Abstract: This paper studies covariate-adjusted estimation in the sharp regression discontinuity (RD) design with a multinomial outcome. We propose a two-step local (quasi-)likelihood estimator for the RD treatment effects that incorporates observed covariates via a semiparametric extension of the multinomial logit model. We establish the asymptotic properties of the estimator, including a distributional result that facilitates simultaneous, covariate-adjusted inference on the multinomial RD effects under general rate conditions. For implementation, we propose an asymptotic mean squared error bandwidth selection rule for point estimation and develop inference procedures that remain valid under misspecification of the likelihood function—a novel contribution to the local likelihood RD framework. Simulation evidence demonstrates that the proposed covariate-adjusted methods outperform standard local likelihood RD techniques, yielding improvements in estimation accuracy and confidence interval length. We illustrate the practical utility of our approach through an empirical application that examines the effect of increased healthcare expenditures on infant mortality, using the covariate-adjusted methods to conduct sensitivity analysis within the local likelihood RD framework. All methods are implemented in an R package available as a supplement to the paper.

Abstract:  This paper studies estimation and inference for local multinomial treatment effects in the fuzzy regression discontinuity (RD) design. Extending the seminal treatment of the sharp RD design with an unordered multinomial outcome, we show the fuzzy multinomial RD estimands are identified under standard instrumental variables and local smoothness assumptions proposed in the literature. We propose a local (quasi-)likelihood estimator for the fuzzy multinomial RD effects and derive its asymptotic properties, including a joint distributional approximation for a bias-corrected estimator that enables valid inference across outcome categories even under large bandwidth choices. Leveraging these theoretical results, we construct confidence sets for the multinomial RD effects that are robust to misspecification of the likelihood function. These misspecification-robust confidence sets, developed for both the sharp and fuzzy multinomial treatment effects, represent a novel contribution to the local likelihood RD framework. We propose optimal bandwidth selection rules for both point and bias estimation in the fuzzy design, and develop an automatic data-driven bandwidth selection algorithm for implementation. All methods are available in an accompanying R package. A simulation study and empirical application are forthcoming.