📈 economics

Donut Regression Discontinuity Designs

This paper provides a theoretical framework for donut regression discontinuity designs, demonstrating how excluding observations near the cutoff impacts bias and variance while showing that existing bias-aware inference methods remain valid and formalizing specification tests for comparing conventional and donut estimators.

Original authors: Cladia Noack, Chistoph Rothe

Published 2026-08-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cladia Noack, Chistoph Rothe

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 a detective trying to figure out why something happened, but you can't run a controlled experiment. You have to rely on "observational data"—just watching what happens in the real world. In the world of economics and social science, there is a clever trick called a Regression Discontinuity (RD) design. Think of it like a strict bouncer at an exclusive club. The bouncer has a rule: "If you are 18 years old or older, you get in; if you are 17 years and 364 days old, you stay out."

Because the rule is so sharp, the people who are just barely old enough (18 years and 1 day) and the people who are just barely too young (17 years and 364 days) are practically identical twins in every way except for that one day of age. If you compare their outcomes—say, how much money they make later in life—you can isolate the effect of being 18. It's like a natural experiment where the only thing that changes is the "treatment" (getting into the club).

But here's the catch: sometimes the data gets messy. People might lie about their age, or the bouncer might get confused and let in a 17-year-old who looks 18, or maybe people who are exactly 18 years old are just weirdly different from everyone else. When researchers get suspicious that the data right at the "cutoff" line is broken or manipulated, they often use a strategy called a "Donut" RD design. This is where they throw away all the data points that are too close to the line, creating a hole in the middle of their data—like a donut. They then try to guess what would have happened at the center by looking at the people who are a bit further away.

The problem is, for a long time, nobody really knew the math behind this "donut" hole. Researchers were just guessing how big the hole should be and whether their results were trustworthy. They were eating the donut without knowing if the hole was too big or if the glaze was melting.

The Paper's Story: The Donut Math

In this paper, Claudia Noack and Christoph Rothe decide to stop guessing and start doing the heavy math. They treat the "donut" not just as a heuristic trick, but as a formal statistical tool. They ask: If we cut out the middle of the data, what happens to our estimates? Do we get more wrong answers? Do our confidence intervals (the range where we think the truth lies) get wider?

Their findings are a mix of "be careful" and "here's how to do it right."

First, they discovered that cutting out the data near the line does make your estimates less precise. It's like trying to guess the temperature at the center of a room by only measuring the corners. You have to guess (extrapolate) more, which introduces more error. They found that if you remove a small "donut" (say, observations within 10% of your usual data window), your estimate's bias (how far off you might be) can jump by 41% to 63%, and your variance (how much your answer might bounce around if you did the study again) can increase by 53% to 61%. In plain English: the donut makes your answer noisier and potentially more skewed.

However, the authors also found a silver lining. They showed that a specific type of statistical tool, called "bias-aware" confidence intervals, works perfectly fine even with the donut hole. These are special intervals that are a bit wider to account for the fact that you might be slightly off. The paper proves that you don't need to invent new math for donuts; you just need to use these existing, robust tools. But there's a cost: because of the extra uncertainty, these confidence intervals get 26% to 33% longer when you use a donut. It's like the bouncer saying, "We aren't sure exactly who gets in, so we'll give you a wider range of possibilities."

The paper also tackles the question of why researchers use donuts in the first place. Often, they use them as a "diagnostic test." If the result from the "donut" data (ignoring the messy middle) is totally different from the result using all the data, it's a red flag that something is wrong with the data near the cutoff. The authors developed new, formal tests to check this. They compared two ways of doing this check:

The Old Way: Compare the "Donut" result to the "Full Data" result.
The New Way: Compare the "Donut" result to a result calculated only from the data inside the hole (the "within-donut" data).

Through simulations and math, they found that the New Way is generally a better detective. It is more powerful at spotting when the data near the cutoff is actually broken. In their simulations, the new test caught the problems much more often than the old method.

Finally, the authors applied their new rules to two real-world examples. One was about teacher contracts in Pakistan, where a policy change created a cutoff. They found that when they used the donut method, the results changed drastically, suggesting the original data near the cutoff was indeed problematic. The other example was about low birthweight babies (around 1,500 grams). They confirmed that rounding errors in birth weights were messing up the data right at the 1,500g mark, and their new tests helped quantify exactly how much that mattered.

