📈 economics

rd2d: Causal Inference in Boundary Discontinuity Designs

This paper introduces \pkg{rd2d}, a comprehensive software package for R, Python, and Stata that implements local polynomial estimation and inference for Boundary Discontinuity designs, offering features such as automatic bandwidth selection, robust bias correction, and cluster-robust inference, which are demonstrated through an empirical application to Opportunity Zones.

Original authors: Matias D. Cattaneo, Rocio Titiunik, Ruiqi Rae Yu

Published 2026-05-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matias D. Cattaneo, Rocio Titiunik, Ruiqi Rae Yu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out if a new policy actually helps people. In the world of statistics, researchers often use a clever trick called a "Regression Discontinuity" (RD) design. Think of this like a cliff edge.

In a standard RD design, imagine a single line on a map. If you live just to the left of the line, you get a scholarship (Treatment). If you live just to the right, you don't (Control). Because the people living right next to the line are almost identical in every way except for which side of the line they fall on, you can compare them to see if the scholarship actually works.

The Problem: The World is 2D, Not 1D
The paper rd2d: Causal Inference in Boundary Discontinuity Designs points out that real life isn't just a single line; it's a whole map. Sometimes, eligibility for a program depends on two things at once.

For example, to qualify for an "Opportunity Zone" (a tax break for poor areas), a neighborhood might need to be:

Poor enough (Low Income), AND
Have a high enough poverty rate.

This creates a boundary that looks like an "L" shape on a map, not a straight line. This is called a Boundary Discontinuity (BD) design. The challenge is: How do you compare neighborhoods that are right next to this weird, two-dimensional "L" shaped line?

The Solution: The rd2d Software
The authors (Cattaneo, Titiunik, and Yu) have built a new software tool called rd2d (available for R, Python, and Stata) to solve this math puzzle. They offer two main ways to look at the data, like two different pairs of glasses:

1. The "Location" Glasses (The Direct Approach)

This method looks at the map exactly as it is. It treats the neighborhood's poverty score and income score as two separate coordinates (like Latitude and Longitude).

How it works: It draws a tiny circle around every point on that "L" shaped boundary and compares the people inside the circle on the "Treatment" side to those on the "Control" side.
Why it's good: It uses all the information available. It's like looking at a high-resolution photo where you can see every detail of the terrain.
The Catch: It's computationally heavy because it has to do complex math in two dimensions at once.

2. The "Distance" Glasses (The Simplified Approach)

This method simplifies the map. Instead of looking at two scores, it asks: "How far is this neighborhood from the boundary line?"

How it works: It converts the 2D map into a single line. If you are 1 mile inside the "Treatment" zone, your score is +1. If you are 1 mile outside, your score is -1.
Why it's good: It turns a complex 2D problem back into a simple 1D problem (like the original cliff edge). It's easier to calculate and easier to explain.
The Catch: It loses some detail. If the boundary line has a sharp corner (a "kink"), this method might get confused, just like a GPS that tries to drive a car around a sharp turn using only a straight road map.

What the Software Actually Does
The rd2d package doesn't just draw pictures; it does heavy statistical lifting:

Automatic Tuning: It automatically decides how big a "magnifying glass" (bandwidth) to use. If the glass is too small, the picture is grainy (too much noise). If it's too big, you blur out the details. The software finds the perfect size.
Fuzzy vs. Sharp: Sometimes, crossing the line guarantees you get the treatment (Sharp). Other times, crossing the line just increases your chances of getting it (Fuzzy). The software handles both.
- Analogy: A "Sharp" design is like a vending machine: put in the coin, get the soda. A "Fuzzy" design is like a lottery: buying a ticket increases your odds, but doesn't guarantee a win.
The "Opportunity Zones" Test Drive: The authors tested their software on a real-world example: Opportunity Zones.
- The Setup: They looked at whether being designated as an Opportunity Zone actually created jobs.
- The Result: They found that being eligible did strongly increase the chance of getting the designation (the "First Stage" worked). However, looking at the actual job growth, the software found no significant evidence that the zones created jobs in the early years. The effect was too small to distinguish from random noise.
- The Comparison: They ran the analysis using both the "Location" and "Distance" methods. Both methods agreed: The policy worked to get the designation, but the early job growth results were inconclusive.

Why This Matters
Before this paper, researchers had to invent their own, often inconsistent, ways to handle these 2D boundaries. Some would just ignore one of the scores; others would use messy approximations.

The rd2d package provides a standardized, rigorous toolkit. It allows researchers to:

Map out the exact "Boundary Average Treatment Effect" (BATEC)—essentially, a map showing where the policy works and where it doesn't.
Summarize that map into a single number (the average effect) or find the "best case" scenario (the largest effect).
Do all this with mathematically proven confidence intervals, so we know if the results are real or just luck.

In a Nutshell
This paper introduces a new, versatile tool for measuring cause-and-effect when the rules for getting a treatment depend on two factors at once. It offers two ways to look at the data (direct 2D mapping or simplified 1D distance), provides automatic tools to handle the math, and was successfully tested on Opportunity Zones, revealing that while the zones were easy to get, their early impact on jobs was hard to prove.

Technical Summary: rd2d: Causal Inference in Boundary Discontinuity Designs

Problem Statement

Regression Discontinuity (RD) designs are a standard tool for causal inference, typically relying on a scalar score and a single cutoff. However, many empirical settings involve treatment assignment determined by a multidimensional score relative to a one-dimensional boundary. These Boundary Discontinuity (BD) designs, also known as multi-score RD designs, include geographic RD designs and eligibility rules based on multiple thresholds (e.g., poverty rate and median income).

