Method·advanced·11 min read

Design-BasedEstablished

Regression Discontinuity Design – Fuzzy

When crossing the cutoff changes the probability of treatment (not a guarantee), use fuzzy RDD — essentially IV at the cutoff.

When to UseWhen crossing the cutoff changes the probability of treatment but does not determine it perfectly — there is noncompliance at the threshold. Essentially IV at the cutoff.

AssumptionThe running variable cannot be precisely manipulated, the cutoff is a valid instrument (exclusion restriction holds at the threshold), and the first stage (jump in treatment probability at the cutoff) is sufficiently strong.

MistakeApplying sharp RDD methods when there is substantial noncompliance at the threshold, or not reporting and testing the first-stage jump in treatment probability.

Reading Time~11 min read · 11 sections · 8 interactive exercises

One-Line Implementation

Rrdrobust(y = df$y, x = df$x, c = 0, fuzzy = df$treatment)

Statardrobust y x, c(0) fuzzy(treatment)

Pythonrdrobust(y=df['y'], x=df['x'], c=0, fuzzy=df['treatment'])

Download Full Analysis Code

Complete scripts with diagnostics, robustness checks, and result export.

Motivating Example: Maimonides' Rule and Class Size

The 12th-century scholar Maimonides wrote that a class should not exceed 40 students. Israel adopted this as policy: when enrollment in a grade exceeds 40, the school must split into two classes. At 41 students, average class size drops from 41 to about 20.5.

Angrist and Lavy (1999) exploited this rule to estimate the effect of class size on student achievement. But here is the complication: the rule creates a discontinuity in expected class size, not a perfect deterministic assignment. Some schools do not comply perfectly — they might get exemptions, combine classes, or handle the split differently. Crossing the 40-student threshold changes the probability of being in a small class, but does not guarantee it.

This setting is a fuzzy RDD. The running variable (enrollment) crosses a cutoff (40), and the probability of treatment (small class) jumps — but not from 0 to 1. It jumps from, say, 0.1 to 0.7. To handle this non-compliance, you use the cutoff as an instrument for actual treatment.

AOverview

Sharp RDD (Quick Review)

In a sharp RDD, treatment is a deterministic function of the running variable:

D_i = \mathbf{1}(X_i \geq c)

Everyone above the cutoff is treated; everyone below is not. The treatment effect is identified by the jump in the outcome at $c$ .

Fuzzy RDD

In a fuzzy RDD, the cutoff changes the probability of treatment, but not deterministically:

\lim_{x \downarrow c} P(D_i = 1 \mid X_i = x) \neq \lim_{x \uparrow c} P(D_i = 1 \mid X_i = x)

There is a jump in the first stage (the probability of treatment) at the cutoff, but it is less than one. Some units above the cutoff do not receive treatment, and some below do.

The Key Insight: Fuzzy RDD = IV at the Cutoff

The fuzzy RDD estimator is the ratio of two jumps:

\tau_{FRD} = \frac{\text{Jump in outcome at } c}{\text{Jump in treatment probability at } c} = \frac{\lim_{x \downarrow c} E[Y_i \mid X_i = x] - \lim_{x \uparrow c} E[Y_i \mid X_i = x]}{\lim_{x \downarrow c} E[D_i \mid X_i = x] - \lim_{x \uparrow c} E[D_i \mid X_i = x]}

This ratio is exactly the Wald/IV estimator (the same logic underlying instrumental variables estimation), with the instrument being the indicator for crossing the cutoff: $Z_i = \mathbf{1}(X_i \geq c)$ .

Common Confusions

BIdentification

The Three Requirements

Fuzzy RDD requires:

Relevance — The cutoff causes a jump in the probability of treatment. This jump is the first stage, and it must be meaningfully large.
Continuity of potential outcomes — $E[Y_i(0) \mid X_i = x]$ and $E[Y_i(1) \mid X_i = x]$ are continuous at $c$ . Informally: absent the treatment jump, the outcome would not jump at the cutoff.
No manipulation — Units cannot precisely control the running variable to sort above or below the cutoff.

Monotonicity

Additionally, fuzzy RDD requires a monotonicity assumption: crossing the cutoff can only increase (or only decrease) the probability of treatment for any individual. There are no "defiers" — units for whom crossing the cutoff would reduce their treatment probability.

Formal Estimand

The fuzzy RDD identifies:

\tau_{FRD} = E[Y(1) - Y(0) \mid \text{complier at } X = c]

This estimand is the treatment effect for the subpopulation of compliers — those whose treatment status would change if they moved from just below to just above the cutoff — evaluated right at the cutoff.

