1 Introduction The propensity score, which is defined as the conditional probability of receiving treatment given covariates, is one of the most widely used tools for causal inference. Part of its popularity can be credited to the seminal result of Rosenbaum and Rubin (1983): if the treatment assignment is independent of the potential outcomes conditional on a vector of covariates, then one can obtain unbiased and consistent estimators of different treatment effect measures by adjusting for the propensity score alone, greatly reducing the dimensionality of the underlying problem. Several methods that exploit this important insight are now an essential part of the applied researcher’s toolkit. Examples include matching, see e.g. Rosenbaum and Rubin (1985), Heckman et al. (1997) and Abadie and Imbens (2016); inverse probability weighting (IPW), see e.g. Rosenbaum (1987), Hirano et al. (2003) and Donald and Hsu (2014); regression methods, see e.g. Hahn (1998) and Firpo (2007); and many others. For literature reviews, see Heckman and Vytlacil (2007) and Imbens and Wooldridge (2009). Despite their popularity, a main concern of these methods is that the propensity score is usually unknown, and therefore has to be estimated. Given the high dimensionality of available covariates, researchers are usually coerced to adopt a parametric model for the propensity score since nonparametric estimation methods suffer from the “curse of dimensionality”, implying that the resulting treatment effect estimators can have considerably poor properties, even for large sample sizes. Such a common practice raises the important issue of model misspecification. Indeed, as shown by Fr¨ olich (2004), Millimet and Tchernis (2009), Huber et al. (2013) and Busso et al. (2014), propensity score misspecifications can lead to misleading treatment effect estimates. In this paper we propose new specification tests for parametric propensity score mod- els. Our proposal builds on the common practice of comparing the density of the propensity score between treated and control groups to determine the covariate overlap region, see e.g. Heckman, Ichimura, Smith and Todd (1998). However, instead of comparing conditional densi- ties, we focus on comparing conditional cumulative distribution functions (CDFs). In particular, we derive a restriction between the propensity score CDFs among treated and control groups that gives information on overlapping $^{1}$, show that such a restriction is equivalent to a particular infinite number of unconditional moment conditions, and develop tests based upon it. 1 We thank an anonymous referee for making this suggestion. 2