While classical scalar RD theory is well-established, BD designs have historically lacked systematic methodological treatment. Empirical applications often employ ad-hoc reductions of the boundary problem without formal guidance on:

Which causal parameter is being estimated.
How to summarize treatment-effect heterogeneity along the boundary.
When inference is valid uniformly across multiple boundary locations.

Existing approaches often reduce the bivariate score to a univariate signed distance, but the validity of this reduction depends heavily on the geometry of the boundary (e.g., kinks or irregularities), which can introduce bias even if the underlying conditional expectations are smooth.

Methodology

The paper introduces rd2d, a software package for R, Python, and Stata that implements recent theoretical developments in BD designs. The methodology is built on two primary approaches:

1. Location-Based Approach

This approach uses the bivariate score $X_i = (X_{1i}, X_{2i})^\top$ directly.

Estimation: It fits local polynomial regressions on both sides of the boundary $B$ using a joint treatment-interacted regression representation. This allows for the estimation of the Boundary Average Treatment Effect Curve (BATEC), $\tau(x)$ , which captures local treatment effects at every point $x$ on the boundary.
Inference: It employs robust bias-corrected (RBC) inference, extending the work of Calonico et al. (2014, 2018, 2020, 2022) to boundary-indexed estimators. This includes pointwise confidence intervals and uniform confidence bands over the boundary.
Aggregation: The framework supports scalar summaries of the BATEC, specifically the Weighted Boundary Average Treatment Effect (WBATE) and the Largest Boundary Average Treatment Effect (LBATE).
Fuzzy Designs: For settings with imperfect treatment take-up, the package estimates the reduced-form (ITT), first-stage (FS), and the fuzzy BATEC (Wald ratio), along with their respective uniform bands and aggregate summaries.

2. Distance-Based Approach

This approach replaces the bivariate score with a scalar signed distance to each point on the boundary.

Estimation: It treats the problem as a univariate RD design at each boundary point.
Validity: Theoretical results (Cattaneo et al., 2026b) clarify that distance-based methods recover the same causal targets as location-based methods only under specific geometric conditions. If the boundary has kinks or irregularities, the distance-based estimand may differ from the true BATEC due to averaging over level sets of the distance function.
Implementation: The package allows users to specify known kink locations or treat them as unknown to adjust bandwidth selection and bias correction orders accordingly.

Key Implementation Features

Joint vs. Separate Fitting: The default fitmethod = "joint" uses a treatment-interacted regression, which is crucial for cluster-robust inference when clusters (e.g., school districts) span both sides of the boundary. The separate option is retained for replication of earlier conventions.
Automatic Bandwidth Selection: Implements various selectors (MSE, CER, integrated, pointwise) with options for separate or common bandwidths across treatment sides and score coordinates.
Covariate Adjustment: Allows for efficiency improvements by including predetermined covariates with common coefficients across treatment sides, integrated into both estimation and bandwidth selection.
Regularization: Includes safeguards for covariance matrix regularization (for uniform inference), mass-point checks, and small bias regularization to prevent excessively large bandwidths.

Empirical Application

The paper illustrates the package using the Opportunity Zones program.

Setup: Eligibility is determined by a bivariate score (poverty rate and median family income) forming an L-shaped boundary. Designation is fuzzy (not all eligible tracts are designated).
Results:
- First Stage: Both location-based and distance-based methods show a strong, statistically significant first-stage effect of eligibility on designation across the boundary.
- Reduced Form: The effect of eligibility on early workplace-job growth is small and statistically insignificant.
- Fuzzy Effect: Consequently, the fuzzy BATEC (Wald ratio) is imprecise and statistically insignificant. Neither method finds evidence of a significant treatment effect on job growth.
- Comparison: The distance-based first-stage estimates showed slightly more variability near certain boundary locations compared to the location-based estimates, consistent with the theoretical sensitivity to boundary geometry.

Key Contributions

Software Implementation: The primary contribution is the rd2d package, which operationalizes complex theoretical results for applied researchers across three major statistical platforms (R, Python, Stata).
Unified Framework: It provides a coherent framework for estimating and inferring causal effects in BD designs, covering sharp and fuzzy designs, pointwise and uniform inference, and scalar summaries (WBATE, LBATE).
Methodological Clarity: By implementing both location-based and distance-based approaches, the software allows researchers to diagnose the impact of boundary geometry on their estimates. It highlights that while distance-based methods are simpler, their validity is more contingent on the smoothness of the boundary.
Robust Inference Tools: The package implements advanced inference techniques, including uniform confidence bands and cluster-robust variance estimation, which are essential for analyzing treatment effect heterogeneity along a boundary.

Significance and Claims

The paper positions rd2d as a necessary tool for the growing literature on multi-score and geographic RD designs. It claims to fill a gap where empirical work previously lacked formal guidance on parameter identification and inference validity.

The authors emphasize that while distance-based methods are valuable for their simplicity and interpretability, the location-based approach serves as the main benchmark whenever the bivariate score is available, particularly when the boundary geometry is complex. The software is designed to facilitate transparent reporting of boundary-specific effects, aggregate summaries, and the necessary diagnostics (e.g., bandwidths, effective sample sizes, and covariance regularization warnings) to ensure robust empirical conclusions.

The paper concludes that the package enables researchers to separate reduced-form, first-stage, and fuzzy effects along multidimensional boundaries, as demonstrated in the Opportunity Zones application, and that the provided syntax and documentation aim to standardize the reporting of BD designs in empirical research.