This paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity. The first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions. These conditions are typically referred to as completeness conditions. The second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions. For each of these hypothesis testing problems, we provide conditions under which any test will have power no greater than size against any alternative. In this sense, we conclude that no nontrivial tests for these hypothesis testing problems exist.

This paper studies the behavior, under local misspecification, of several confidence sets (CSs) commonly used in the literature on inference in moment (in)equality models. We propose the amount of asymptotic confidence size distortion as a criterion to choose among competing inference methods. This criterion is then applied to compare across test statistics and critical values employed in the construction of CSs. We find two important results under weak assumptions. First, we show that CSs based on subsampling and generalized moment selection (Andrews and Soares (2010)) suffer from the same degree of asymptotic confidence size distortion, despite the fact that asymptotically the latter can lead to CSs with strictly smaller expected volume under correct model specification. Second, we show that the asymptotic confidence size of CSs based on the quasi-likelihood ratio test statistic can be an arbitrary small fraction of the asymptotic confidence size of CSs based on the modified method of moments test statistic.

This paper develops a theory of randomization tests under an approximate symmetry as-sumption. Randomization tests provide a general means of constructing tests that control size in finite samples whenever the distribution of the observed data exhibits symmetry under the null hypothesis. Here, by exhibits symmetry we mean that the distribution remains invariant under a group of transformations. In this paper, we provide conditions under which the same construction can be used to construct tests that asymptotically control the probability of a false rejection whenever the distribution of the observed data exhibits approximate symmetry in the sense that the limiting distribution of a function of the data exhibits symmetry under the null hypothesis. An important application of this idea is in settings where the data may be grouped into a fixed number of “clusters ” with a large number of observations within each cluster. In such settings, we show that the distribution of the observed data satisfies our ap-proximate symmetry requirement under weak assumptions. In particular, our results allow for the clusters to be heterogeneous and also have dependence not only within each cluster, but also across clusters. This approach enjoys several advantages over other approaches in these settings. Among other things, it leads to a test that is asymptotically similar, which, as shown in a simulation study, translates into improved power at many alternatives. Finally, we use our results to revisit the analysis of Angrist and Lavy (2009), who examine the impact of a cash award on exam performance for low-achievement students in Israel.