Regression Discontinuity Designs With a Continuous Treatment

Abstract

The standard regression discontinuity (RD) design deals with a binary treatment. Many empirical applications of RD designs involve continuous treatments. This article establishes identification and robust bias-corrected inference for such RD designs. Causal identification is achieved by using any changes in the distribution of the continuous treatment at the RD threshold (including the usual mean change as a special case). We discuss a double-robust identification approach and propose an estimand that incorporates the standard fuzzy RD estimand as a special case. Applying the proposed approach, we estimate the impacts of bank capital on bank failure in the pre-Great Depression era in the United States. Our RD design takes advantage of the minimum capital requirements, which change discontinuously with town size.

Keywords:

• Continuous treatment
• Treatment quantiles
• Rank invariance
• Rank similarity
• Double-robust identification

Supplementary Materials

The supplemental appendix provides proofs of the theoretical results presented in the article, preliminary lemmas, alternative inference based on undersmoothing, and details of estimating the biases and variances of the proposed estimators as well as the AMSE optimal bandwidths. It also provides data description and additional results for the empirical application.