Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Matias D. Cattaneo, Rocio Titiunik, Ruiqi Rae Yu

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.

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

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.

Paper Structure

This paper contains 21 sections, 6 theorems, 33 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Organization
  5. Setup and Assumptions
  6. Identification and Interpretation
  7. Approximation Bias
  8. Estimation and Inference
  9. Convergence Rates
  10. Uncertainty Quantification
  11. Discussion and Implementation
  12. Distance-based Minimax Convergence Rate
  13. Numerical Results
  14. Simulation Study
  15. Empirical Application
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose Assumptions assump: DGP(i)--(iii) and assump: Kernel, Distance, and Boundary(i)--(ii) hold. Then, for all $\mathbf{x}\in\mathcal{B}$.

Figures (4)

  • Figure 1: Lack of smoothness of distance-based conditional expectation near a kink. Note: Analytic example of $\theta_{1,\mathbf{b}}(r) = \mathbb{E}[Y(1)|D_i(\mathbf{b})=r]$, $r\geq0$, for distance transformation $D_i(\mathbf{b})=\mathcal{d}(\mathbf{X}_i, \mathbf{b}) = \|\mathbf{X}_i - \mathbf{b}\|$ to point $\mathbf{b}\in\mathcal{B}$ near a kink point on the boundary, based on location $\mathbf{X}_i=(X_{1i},X_{2i})^\top$. The induced univariate conditional expectation $r\mapsto\theta_{1,\mathbf{b}}(r)$ is continuous but not differentiable at $r=r_3$.
  • Figure 2: Scatter Plot and Selected Distance-Based RD Plots (SPP Data)
  • Figure 3: Average of BATEC Estimators (Simulation Results). Notes: (i) Smooth denotes using bandwidths $h=\widehat{h}_{\mathtt{MSE},\mathbf{x}}$ under the assumption that the boundary is smooth; (ii) Adaptive denotes using bandwidths $\widehat{h}_{\mathtt{kink},\mathbf{x}}(\mathcal{B})$ that adapts to the kink given information about the kink location; (iii) Unknown Kink denotes using the bandwidth $h = \widehat{\mathtt{C}} \cdot n^{-1/4}$ under kink rates; (iv) Rdrobust denotes the (incorrect) univariate MSE optimal bandwidth $h_{\mathtt{1d},\mathbf{x}}$ from rdrobust; and (v) Population denotes the population BATEC causal parameter, $\tau(\mathbf{x})$, calibrated using the SPP dataset (see Table \ref{['tab:true-params']}).
  • Figure 4: BATEC Estimation and Inference (SPP Empirical Application)

Theorems & Definitions (6)

  • Theorem 1: Identification
  • Theorem 2: Approximation Bias: Uniform Guarantee
  • Theorem 3: Approximation Bias: Smooth Boundary
  • Theorem 4: Convergence Rates
  • Theorem 5: Statistical Inference
  • Theorem 6: Distance-based Minimax Convergence Rate