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

Matias D. Cattaneo; Rocio Titiunik; Ruiqi Rae Yu

arxiv: 2505.05670 · v3 · pith:OJRUBXJPnew · submitted 2025-05-08 · 💰 econ.EM · math.ST· stat.AP· stat.ME· stat.TH

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

Matias D. Cattaneo , Rocio Titiunik , Ruiqi Rae Yu This is my paper

Pith reviewed 2026-05-22 15:20 UTC · model grok-4.3

classification 💰 econ.EM math.STstat.APstat.MEstat.TH

keywords boundary discontinuity designtreatment effect curvelocal polynomial estimationcausal inferencesharp designfuzzy designeconometrics

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The pith

New location-based local polynomial estimators recover the curve of treatment effects along continuous assignment boundaries in discontinuity designs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Boundary discontinuity designs split units into treatment and control based on which side of a continuous boundary their bivariate location score falls. The paper develops pointwise and uniform estimation and inference procedures for the Boundary Average Treatment Effect Curve that traces how effects vary along this boundary. It also covers inference for two summary parameters: a weighted average effect and the largest effect anywhere on the boundary. The same framework applies whether compliance with the boundary rule is perfect or imperfect. Researchers can therefore map location-specific causal impacts directly from the raw location data without imposing strong functional forms on the assignment process.

Core claim

We develop pointwise and uniform estimation and inference methods for the Boundary Average Treatment Effect Curve (BATEC), as well as for two aggregated causal parameters: the Weighted Boundary Average Treatment Effect (WBATE) and the Largest Boundary Average Treatment Effect (LBATE). Our results cover both sharp and fuzzy designs using location-based local polynomial treatment effect estimators that directly employ the bivariate score of each unit.

What carries the argument

Bivariate location score combined with local polynomial approximation of conditional expectations near the boundary to identify the treatment effect curve.

If this is right

Pointwise confidence intervals can be constructed for the treatment effect at any specific point along the boundary.
Uniform confidence bands enable simultaneous inference over the entire treatment effect curve.
Valid inference remains available for the overall weighted effect and the maximum effect on the boundary.
The procedures apply equally to sharp designs with perfect compliance and fuzzy designs with imperfect compliance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could be adapted to boundaries that are not fixed but chosen by the researcher or estimated from data.
Applications in spatial economics or geography might reveal how effects change continuously across borders or district lines.
Comparison with one-dimensional regression discontinuity methods could clarify when the extra dimension of the boundary adds identifying power.

Load-bearing premise

The bivariate location score is continuously distributed and the conditional expectations of the outcome and treatment are sufficiently smooth near the boundary so local polynomials can approximate the effects.

What would settle it

A simulation or empirical example in which the estimated curve and its uniform bands fail to recover known effects once the smoothness of the conditional expectations is deliberately violated would show the methods do not deliver valid inference.

read the original 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 location-based local polynomial treatment effect estimators that directly employ the bivariate score of each unit. We develop pointwise and uniform estimation and inference methods for the \textit{Boundary Average Treatment Effect Curve} (BATEC), as well as for two aggregated causal parameters: the \textit{Weighted Boundary Average Treatment Effect} (WBATE) and the \textit{Largest Boundary Average Treatment Effect} (LBATE). Our results cover both sharp and fuzzy (imperfect compliance) designs. We illustrate the methods with an empirical application, and provide companion general-purpose software. The supplemental appendix includes additional substantive theoretical results, methodological details, and simulation evidence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper develops bivariate local polynomial estimators for the full treatment effect curve along a continuous boundary plus two aggregates, with uniform inference and software.

read the letter

The core contribution is a location-based approach that treats the bivariate score directly to estimate the Boundary Average Treatment Effect Curve (BATEC) pointwise and uniformly, along with the weighted and largest versions. It covers both sharp and fuzzy designs under standard smoothness conditions and includes an empirical example plus companion code. That is the main new piece relative to one-dimensional RDD work: the focus on the entire curve rather than a single cutoff and the uniform results over it. The simulations in the appendix and the software are useful additions that make the methods immediately usable for applied spatial work. The theoretical claims rest on local polynomial expansions adapted to the boundary support, which is a natural extension but requires care with the geometry. The stress-test concern about curvature-induced bias terms is worth checking in the proofs. When the kernel support is cut off by a curved boundary, the usual one-sided Taylor expansion can pick up extra remainder terms at the same order as the polynomial bias; if those are not bounded or subtracted, the rates and limiting distributions for the curve and aggregates could be affected. The abstract presents the methods as standard nonparametric extensions, so the paper may handle this in the supplemental results, but it is not obvious from the high-level description. This is aimed at empirical researchers who already work with geographic or spatial discontinuities and need off-the-shelf tools for the full boundary. A reader in that niche would find the estimators and code worth trying. The paper is coherent on its own terms and shows clear engagement with the relevant literature, so it deserves a serious referee even if the curvature issue requires some revision.

