Approximate Permutation Tests and Induced Order Statistics in the Regression Discontinuity Design
Review of Economic Studies · 2017 · Ivan A. Canay、Vishal Kamat
中文摘要
在断点回归设计（RDD）中，常见的做法是通过检验基线协变量的均值在运行变量的断点（或阈值）处是否不发生变化，来评估该设计的可信度。这一做法部分源于 Lee（2008）所推导出的更强推论：在某些条件下，RDD 中基线协变量的分布必须在断点处连续。我们针对基线协变量分布在该断点处连续这一原假设，提出了一种基于所谓诱导次序统计量的置换检验；并引入一个新的渐近框架来分析其性质。该渐近框架旨在近似一种小样本现象：即使观测总数 n 可能很大，断点附近的有效观测数往往很小。因此，尽管 RDD 中的传统渐近理论要求随着 n →∞ 断点附近的观测数不断增长，我们的框架则在 n →∞ 时保持断点附近的观测数 q 固定。该新检验易于实施，在弱条件下渐近有效，在比其渐近有效性所需条件更强的条件下具有有限样本有效性，并且相对于基于均值的检验具有更优的检验功效。在一项模拟研究中，我们发现该新检验在各种设计中都能相当好地控制检验水平。随后，我们用该检验来评估 Lee（2008）中设计的合理性，该文是 RDD 用于研究在位优势的一个著名应用。
Abstract
In the regression discontinuity design (RDD), it is common practice to assess the credibility of the design by testing whether the means of baseline covariates do not change at the cut-off (or threshold) of the running variable.This practice is partly motivated by the stronger implication derived by Lee (2008), who showed that under certain conditions the distribution of baseline covariates in the RDD must be continuous at the cut-off.We propose a permutation test based on the so-called induced ordered statistics for the null hypothesis of continuity of the distribution of baseline covariates at the cut-off; and introduce a novel asymptotic framework to analyse its properties.The asymptotic framework is intended to approximate a small sample phenomenon: even though the total number n of observations may be large, the number of effective observations local to the cut-off is often small.Thus, while traditional asymptotics in RDD require a growing number of observations local to the cut-off as n →∞, our framework keeps the number q of observations local to the cut-off fixed as n →∞.The new test is easy to implement, asymptotically valid under weak conditions, exhibits finite sample validity under stronger conditions than those needed for its asymptotic validity, and has favourable power properties relative to tests based on means.In a simulation study, we find that the new test controls size remarkably well across designs.We then use our test to evaluate the plausibility of the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.