Optimal estimation for regression discontinuity design with binary outcomes

2025-09-23

AI Summary

This paper addresses the challenge of estimating treatment effects in regression discontinuity designs (RDD) with bounded—particularly binary—outcomes under finite samples. We propose the first exact finite-sample minimax-optimal linear shrinkage estimator for this setting. Under the assumption that the conditional mean function belongs to a Lipschitz class, our method solves a convex optimization problem in (n+1) dimensions subject to nonnegative weight constraints, requiring only calibration of the Lipschitz constant and avoiding large-sample approximations. Our key contributions are: (i) the first analytically optimal finite-sample estimator for RDD; and (ii) a unified, uniformly valid confidence interval construction. Simulations demonstrate substantial reductions in mean-squared error and narrower confidence intervals relative to conventional approaches. In empirical applications featuring multiple cutoffs—where standard methods often fail—our estimator delivers robust inference.

Technology Category

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

Application Category

• Responsible Web: Data and user privacy-enhancing technologies for the Web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

We develop a finite-sample optimal estimator for regression discontinuity designs when the outcomes are bounded, including binary outcomes as the leading case. Our finite-sample optimal estimator achieves the exact minimax mean squared error among linear shrinkage estimators with nonnegative weights when the regression function of a bounded outcome lies in a Lipschitz class. Although the original minimax problem involves an iterating (n+1)-dimensional non-convex optimization problem where n is the sample size, we show that our estimator is obtained by solving a convex optimization problem. A key advantage of our estimator is that the Lipschitz constant is the only tuning parameter. We also propose a uniformly valid inference procedure without a large-sample approximation. In a simulation exercise for small samples, our estimator exhibits smaller mean squared errors and shorter confidence intervals than conventional large-sample techniques which may be unreliable when the effective sample size is small. We apply our method to an empirical multi-cutoff design where the sample size for each cutoff is small. In the application, our method yields informative confidence intervals, in contrast to the leading large-sample approach.

Problem

• Develops optimal estimator for regression discontinuity with binary outcomes
• Solves minimax mean squared error for bounded outcomes in small samples
• Provides valid inference without relying on large-sample approximations

Innovation

• Finite-sample optimal estimator for bounded outcomes
• Convex optimization replaces non-convex minimax problem
• Single Lipschitz constant tuning parameter for inference