Local-Polynomial Estimation for Multivariate Regression Discontinuity Designs

TL;DR

This paper addresses estimating heterogeneous treatment effects in multivariate regression discontinuity designs with multiple running variables by proposing a multivariate local-linear RD estimator that uses dimension-specific bandwidths and an MSE-optimal bandwidth selector. It provides a bias-corrected inference framework and proves asymptotic normality, showing through simulations that the method yields smaller RMSE and often shorter confidence intervals than rdrobust. The authors demonstrate the approach in two empirical contexts—a Colombian scholarship design with dual thresholds and a Lee-style covariate design—highlighting richer boundary heterogeneity and robustness to scaling. Overall, the work offers a principled kernel-based remedy to the suboptimality of distance-based approaches in multivariate RD and facilitates more precise, boundary-specific inference and heterogeneity discovery.

Abstract

We study a multivariate regression discontinuity design in which treatment is assigned by crossing a boundary in the space of multiple running variables. We document that the existing bandwidth selector is suboptimal for a multivariate regression discontinuity design when the distance to a boundary point is used for its running variable, and introduce a multivariate local-linear estimator for multivariate regression discontinuity designs. Our estimator is asymptotically valid and can capture heterogeneous treatment effects over the boundary. We demonstrate that our estimator exhibits smaller root mean squared errors and often shorter confidence intervals in numerical simulations. We illustrate our estimator in our empirical applications of multivariate designs of a Colombian scholarship study and a U.S. House of representative voting study and demonstrate that our estimator reveals richer heterogeneous treatment effects with often shorter confidence intervals than the existing estimator.

Figures (10)

• Figure 1: Illustration of $\mathcal{T}$.
• Figure 2: A scatter plot with joint density estimates in solid contour plot curves. The $x$-axis represents the SISBEN score minus the policy cutoff; the $y$-axis represents the SABER11 score minus the policy cutoff. Circles over the boundary represent 30 points to evaluate in the simulation, where we use the filled $28$ points for the empirical analysis later. Positive scores in both measures imply that the requirements are satisfied.
• Figure 3: 3D plots of the mean functions at four boundary points. The horizontal line is the boundary; the center circle is the evaluation point. We rotate the axes so that the $X$-axis aligns with the boundary and the sign of $Y$-axis value determines the treatment status. See Appendix \ref{['sec.polynomialShapes']} for the exact polynomial shapes used and supports for each design.
• Figure 4: Histograms of point estimates with trimming of 1% tail realizations. Light-colored distributions are of our estimator; dark-colored distributions are of the rdrobust.
• Figure 5: $95\%$ confidence intervals over the boundary points. Dark-colored ranges are of our estimates. Light-colored ranges are of rdrobust estimates. The Left panel (a) is for exceeding the merit threshold among the need-eligible students; the right panel (b) is for exceeding the need threshold among the merit-eligible students.

Theorems & Definitions (9)

• Proposition 2.1
• Proposition 2.2
• proof
• Theorem 2.1: Asymptotic normality of local-linear estimators
• Theorem B.1: Asymptotic normality of local-polynomial estimators
• proof
• Remark B.1: General form of the MSE of $\widehat{\partial_{j_1\dots j_L}m(r)}$
• Proposition F.1
• proof