Empirical Likelihood Covariate Adjustment for Regression Discontinuity Designs

TL;DR

This paper introduces an empirical likelihood covariate adjustment method for regression discontinuity designs that improves estimator efficiency, robustness, and inference accuracy by optimally balancing covariates using entropy balancing weights.

Contribution

It proposes a novel entropy balancing approach within an empirical likelihood framework for covariate adjustment in RD designs, enhancing efficiency and robustness.

Findings

Estimator has asymptotic variance no larger than standard methods

Likelihood ratio confidence sets can be analytically corrected for better coverage

Method is robust to covariate perturbations and applicable to other RD settings

Abstract

This paper proposes a versatile covariate adjustment method that directly incorporates covariate balance in regression discontinuity (RD) designs. The new empirical entropy balancing method reweights the standard local polynomial RD estimator by using the entropy balancing weights that minimize the Kullback--Leibler divergence from the uniform weights while satisfying the covariate balance constraints. Our estimator can be formulated as an empirical likelihood estimator that efficiently incorporates the information from the covariate balance condition as correctly specified over-identifying moment restrictions, and thus has an asymptotic variance no larger than that of the standard estimator without covariates. We demystify the asymptotic efficiency gain of Calonico, Cattaneo, Farrell, and Titiunik (2019)'s regression-based covariate-adjusted estimator, as their estimator has the same asymptotic variance as ours. Further efficiency improvement from balancing over sieve spaces is possible if our entropy balancing weights are computed using stronger covariate balance constraints that are imposed on functions of covariates. We then show that our method enjoys favorable second-order properties from empirical likelihood estimation and inference: the estimator has a small (bounded) nonlinearity bias, and the likelihood ratio based confidence set admits a simple analytical correction that can be used to improve coverage accuracy. The coverage accuracy of our confidence set is robust against slight perturbation to the covariate balance condition, which may happen in cases such as data contamination and misspecified "unaffected" outcomes used as covariates. The proposed entropy balancing approach for covariate adjustment is applicable to other RD-related settings.