Econometrica: May, 2017, Volume 85, Issue 3
Randomization Tests under an Approximate Symmetry Assumption
Ivan A. Canay, Joseph P. Romano, Azeem M. Shaikh
This paper develops a theory of randomization tests under an approximate symmetry assumption. 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 approximate 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.
Supplement to "Randomization Tests under an Approximate Symmetry Assumption"
This document provides additional results for the authors' paper "Randomization Tests under an Approximate Symmetry Assumption". It includes an application to time series regression, Monte Carlo simulations, an empirical application revisiting the analysis of Angrist and Lavy (2009), the proof of Theorem 2.1, and three auxiliary lemmas.