with our asymptotic results. Finally, our empirical application concerning the effect of trade on the environment shows the feasibility and appeal of our tests in relevant scenarios. Given that the validity of many policy evaluation procedures relies on the correct specification of the propensity score, we argue that the tests proposed in this article are important additions to the applied researchers’ toolkit. We would like to mention that, in general, our tests should be seen as a “model validation” and not a “model selection” procedure. Once a propensity score model is selected, our specifica- tion tests can provide evidence of its reliability or lack thereof. In case the putative propensity score model is rejected, one can consider more flexible specifications. For instance, one can add additional interaction terms into the original model, consider semiparametric single-index or partially linear models, among other possible strategies. Having said so, we emphasize that if one uses our testing procedure as a “model selection” device, one must bear in mind that stan- dard inference procedures for treatment effects may be invalid if one treats the resulting selected propensity score as the “true” one, see e.g. Leeb and P¨ otscher (2005). Thus, in case one uses our proposal for model selection, one must account for the model selection step in order to make valid inference about the treatment effect, see e.g. Belloni et al. (2014), Chernozhukov et al. (2016), and Belloni et al. (2017). A full discussion of this procedure is beyond the scope of this article and we leave it for future research. Finally, we note that results in Lemma 1 can also be used for estimating the propensity score such that, for a given specification q ( X, θ$_{0}$ ), condition E [ D − q ( X, θ$_{0}$ ) | q ( X, θ$_{0}$ )] = 0 a.s. is di- rectly imposed. One could use the minimum distance method described in Dominguez and Lobato (2004) to estimate θ$_{0}$ , for example. A detailed discussion of this estimator is beyond the scope of this article and is deferred to future work. We provide the proofs of our main theoretical results in this appendix. We first prove Lemma 1. Proof of Lemma 1: To begin, let F ( u ) = P ( p ( X ) ≤ u ), and for d = { 0 , 1 } , F$_{d}$ ( u ) = P ( p ( X ) ≤ u | D = d ) . Denote the density of F ( u ), F$_{1}$ ( u ) and F$_{0}$ ( u ) by f ( u ), f$_{1}$ ( u ), and f$_{0}$ ( u ). The proof of (2.1) follows from Lemma 3.1 in Shaikh et al. (2009), as they proved that, for all 30 Appendix A: Mathematical Proofs