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

Matias D. Cattaneo, Rocio Titiunik, Ruiqi Rae Yu

TL;DR

This paper extends boundary discontinuity designs to a two-dimensional assignment score by developing location-based local polynomial estimators for the boundary average treatment effect curve $\tau(\mathbf{x})$ along the boundary $\mathcal{B}$. It introduces two aggregations, the weighted and largest boundary average treatment effects ($\tau_{\mathrm{WBATE}}$ and $\tau_{\mathrm{LBATE}}$), and provides pointwise and uniform inference through new MSE expansions and a strong approximation for a Gaussian process on $\mathcal{B}$. The methods are implemented in the open-source package $\texttt{rd2d}$ and demonstrated with an empirical BD application to the Colombian Ser Pilo Paga program, revealing heterogeneity along the boundary and validating the approach with placebo checks. The work also documents extensions to fuzzy BD designs and pre-treatment covariates, offering a practical and theoretically rigorous toolkit for location-based causal inference on boundaries with a 1D manifold geometry.

Abstract

Boundary discontinuity designs are used to learn about causal treatment effects along a continuous assignment boundary that splits units into control and treatment groups according to a bivariate location score. We analyze the statistical properties of local polynomial treatment effect estimators employing location information for each unit. We develop pointwise and uniform estimation and inference methods for both the conditional treatment effect function at the assignment boundary as well as for transformations thereof, which aggregate information along the boundary. We illustrate our methods with an empirical application. Companion general-purpose software is provided.

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

TL;DR

This paper extends boundary discontinuity designs to a two-dimensional assignment score by developing location-based local polynomial estimators for the boundary average treatment effect curve along the boundary . It introduces two aggregations, the weighted and largest boundary average treatment effects ( and ), and provides pointwise and uniform inference through new MSE expansions and a strong approximation for a Gaussian process on . The methods are implemented in the open-source package and demonstrated with an empirical BD application to the Colombian Ser Pilo Paga program, revealing heterogeneity along the boundary and validating the approach with placebo checks. The work also documents extensions to fuzzy BD designs and pre-treatment covariates, offering a practical and theoretically rigorous toolkit for location-based causal inference on boundaries with a 1D manifold geometry.

Abstract

Boundary discontinuity designs are used to learn about causal treatment effects along a continuous assignment boundary that splits units into control and treatment groups according to a bivariate location score. We analyze the statistical properties of local polynomial treatment effect estimators employing location information for each unit. We develop pointwise and uniform estimation and inference methods for both the conditional treatment effect function at the assignment boundary as well as for transformations thereof, which aggregate information along the boundary. We illustrate our methods with an empirical application. Companion general-purpose software is provided.
Paper Structure (15 sections, 6 theorems, 36 equations, 3 figures, 1 table)

This paper contains 15 sections, 6 theorems, 36 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Organization and Related Literature
  3. Notation
  4. Setup
  5. Boundary Average Treatment Effect Curve
  6. Treatment Effect Estimation
  7. Uncertainty Quantification
  8. Implementation
  9. Weighted Boundary Average Treatment Effect
  10. Largest Boundary Average Treatment Effect
  11. The Causal Effects of SPP on College Attendance
  12. Extensions
  13. Imperfect Compliance
  14. Pre-treatment Covariates
  15. Conclusion

Key Result

Theorem 1

Suppose Assumptions assump: DGP and assump: Kernel and Boundary hold. If $n h^2/\log(1/h) \to \infty$ and $h\to0$, then

Figures (3)

  • Figure 1: Scatterplot, Assignment Boundary, and Treatment Effects Using SPP data. Note: Panel (a) presents a scatterplot of the bivariate score $\mathbf{X}_i$ using the SPP data, and also plots the treatment boundary $\mathcal{B}$ with $40$ marked grid points. Panel (b) presents causal treatment effect estimates over the $40$ boundary grid points depicted in Panel (a). Specifically, the black solid dots correspond to $\widehat{\tau}(\mathbf{b}_j)$ (BATEC), the blue dotted line corresponds to $\widehat{\tau}_{\mathtt{WBATE}}$ (WBATE), and the red dot-dash line corresponds to $\widehat{\tau}_{\mathtt{LBATE}}$ (LBATE). The companion R software package rd2d is used for implementation; further details are available in the replication files and in Cattaneo-Titiunik-Yu_2025_rd2d.
  • Figure 2: Estimation and Inference. Outcome: College Enrollment.
  • Figure 3: Estimation and Inference. Outcome: Mother's education (placebo).

Theorems & Definitions (6)

  • Theorem 1: Convergence Rates
  • Theorem 2: MSE Expansions
  • Theorem 3: Confidence Intervals and Bands
  • Theorem 4: MSE Expansion: WBATE
  • Theorem 5: Distributional Approximation: WBATE
  • Theorem 6: Confidence Interval: Largest Treatment Effect