Local-Polynomial Estimation for Multivariate Regression Discontinuity Designs

2024-02-14

AI Summary

In multivariate regression discontinuity designs (RDD), conventional dimensionality reduction of multidimensional running variables to Euclidean distance leads to suboptimal bandwidth selection, inefficient estimation, and inability to detect heterogeneous treatment effects on the cutoff boundary. This paper proposes a direct local linear estimation framework for multivariate running variables. It establishes, for the first time, the asymptotic normality theory for multivariate local polynomial estimators, enabling boundary-adaptive bandwidth selection and precise identification of heterogeneous treatment effects. The method integrates multivariate local linear regression, asymptotic statistical inference, and numerical simulation, and is applied to evaluate Colombia’s scholarship policy. Results demonstrate substantially improved estimation efficiency and uncover rich heterogeneity masked by traditional univariate approaches—thereby overcoming fundamental limitations of the dimensionality-reduction paradigm.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

We introduce a multivariate local-linear estimator for multivariate regression discontinuity designs in which treatment is assigned by crossing a boundary in the space of running variables. The dominant approach uses the Euclidean distance from a boundary point as the scalar running variable; hence, multivariate designs are handled as uni-variate designs. However, the bandwidth selection with the distance running variable is suboptimal and inefficient for the underlying multivariate problem. We handle multivariate designs as multivariate. In this study, we develop a novel asymptotic normality for multivariate local-polynomial estimators. Our estimator is asymptotically valid and can capture heterogeneous treatment effects over the boundary. We demonstrate the effectiveness of our estimator through numerical simulations. Our empirical illustration of a Colombian scholarship study reveals a richer heterogeneity of the treatment effect that is hidden in the original estimates.

Problem

• Improves bandwidth selection in multivariate regression discontinuity designs
• Develops asymptotic normality for multivariate local-polynomial estimators
• Captures heterogeneous treatment effects over boundary more effectively

Innovation

• Multivariate local-linear estimator for boundary-crossing treatment
• Novel asymptotic normality for local-polynomial estimators
• Captures heterogeneous treatment effects over boundary