Estimation and Inference in Boundary Discontinuity Designs: Distance-Based Methods

TL;DR

This paper develops nonparametric distance-based local polynomial methods for estimating treatment effects at boundaries, providing theoretical guarantees, practical guidance, and software for causal inference in boundary discontinuity designs.

Contribution

It introduces new boundary-robust estimation and inference techniques, including bandwidth selection rules, with theoretical validation and software implementation.

Findings

Derived bounds on estimator bias and convergence rates.

Established distributional approximations for valid inference.

Provided practical bandwidth selection rules for irregular boundaries.

Abstract

We study nonparametric distance-based (isotropic) local polynomial methods for estimating the boundary average treatment effect curve, a causal functional that captures treatment effect heterogeneity in boundary discontinuity designs. We establish identification, estimation, and inference results both pointwise and uniformly along the treatment assignment boundary. We show that the geometric regularity of the boundary, a one-dimensional manifold, plays a central role in determining feasible convergence rates and valid inference procedures. Our theoretical contributions are threefold. First, we derive uniform lower and upper bounds on the convergence rate of the misspecification bias of isotropic local polynomial estimators. Second, we obtain uniform distributional approximations that justify boundary-robust inference. Third, we establish minimax lower bounds for a broad class of nonparametric isotropic regression estimators. These results yield practical guidance for empirical implementation, including new bandwidth selection rules that adapt to local irregularities of the treatment-assignment boundary. We illustrate the proposed methods using simulation evidence and an empirical application, and provide companion general-purpose software.