In contrast to existing proposals, our tests are fully data-driven, do not require user-chosen tuning parameters such as bandwidths, and are able to detect a broad class of local alternatives converging to the null at the parametric rate n − 1 / $^{2}$, with n the sample size. Furthermore, our tests do not suffer from the “curse of dimensionality” when the vector of covariates is of high dimensionality, and have greater power than competing tests for many alternatives. Of course, such power gains do not come without a cost: there exist some classes of alternative hypotheses against which our tests have trivial power. Nonetheless, we believe that such a compromise is reasonable since, as pointed out by Janssen (2000) and Escanciano (2009 a ), achieving reasonable power over all possible directions seems hopeless. The proposal closest to ours is Shaikh et al. (2009). Despite using a similar characterization of the null hypothesis as Shaikh et al. (2009), our proposal greatly differs from theirs. Whereas Shaikh et al. (2009) adopts the local smoothing approach, see e.g. Hardle and Mammen (1993), Zheng (1996), Fan and Li (1996) and Li and Wang (1998), we adopt the integrated conditional moment (ICM) approach, see e.g. Bierens (1982, 1990), Bierens and Ploberger (1997), Stute (1997) and Escanciano (2006 a ). As a consequence, our approach inherits some advantages of ICM when compared to Shaikh et al. (2009). First, our tests do not require delicate bandwidth choice, unlike Shaikh et al. (2009)’s test whose performance can be sensitive to it. Second, in contrast with Shaikh et al. (2009), our approach has power against local alternatives converging to the null at the parametric rate. Another popular procedure to assess misspecification of the propensity score model is to use “balancing” tests. Initially proposed by Rosenbaum and Rubin (1985), these tests consist of assessing if each covariate is independent of the treatment assignment, conditional on the propensity score. This is often implemented examining whether moments (usually just the mean) of the observable characteristics between the two “matched” or “weighted” groups are the same; see e.g. Dehejia and Wahba (2002) and Smith and Todd (2005). One should bear in mind that because “balancing” tests are usually based on a finite number of orthogonality conditions, there are uncountably many directions of misspecification that cannot be detected with these tests. Furthermore, as shown by Lee (2013), balancing tests may have size distortions due to the “multiple testing problem”, the failure to account for the estimation effect of the propensity score, and poor covariate overlap. Such drawbacks put at stake the reliability of many of these procedures. Our proposal, on the other hand, does not suffer from these. Our paper also contributes to the literature on ICM tests. What appears distinctive to 3