CVisual Intuition

Imagine two graphs stacked vertically. The top graph shows the outcome (test scores) as a function of the running variable (enrollment). You see a cloud of data points with a fitted curve on each side of the cutoff. At the cutoff, the outcome might show a jump (the reduced form).

The bottom graph shows the treatment (actual class size) as a function of enrollment. Here, you clearly see a jump at 40: the probability of being in a small class increases sharply. This jump is the first stage.

The fuzzy RDD estimate is the ratio of the top jump to the bottom jump. If test scores jump by 3 points and the probability of small-class treatment jumps by 0.6, the fuzzy RDD estimate is $3 / 0.6 = 5$ points — the causal effect of small classes for compliers at the cutoff.

If the bottom graph shows no jump (flat first stage), you have no instrument and cannot identify the treatment effect. The strength of the first stage is critical.

DMathematical Derivation

Don't worry about the notation yet — here's what this means in words: The fuzzy RDD estimate divides the jump in the outcome by the jump in treatment probability at the cutoff, just like an IV Wald estimator applied locally.

Define the instrument $Z_i = \mathbf{1}(X_i \geq c)$ . The local Wald estimator is:

\hat{\tau}_{FRD} = \frac{\hat{\tau}_Y}{\hat{\tau}_D}

where:

\hat{\tau}_Y = \lim_{x \downarrow c} \hat{E}[Y_i \mid X_i = x] - \lim_{x \uparrow c} \hat{E}[Y_i \mid X_i = x]

\hat{\tau}_D = \lim_{x \downarrow c} \hat{E}[D_i \mid X_i = x] - \lim_{x \uparrow c} \hat{E}[D_i \mid X_i = x]

In practice, both limits are estimated using local polynomial regressions within a bandwidth $h$ of the cutoff. The most common approach is local linear regression, weighted by a kernel function.

Implementation via 2SLS within the bandwidth:

For observations within bandwidth $h$ of $c$ , estimate:

First stage:

D_i = \gamma_0 + \gamma_1 Z_i + \gamma_2(X_i - c) + \gamma_3 Z_i(X_i - c) + v_i

Second stage:

Y_i = \beta_0 + \beta_1 \hat{D}_i + \beta_2(X_i - c) + \beta_3 Z_i(X_i - c) + \varepsilon_i

The coefficient $\hat{\beta}_1$ is the fuzzy RDD estimate. The slope interaction uses the exogenous cutoff indicator $Z_i = \mathbf{1}\{X_i \geq c\}$ rather than $\hat{D}_i$ , because $D_i(X_i - c)$ would be a second endogenous regressor requiring its own first stage. The $\gamma_1$ coefficient from the first stage is the jump in treatment probability.

Bandwidth selection: Imbens and Kalyanaraman (2012) and Calonico et al. (2014) provide data-driven bandwidth selectors that balance bias and variance.

EImplementation

1# Requires: rdrobust, rddensity
2library(rdrobust)
3library(rddensity)
4
5# --- Step 1: Fuzzy RDD Estimation ---
6# rdrobust() with fuzzy= estimates the LATE at the cutoff using
7# a local IV approach: crossing the cutoff instruments treatment take-up.
8# y = outcome, x = running variable, c = cutoff value.
9# fuzzy = the endogenous treatment variable (not everyone complies).
10frd <- rdrobust(y = df$test_score, x = df$enrollment, c = 40,
11              fuzzy = df$small_class)
12# Output: conventional, bias-corrected, and robust estimates + CIs.
13# The estimate is the LATE for compliers at the cutoff.
14summary(frd)
15
16# --- Step 2: First-Stage Discontinuity ---
17# Check that the treatment probability jumps at the cutoff.
18# A large, significant jump confirms the instrument (cutoff) is strong.
19# A weak first stage leads to imprecise and potentially biased estimates.
20first_stage <- rdrobust(y = df$small_class, x = df$enrollment, c = 40)
21summary(first_stage)
22
23# --- Step 3: RD Plot ---
24# Visualize the discontinuity with binned scatter plots on each side.
25# nbins = c(20, 20) uses 20 bins on each side of the cutoff.
26# Look for a visible jump in the outcome at c = 40.
27rdplot(y = df$test_score, x = df$enrollment, c = 40, nbins = c(20, 20))
28
29# --- Step 4: Manipulation Test ---
30# McCrary density test checks for sorting around the cutoff.
31# If agents can manipulate enrollment to cross the threshold,
32# the RDD assumption (local randomization) is violated.
33# A significant result (low p-value) is evidence of manipulation.
34manip <- rddensity(X = df$enrollment, c = 40)
35summary(manip)

Requiresrdrobust rddensity

FDiagnostics

First-Stage Strength

The first stage must show a clear, statistically significant jump in treatment probability at the cutoff. Report:

The size of the jump (coefficient on the cutoff indicator in the first-stage regression)
The effective F-statistic
A plot of treatment probability against the running variable

If the first stage is weak (small jump), the fuzzy RDD estimate will be imprecise and potentially biased, just like weak-instrument IV. Sensitivity analysis across different bandwidths can help assess the robustness of the result.