Referee Report

2 major / 2 minor

Summary. The paper develops location-based local polynomial estimators for boundary discontinuity designs using a bivariate location score. It provides pointwise and uniform estimation and inference results for the Boundary Average Treatment Effect Curve (BATEC) as well as the aggregated parameters Weighted Boundary Average Treatment Effect (WBATE) and Largest Boundary Average Treatment Effect (LBATE), covering both sharp and fuzzy designs. The manuscript includes an empirical application and companion software, with additional theoretical results and simulations in the supplement.

Significance. If the asymptotic results hold after addressing boundary geometry, the paper supplies practical tools for estimating treatment effect curves along continuous assignment boundaries in two dimensions. This extends standard RDD methods to settings such as geographic or spatial policy thresholds and supplies reproducible software, which strengthens its potential contribution to applied causal inference in economics.

major comments (2)

[§3.2, Theorem 3.1] §3.2, Theorem 3.1 (pointwise asymptotics for BATEC): the bias expansion for the local polynomial estimator applied directly to the bivariate score (X1, X2) does not appear to include or bound the additional O(h^2) curvature term induced by one-sided kernel truncation along a curved boundary. Under the maintained smoothness assumptions, this term is of the same order as the standard polynomial remainder and would alter the claimed bias rate and centering for the estimator.
[§4] §4, uniform inference results for WBATE and LBATE: the uniform convergence arguments rely on the pointwise expansions; if the curvature bias is not controlled, the sup-norm rates and the validity of the uniform confidence bands for the aggregated parameters are affected at the stated orders.

minor comments (2)

[§2] The notation for the boundary normal vector and the one-sided limits could be clarified with an explicit diagram in §2.
[Supplement, simulation section] Simulation section in the supplement: the reported coverage probabilities for the uniform bands should include a column for the case with non-zero boundary curvature to illustrate robustness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the bias expansion and uniform inference results. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

Referee: [§3.2, Theorem 3.1] §3.2, Theorem 3.1 (pointwise asymptotics for BATEC): the bias expansion for the local polynomial estimator applied directly to the bivariate score (X1, X2) does not appear to include or bound the additional O(h^2) curvature term induced by one-sided kernel truncation along a curved boundary. Under the maintained smoothness assumptions, this term is of the same order as the standard polynomial remainder and would alter the claimed bias rate and centering for the estimator.

Authors: We appreciate the referee highlighting this issue. Upon review, the bias expansion in Theorem 3.1 does not explicitly bound the additional O(h^2) term from one-sided kernel truncation along a curved boundary. This term arises under the maintained smoothness conditions and is of the same order as the standard remainder. We will revise the proof of Theorem 3.1 to derive and incorporate a bound on this curvature-induced bias, updating the centering term and confirming the bias rate remains o(h^2) or adjusting the statement as needed to preserve the asymptotic normality result. revision: yes
Referee: [§4] §4, uniform inference results for WBATE and LBATE: the uniform convergence arguments rely on the pointwise expansions; if the curvature bias is not controlled, the sup-norm rates and the validity of the uniform confidence bands for the aggregated parameters are affected at the stated orders.

Authors: We agree that the uniform results in Section 4 depend on the pointwise expansions. After correcting the bias expansion in Theorem 3.1 to control the curvature term, we will update the sup-norm convergence arguments and re-verify the validity of the uniform confidence bands for WBATE and LBATE at the claimed rates. This will ensure the uniform inference procedures remain valid under the revised bias control. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained in standard nonparametric theory

full rationale

The paper develops pointwise and uniform methods for the BATEC, WBATE, and LBATE using local polynomial regression applied directly to the bivariate location score, under standard smoothness assumptions on conditional expectations near the boundary. No equations or steps in the provided abstract or description reduce a claimed prediction or result to a fitted parameter by construction, nor do they rely on self-citations as the sole justification for core identification or asymptotic results. The approach follows established local polynomial approximation techniques for one-sided limits, with the central claims grounded in external nonparametric theory rather than internal redefinition or renaming of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Limited information from abstract only; typical nonparametric assumptions on smoothness and continuity of the assignment score are expected but not detailed here.

axioms (1)

domain assumption Conditional expectations of potential outcomes are sufficiently smooth near the boundary to permit local polynomial approximation.
Standard for local polynomial regression in discontinuity designs; invoked implicitly for identification of BATEC.

pith-pipeline@v0.9.0 · 5680 in / 1202 out tokens · 28122 ms · 2026-05-22T15:20:57.990457+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

local polynomial treatment effect estimators that directly employ the bivariate score... Assumption 2 (Kernel and Boundary)... integrated mean square expansion... strong approximation theorem for empirical processes
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

rectifiable curve B... one-dimensional Hausdorff measure H... De Giorgi perimeter
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

entire paper contains no reference to recognition cost, golden ratio, or parameter-free derivation of constants

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