Advances in Regression Discontinuity Designs

Paper Session

Sunday, Jan. 8, 2017 1:00 PM – 3:00 PM

Hyatt Regency Chicago, Water Tower
Hosted By: Econometric Society
  • Chair: Matias Cattaneo, University of Michigan

Regression Discontinuity Designs Using Covariates

Matias Cattaneo
,
University of Michigan
Sebastian Calonico
,
University of Miami
Max H. Farrell
,
University of Chicago
Rocio Titiunik
,
University of Michigan

Abstract

We study identification, estimation, and inference in Regression Discontinuity (RD) designs when additional covariates are included in the estimation. Standard RD estimation and inference is based on nonparametric local polynomial methods using two variables: the outcome variable and the running variable that determines treatment assignment. Applied researchers often include additional ``pre-intervention'' covariates in their specifications to increase efficiency. However, no results justifying covariate adjustment have been formally derived in the RD literature, leaving applied researchers with little practical guidance and leading to a proliferation of ad-hoc methods that may result in invalid estimation and inference. We examine the properties of a local polynomial estimator that incorporates discrete and continuous covariates in an additive separable, linear-in-parameters way and imposes a common (likely misspecified) covariate effect on both sides of the cutoff. Under intuitive, minimal assumptions, we show that this covariate-adjusted RD estimator remains consistent for the standard RD treatment effect, while also providing point estimation and inference improvements. In contrast, we show that estimating a specification with interactions between treatment status and the covariates leads to an estimator that is inconsistent in general. We present new asymptotic mean squared error expansions, optimal bandwidth choices, optimal point estimators, robust nonparametric inference procedures based on bias-correction techniques, and heteroskedasticity-consistent standard errors. Our results cover sharp, fuzzy, and kink RD designs, and we also discuss extensions to clustered data. Finally, we present two empirical illustrations where we find 5\% to 10\% reduction in confidence interval length, and an extensive simulation study. All methods are implemented in companion \texttt{R} and \texttt{Stata} software packages.

Approximate Permutation Tests and Induced Order Statistics in the Regression Discontinuity Design

Vishal Kamat
,
Northwestern University
Ivan Canay
,
Northwestern University

Abstract

This paper proposes an asymptotically valid permutation test for a testable implication of the identification assumption in the regression discontinuity design (RDD). Here, by testable implication, we mean the requirement that the distribution of observed baseline covariates should not change discontinuously at the threshold of the so-called running variable. This contrasts to the common practice of testing the weaker implication of continuity of the means of the covariates at the threshold. When testing our null hypothesis using observations that are “close” to the threshold, the standard requirement for the finite sample validity of a permutation test does not necessarily hold. We therefore propose an asymptotic framework where there is a fixed number of closest observations to the threshold with the sample size going to infinity, and propose a permutation test based on the so-called induced order statistics that controls the limiting rejection probability under the null hypothesis. In a simulation study, we find that the new test controls size remarkably well in most designs. Finally, we use our test to evaluate the validity of the design in Lee (2008), a well-known application of the RDD to study incumbency advantage.

Simple and Honest Confidence Intervals in Nonparametric Regression

Timothy Armstrong
,
Yale University
Michal Kolesar
,
Princeton University

Abstract

We consider the problem of constructing honest confidence intervals (CIs) for a scalar parameter of interest, such as the regression discontinuity parameter, in nonparametric regression based on kernel or local polynomial estimators. To ensure that our CIs are honest, we derive and tabulate novel critical values that take into account the possible bias of the estimator upon which the CIs are based. We give sharp efficiency bounds of using different kernels, and derive the optimal bandwidth for constructing honest CIs. We show that using the bandwidth that minimizes the maximum mean-squared error results in CIs that are nearly efficient and that in this case, the critical value depends only on the rate of convergence. For the common case in which the rate of convergence is n^(4/5), the appropriate critical value for 95% CIs is 2.18, rather than the usual 1.96 critical value. We illustrate our results in a Monte Carlo study and an empirical application.

Bounds on Treatment Effects in Regression Discontinuity Designs under Manipulation of the Running Variable, with an Application to Unemployment Insurance in Brazil

Christophe Rothe
,
Columbia University
Francois Gerard
,
Columbia University
Miikka Rokkanen
,
Columbia University

Abstract

A key assumption in regression discontinuity analysis is that units cannot affect the value of their running variable through strategic behavior, or manipulation, in a way that leads to sorting on unobservable characteristics around the cutoff. Standard identification arguments break down if this condition is violated. This paper shows that treatment effects remain partially identified under weak assumptions on individuals' behavior in this case. We derive sharp bounds on causal parameters for both sharp and fuzzy designs, and show how additional structure can be used to further narrow the bounds. We use our methods to study the disincentive effect of unemployment insurance on (formal) reemployment in Brazil, where we find evidence of manipulation at an eligibility cutoff. Our bounds remain informative, despite the fact that manipulation has a sizable effect on our estimates of causal parameters.
JEL Classifications
  • C0 - General