Manipulation Test

Use the McCrary (2008) density test or the Cattaneo et al. (2020) test to check whether units bunch on one side of the cutoff. If the density is discontinuous at $c$ , units may be manipulating the running variable.

Covariate Balance at the Cutoff

Pre-determined covariates should not jump at the cutoff. Run the RDD specification replacing the outcome with each covariate. Significant jumps suggest either manipulation or a violation of the continuity assumption.

Bandwidth Sensitivity

Show that results are robust to different bandwidth choices: the optimal bandwidth, half the bandwidth, and twice the bandwidth. The rdrobust package produces bias-corrected estimates that are less sensitive to bandwidth choice.

Interpreting Your Results

The fuzzy RDD estimate is the LATE for compliers at the cutoff — the causal effect of treatment for units whose treatment status changes due to the cutoff rule.
This estimate is not the average treatment effect (ATE) for the full population. Compliers at the cutoff may be a special group.
The reduced form (the jump in outcome at the cutoff, without dividing by the first stage) is the ITT analog for RDD. It is generally worth reporting.
If the first-stage jump is large (close to 1), the fuzzy RDD approaches the sharp RDD, and the LATE approaches a local ATE.
It is recommended to present the results graphically. The RD plot is often the most compelling piece of evidence — readers should see the discontinuity.

GWhat Can Go Wrong

Problem	What It Does	How to Fix It
Weak first stage	Imprecise and biased estimate	Report effective F-stat; consider whether the design is viable
Manipulation of running variable	Units sort above/below cutoff, invalidating the design	McCrary/density test; covariate balance checks
Using sharp RDD when fuzzy is needed	Understates the treatment effect (gives ITT, not LATE)	Check for non-compliance; use fuzzy specification
Global polynomial	High-order polynomials produce misleading estimates at the cutoff	Use local linear regression with data-driven bandwidth
Extrapolating away from cutoff	Estimates are only valid at the cutoff	Report the estimate as local; discuss external validity
Donut hole needed	Observations exactly at the cutoff are unusual	Try excluding observations within a small window of the cutoff

What Can Go Wrong

Weak First Stage at the Cutoff

The cutoff induces a large jump in treatment probability (from 0.15 to 0.75)

Fuzzy RDD estimate = 4.8 (SE = 1.2). First-stage jump = 0.60, effective F = 42. Precise and reliable estimate of the LATE for compliers at the cutoff.

What Can Go Wrong

Manipulation of the Running Variable

Students cannot precisely control their entrance exam scores near the scholarship cutoff

McCrary density test p = 0.43. Smooth density across the cutoff. Covariate balance tests show no jumps. The quasi-random assignment near the cutoff is credible.

What Can Go Wrong

Using a Global High-Order Polynomial Instead of Local Regression

Local linear regression within an MSE-optimal bandwidth of 5 points around the cutoff

Fuzzy RDD estimate = 5.2 (robust bias-corrected SE = 1.8). Estimate is driven by observations close to the cutoff where the design is most credible.

As (Gelman & Imbens, 2019) argue, high-order polynomial regressions in RDD are unreliable and should generally be avoided.

Concept Check

In Angrist and Lavy's class size study, suppose the reduced form (jump in test scores at enrollment = 40) is 2.5 points, and the first stage (jump in the probability of having a small class) is 0.50. What is the fuzzy RDD estimate, and what does it represent?

5.0 points — the average effect of small classes for all students.5.0 points — the effect of small classes for compliers at enrollment = 40.2.5 points — the intent-to-treat effect.1.25 points — the effect scaled by non-compliance.

HPractice

Concept Check

A scholarship is awarded to students scoring above 80 on an entrance exam. However, some students above 80 decline the scholarship, and some below 80 receive it through appeals. A researcher runs a standard (sharp) RDD, comparing average outcomes just above and below 80. What does this estimate?

The causal effect of receiving the scholarship.The reduced form (intent-to-treat): the effect of being above the cutoff, regardless of whether the student actually received the scholarship.The average treatment effect for all students.Nothing useful — the design is invalid because of non-compliance.

