Randomization Tests Under an Approximate Symmetry Assumption

Abstract

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.

Synthetic Control and Inference

“…For review of this property, we will borrow the short discussion in Canay et al (2017). 4 Suppose that a researcher observes a vector of observations X, whose joint distribution is P. The objective is to test whether P ∈ P 0 , where P 0 is a collection of probability distributions such that the distribution of X is equal to that of gX for every g in G, where G is a finite collection of transformations.…”

Approximate permutation tests and induced order statistics in the regression discontinuity design

“…We first derive a result on the asymptotic properties of induced order statistics in (7) that may be of independent interest and provides an important milestone in proving the asymptotic validity of our test. We then use this intermediate result and use arguments similar to those in Canay et al (2014) to prove our main theorem.…”

“…However, if each of these conditional cdfs are continuous in Z, it follows a sample from H 0 (w|Z − n,(j) ) exhibits a similar behavior to a sample from H 0 (w|0), at least for n sufficiently large. This insight of approximating randomization tests when the randomization hypothesis does not hold in finite samples, but is satisfied in the limit, was first developed by Canay et al (2014) in a context where the group of transformations G was essentially sign-changes. In this paper we show that such insight can be applied to permutations tests under a set of conditions that capture the fact that several test statistics for our problem, including the one in (10), are discontinuous and do not satisfy the no-ties condition required in Canay et al (2014).…”

“…Section 2 introduces the notation and the hypotheses of interest. Section 3 describes our permutation test based on a fixed number of closest observations and highlights how it does not directly map into the framework of Canay et al (2014).…”

“…These asymptotics are intended to capture a situation where the number of effective observations close to the threshold is small relative to the sample size. Formally, we exploit the framework, with novel additions, from Canay, Romano and Shaikh (2014), which first developed the insight of approximating randomization tests in this manner. Further, in an important intermediate stage, we use induced order statistics, c.f.…”