In short, Noack and Rothe didn't just say "donuts are okay." They gave researchers a manual: "If you use a donut, expect your answers to be a bit noisier and your safety ranges to be wider. But if you use these specific tools and this new testing method, you can still trust your conclusions." They turned a popular but messy kitchen trick into a precise, scientific instrument.

Technical Summary: Donut Regression Discontinuity Designs

1. Problem Statement

Regression Discontinuity (RD) designs are a standard framework for identifying causal effects from observational data. However, empirical researchers frequently encounter concerns regarding the validity of the running variable near the cutoff, including sorting, manipulation, measurement error, or data heaping (e.g., rounding to focal values). To address these issues, researchers often employ "donut" RD designs, which exclude observations within a neighborhood (the "donut hole") around the treatment threshold.

Despite their widespread use, the implementation of donut RD designs has largely been heuristic. There is limited formal guidance on how trimming observations affects the bias, variance, and statistical inference of the estimator. Specifically, it remains unclear:

How the exclusion of data near the cutoff alters the trade-off between bias and variance.
Whether standard errors and confidence intervals reported by conventional software remain valid.
How to formally test whether the difference between a conventional RD estimate and a donut RD estimate is statistically significant or merely due to sampling variation.
How to interpret the magnitude of differences between these two estimators in the context of specification testing.

2. Methodology and Theoretical Framework

The authors analyze donut RD designs within the local linear estimation framework. They consider a sharp RD design where the running variable $X_i$ determines treatment $T_i = \mathbb{1}\{X_i \geq 0\}$ .

2.1. The Donut Estimator

The donut RD estimator, $\hat{\tau}(h, d)$ , is defined as a local linear regression that excludes observations where $|X_i| < d$ , where $d$ is the donut size and $h$ is the bandwidth ( $d < h$ ). This estimator nests the conventional RD estimator as a special case where $d=0$ .

2.2. Estimands and Identification

The paper distinguishes between two potential goals of using a donut design:

Causal Estimation: The researcher posits a hypothetical function $\mu^*(x)$ that coincides with the observed conditional expectation $\mu(x)$ outside the donut ( $|x| \geq d$ ) but may differ inside. The target parameter is the jump in $\mu^*(x)$ at zero, $\tau^* = \mu^*_+ - \mu^*_-$ .
Diagnostic/Specification Testing: The researcher tests the null hypothesis that the observed data inside the donut is compatible with the smoothness assumptions governing the data outside the donut. Under the null, $\mu(x) = \mu^*(x)$ for all $x$ , implying $\tau = \tau^*$ .

2.3. Asymptotic Regimes

The analysis considers two asymptotic regimes regarding the donut size $d$ relative to the bandwidth $h$ :

Small Donut Asymptotics: $d = ch$ where $c \in [0, 1)$ is fixed and $h \to 0$ . This models scenarios where the excluded neighborhood is small relative to the estimation window.
Large Donut Asymptotics: $d$ is fixed and $h \to d$ . This models scenarios where the estimation window is only slightly larger than the excluded gap.

2.4. Key Technical Tools

The authors derive asymptotic approximations for the bias and variance of the donut estimator. They show that the donut estimator behaves like a conventional estimator with a modified kernel function $K_c(u) = K(u)\mathbb{1}\{|u| > c\}$ . Consequently, the asymptotic properties depend on new kernel constants $B_K(c)$ and $S_K(c)$ , which are integrals over the truncated support $[c, 1]$ .

For inference, the paper utilizes bias-aware confidence intervals (Armstrong and Kolesár, 2018; Kolesár and Rothe, 2018). These intervals explicitly account for the worst-case bias allowed by a smoothness bound $M$ , rather than relying on higher-order bias correction which may be sensitive to trimming.

3. Key Contributions and Results

3.1. Point Estimation: Bias and Variance Trade-offs

The paper establishes that donut trimming introduces substantial additional uncertainty relative to conventional RD estimators.

