Nonlinear and Nonseparable Structural Functions in Fuzzy Regression Discontinuity Designs

TL;DR

This paper develops a nonparametric identification and semiparametric estimation method for the structural function in fuzzy regression discontinuity designs with a continuous treatment, allowing for nonlinear and nonseparable models, and demonstrates its application to health outcomes.

Contribution

It introduces a novel three-step semiparametric estimation procedure for nonlinear, nonseparable structural functions in fuzzy RD with continuous treatments, under shape restrictions.

Findings

The structural function can be nonparametrically identified at the RD cutoff.

The proposed estimator achieves the same convergence rate as in binary treatment cases.

Application estimates the causal effect of sleep time on health using natural light discontinuity.

Abstract

Many empirical examples of regression discontinuity (RD) designs concern a continuous treatment variable, but the theoretical aspects of such models are less studied. This study examines the identification and estimation of the structural function in fuzzy RD designs with a continuous treatment variable. The structural function fully describes the causal impact of the treatment on the outcome. We show that the nonlinear and nonseparable structural function can be nonparametrically identified at the RD cutoff under shape restrictions, including monotonicity and smoothness conditions. Based on the nonparametric identification equation, we propose a three-step semiparametric estimation procedure and establish the asymptotic normality of the estimator. The semiparametric estimator achieves the same convergence rate as in the case of a binary treatment variable. As an application of the method, we estimate the causal effect of sleep time on health status by using the discontinuity in natural light timing at time zone boundaries.