from the National Natural Science Foundation of China (Grant No. 71532001) and Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education. Acknowledgements References Abadie, A., and Imbens, G. W. (2016), “Matching on the estimated propensity score,” Econometrica , 84(2), 781–807. Akritas, M. G., and van Keilegom, I. (2001), “Non-Parametric Estimation of the Residual Distribution,” Scandinavian Journal of Statistic , 28(3), 549–567. Belloni, A., Chernozhukov, V., Fernandez-Val, I., and Hansen, C. (2017), “Program Evaluation and Causal Inference With High-Dimensional Data,” Econometrica , 85(1), 233–298. Belloni, A., Chernozhukov, V., and Hansen, C. (2014), “Inference on Treatment Effects after Selection among High-Dimensional Controls,” The Review of Economic Studies , 81(2), 608–650. Bickel, P. J., Ritov, Y., and Stoker, T. M. (2006), “Tailor-made tests for goodness of fit to semiparametric hypotheses,” Annals of Statistics , 34(2), 721–741. Bierens, H. J. (1982), “Consistent model specification tests,” Journal of Econometrics , 20(1982), 105– 134. Bierens, H. J. (1990), “A consistent conditional moment test of functional form,” Econometrica , 58(6), 1443–1458. Bierens, H. J., and Ploberger, W. (1997), “Asymptotic theory of integrated conditional moment tests,” Econometrica , 65(5), 1129–1151. Busso, M., Dinardo, J., and McCrary, J. (2014), “New Evidence on the Finite Sample Properties of Propensity Score Reweighting and Matching Estimators,” The Review of Economics and Statistics , 96(5), 885–895. 44