Increased Bias and Variance: Under small donut asymptotics, excluding observations within 10% of the bandwidth ( $c=0.1$ ) increases the asymptotic bias by 41% to 63% and the variance by 53% to 61%, depending on the kernel function (Uniform, Triangular, or Epanechnikov).
Optimal Bandwidth: The authors characterize the Mean Squared Error (MSE) optimal bandwidth $h_{MSE}(d)$ . They show that for standard kernels, the optimal bandwidth is monotonically increasing in the donut size $d$ .

3.2. Inference: Validity of Bias-Aware Intervals

Robustness of Bias-Aware CIs: The paper demonstrates that bias-aware confidence intervals remain asymptotically valid for donut RD designs under both small and large donut asymptotics without requiring special adjustments.
Interval Length: While valid, the presence of a donut substantially widens these intervals. For $c=0.1$ , the asymptotic length of the confidence interval increases by 26% to 33% compared to the conventional case. This increase is driven primarily by the rise in standard error, though the critical value function also contributes.
Robust Bias Correction (RBC): In contrast, the paper notes that RBC confidence intervals (Calonico et al., 2014) only maintain correct coverage under small donut asymptotics and suffer even greater length inflation (60% to 66% for $c=0.1$ ) due to the sensitivity of higher-order polynomials to trimming.

3.3. Specification Testing

The authors develop formal statistical tests to compare conventional and donut estimators, addressing the high correlation between them.

Test 1 ( $\Delta$ ): Compares the donut estimator $\hat{\tau}(h, d)$ with the conventional estimator $\hat{\tau}(h, 0)$ . The test statistic accounts for the covariance between the two estimators.
Test 2 ( $\Gamma$ ): Compares the donut estimator $\hat{\tau}(h, d)$ with a "within-donut" estimator $\hat{\tau}(d, 0)$ (using only data inside the hole). This approach uses non-overlapping data, resulting in statistically independent estimators.
Power Properties: Theoretical analysis and simulations show that the $\Gamma$ test generally possesses superior local power properties compared to the $\Delta$ test against a class of plausible local alternatives where the conditional expectation behaves differently inside the donut.

3.4. Extensions to Fuzzy RD

The framework is extended to fuzzy RD designs.

Strong First Stage: Under small donut asymptotics with a strong first stage, the fuzzy donut estimator is asymptotically equivalent to a sharp donut estimator applied to an auxiliary outcome variable. Standard bias-aware inference applies.
Weak First Stage: In cases of weak identification or large donuts, point estimation is generally inconsistent. The authors recommend using Anderson–Rubin-type confidence sets and conducting separate specification tests for the first stage and reduced form, adjusting for multiple testing via Bonferroni correction.

4. Empirical Illustrations

The paper illustrates the methods with two applications:

Teacher Contracts (Pakistan): Revisiting Bau and Das (2020), the authors find that donut trimming (excluding a low-hiring period) drastically changes the first-stage estimate of contract assignment probabilities. Specification tests strongly reject the conventional fuzzy RD design, suggesting the results are driven by the excluded period.
Low Birthweight (US): Revisiting Almond et al. (2010) and Barreca et al. (2011) regarding infant mortality at the 1,500g threshold. The study shows that excluding observations near the cutoff (due to rounding/heaping) reduces the estimated effect of medical intervention from 0.95% to 0.16%. Specification tests indicate that the difference between the conventional and donut estimates is statistically significant, supporting concerns about data irregularities at the threshold.

5. Significance and Claims

The paper provides the first formal theoretical foundation for donut RD designs, moving the practice from heuristic application to rigorous statistical inference.

Clarification of Trade-offs: It quantifies the cost of donut trimming, showing that while it may address identification threats (like manipulation), it comes at the price of significantly higher bias and variance.
Valid Inference: It provides a toolkit for valid statistical inference (confidence intervals and hypothesis tests) that correctly accounts for the uncertainty introduced by trimming.
Diagnostic Utility: It formalizes the use of donut designs as diagnostic tools, offering a rigorous way to test whether the data near the cutoff behaves consistently with the rest of the sample.

The authors conclude that while donut trimming is a useful strategy for addressing specific data concerns, researchers must acknowledge the substantial increase in uncertainty. They suggest that the choice of donut size $d$ should rely on subject-matter knowledge rather than purely data-driven selection, particularly when the goal is valid causal inference. In specification testing contexts, however, exploring a range of $d$ values is feasible and informative.