Concept Check

In a fuzzy RDD, the first stage shows that crossing the cutoff increases the probability of treatment from 0.20 to 0.28 — a jump of only 0.08. The effective F-statistic is 4.2. What should you be concerned about?

The first stage is positive, so the design is valid.The first stage is weak — a small treatment jump amplifies noise in the estimate, making it imprecise and potentially biased.You need to use a wider bandwidth to increase the first stage.The jump of 0.08 is too small to detect statistically, so you should use a different cutoff.

Concept Check

A researcher studies the effect of remedial math classes on college GPA. Students scoring below 60 on a placement test are assigned to remediation, but some below 60 skip it and some above 60 voluntarily enroll. She estimates the fuzzy RDD and finds an effect of 0.4 GPA points for compliers at the cutoff. Can she generalize this to all students?

Yes — the estimate is causal, so it applies broadly.No — fuzzy RDD identifies a LATE for compliers at the cutoff, which may not generalize to students far from the threshold or to always-takers and never-takers.Yes — if the bandwidth is wide enough, the estimate covers a broad range of students.No — but she can extrapolate using the slope of the outcome function away from the cutoff.

Concept Check

A researcher fits a fourth-degree global polynomial on both sides of the cutoff to estimate a fuzzy RDD. A reviewer insists she should use local linear regression with data-driven bandwidth instead. Why does the reviewer prefer this approach?

High-order global polynomials are computationally expensive.Global polynomials can produce spurious jumps at the cutoff by overfitting data far from the boundary, while local linear regression focuses on observations near the cutoff where the design is most credible.Local linear regression always gives smaller standard errors.The reviewer is wrong — global polynomials are more flexible and therefore better.

Guided Exercise

Fuzzy RDD: Medicaid Eligibility and Health Outcomes

A health economist studies whether Medicaid enrollment improves health outcomes. Medicaid eligibility is determined by an income threshold: households earning below 138% of the federal poverty level (FPL) are eligible. However, take-up is imperfect — some eligible households do not enroll, and some ineligible households obtain coverage through other programs. The researcher uses income relative to the 138% FPL cutoff as the running variable.

Error Detective

Read the analysis below carefully and identify the errors.

A researcher studies whether receiving a scholarship (given to students scoring above 80 on an entrance exam) affects college completion. They note that some students above 80 do not accept the scholarship and some below 80 receive it through appeals. They run: reg completion `score_above_80`, vce(robust), where `score_above_80` = 1 if entrance score >= 80. They report: 'Crossing the 80-point threshold increases college completion by 12 percentage points (p < 0.01).'

Select all errors you can find:

Using sharp RDD when the design is fuzzy(Estimation approach)

Not using local polynomial regression(Specification)

No manipulation test or bandwidth sensitivity(Diagnostics)

Referee Exercise

Read the paper summary below and write a brief referee critique (2-3 sentences) of the identification strategy.

Paper Summary

A study examines whether receiving need-based financial aid improves college graduation rates. Students with family income below $50,000 are eligible for a grant, but take-up is imperfect: only about 65% of eligible students actually receive the aid (some fail to complete paperwork), and about 10% of ineligible students receive aid through appeals. The authors use a fuzzy RDD with family income as the running variable and the $50,000 threshold as the cutoff. They report a fuzzy RDD estimate that receiving aid increases 6-year graduation rates by 18 percentage points.

Key Table

Variable	Coefficient	Robust SE	p-value
Aid received (fuzzy RDD)	0.180	0.065	0.006
First-stage jump	0.550
Effective F-statistic	31.2
Bandwidth (MSE-optimal)	$8,200
McCrary density test p	0.03
N (within bandwidth)	4,200

Authors' Identification Claim

The $50,000 income threshold creates a quasi-random assignment of financial aid eligibility near the cutoff. The fuzzy RDD accounts for imperfect compliance by instrumenting actual aid receipt with eligibility status.

ISwap-In: When to Use Something Else

Sharp RDD: When compliance at the cutoff is perfect — every unit above the threshold receives treatment, every unit below does not.
IV / 2SLS: When the source of exogenous variation is not a running-variable threshold but a discrete instrument. Fuzzy RDD is a special case of IV where the instrument is the indicator for crossing the cutoff.
Difference-in-differences: When there is temporal variation in treatment adoption rather than a threshold-based assignment rule.
Matching: When there is no threshold-based assignment but rich pre-treatment covariates support a selection-on-observables strategy.

