our approach is that ( i ) we exploit the dimension-reduction coming from our derived restric- tion between propensity score CDFs, and ( ii ) we acknowledge our lack of knowledge of the “true” correct specification of the propensity score, by means of an orthogonal projection onto the tangent space of nuisance parameters. The result of ( i ) and ( ii ) is a test with improved power properties, and with a simple bootstrap implementation. The power improvement due to the dimension reduction has been noticed by Stute and Zhu (2002), Escanciano (2006 a ) and Shaikh et al. (2009), whereas the power improvement due to the use of orthogonal projections has been noticed in different contexts, see e.g. Neyman (1959), and more recently, Bickel et al. (2006) and Escanciano and Goh (2014). To the best of our knowledge, our proposal is the first to incorporate both procedures. As mentioned above, our paper is related to the relatively scarce literature on projection- based specification tests, see e.g. Escanciano (2009 b ) and Escanciano and Goh (2014) for two notable exceptions. Escanciano (2009 b ) proposes a simple bootstrap testing procedure for condi- tional moment restrictions that acknowledges specifically the fact that the nuisance parameters are unknown and introduces the projection methodology, while Escanciano and Goh (2014) propose a projection-based testing procedure for linear quantile regression using a related pro- jection weight function. Our proposal builds on these papers, with the important difference that our test statistics also exploit the dimension-reduction property of the propensity score. The rest of the paper is organized as follows. In Section 2 we present the testing framework and derive the restriction upon which our tests are based. The asymptotic properties of our tests are established in Section 3. We next examine the finite sample properties of our tests by means of a Monte Carlo study in Section 4. We provide an empirical illustration of our procedures in Section 5. Section 6 concludes. Mathematical proofs are gathered in an appendix at the end of the article. Finally, all proposed tests discussed in this article can be implemented via open-source R package pstest, which is freely available from GitHub ( https://github.com/pedrohcgs/pstest ). Let D be a binary random variable that indicates participation in the program, i.e. D = 1 if the individual participates in the treatment and D = 0 otherwise. Define Y (1) and Y (0) as the 2 Testing Framework 2.1 Background 4