JReviewer Checklist

Critical Reading Checklist

0 of 9 items checked0%

Is the fuzzy design clearly justified (non-compliance at the cutoff documented)?
Is the first-stage jump reported and statistically significant?
Is the reduced form (ITT) reported alongside the fuzzy RDD estimate?
Is a manipulation/density test presented?
Are covariate balance tests at the cutoff shown?
Is the RD plot presented (outcome and first stage vs. running variable)?
Is bandwidth sensitivity demonstrated (half, optimal, double)?
Is local linear regression used (not high-order global polynomials)?
Is the estimate correctly interpreted as a LATE for compliers at the cutoff?

Paper Library

Has replication code

Foundational (8)

Battistin, E., & Rettore, E. (2008). Ineligibles and Eligible Non-Participants as a Double Comparison Group in Regression-Discontinuity Designs.

Journal of EconometricsDOI: 10.1016/j.jeconom.2007.05.006

Battistin and Rettore propose using ineligible units and eligible non-participants as a double comparison group in regression-discontinuity designs. This specification-testing strategy allows researchers to assess the validity of RDD assumptions by checking whether the two comparison groups yield consistent estimates, strengthening the credibility of RDD-based inference.

Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014). Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs.

EconometricaDOI: 10.3982/ECTA11757

Calonico, Cattaneo, and Titiunik develop bias-corrected confidence intervals for RDD that address the problem of conventional confidence intervals being invalid when using optimal bandwidth selectors. Their rdrobust software package has become the standard tool for implementing RDD in practice.

Cattaneo, M. D., Jansson, M., & Ma, X. (2020). Simple Local Polynomial Density Estimators.

Journal of the American Statistical AssociationDOI: 10.1080/01621459.2019.1635480

Cattaneo, Jansson, and Ma propose a local polynomial density estimator for manipulation testing in regression discontinuity designs. Implemented in the rddensity package, it provides a modern alternative to the McCrary (2008) density test with better boundary properties.

Dong, Y., & Lewbel, A. (2015). Identifying the Effect of Changing the Policy Threshold in Regression Discontinuity Models.

Review of Economics and StatisticsDOI: 10.1162/REST_a_00510

Dong and Lewbel show that the derivative of the RD treatment effect with respect to the running variable at the cutoff is identified. Under a local policy-invariance interpretation, this derivative can be used to evaluate counterfactual policies that shift the eligibility threshold, broadening the policy relevance of RDD beyond the effect at the existing cutoff.

Hahn, J., Todd, P., & Van der Klaauw, W. (2001). Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design.

EconometricaDOI: 10.1111/1468-0262.00183

Hahn, Todd, and Van der Klaauw provide the formal econometric framework for both sharp and fuzzy regression discontinuity designs. For the fuzzy case, they show that the treatment effect can be identified as the ratio of the discontinuity in the outcome to the discontinuity in the treatment probability, analogous to a Wald estimator.

Imbens, G. W., & Lemieux, T. (2008). Regression Discontinuity Designs: A Guide to Practice.

Journal of EconometricsDOI: 10.1016/j.jeconom.2007.05.001

Imbens and Lemieux provide a comprehensive practical guide to implementing RDD, covering bandwidth selection, functional form, and graphical analysis. Their treatment of fuzzy RDD as a local IV estimator clarifies the interpretation and implementation for applied researchers.

Imbens, G., & Kalyanaraman, K. (2012). Optimal Bandwidth Choice for the Regression Discontinuity Estimator.

Review of Economic StudiesDOI: 10.1093/restud/rdr043

Imbens and Kalyanaraman derive the asymptotically optimal bandwidth for the local linear regression discontinuity estimator and propose a simple data-driven bandwidth selector. The IK bandwidth becomes the standard choice before the Calonico-Cattaneo-Titiunik (2014) refinement.

McCrary, J. (2008). Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test.

Journal of EconometricsDOI: 10.1016/j.jeconom.2007.05.005

McCrary develops the standard test for whether agents are manipulating the running variable to sort around the cutoff. If the density of the running variable shows a discontinuity at the cutoff, the RDD is compromised. This density test is now a routine validity check in all RDD papers.

Application (3)

Angrist, J. D., & Lavy, V. (1999). Using Maimonides' Rule to Estimate the Effect of Class Size on Scholastic Achievement.

Quarterly Journal of EconomicsDOI: 10.1162/003355399556061

Angrist and Lavy exploit a rule that caps class sizes at 40 students, creating discontinuities in class size as enrollment crosses multiples of 40. The imperfect compliance with the rule makes this a fuzzy RDD. This paper is one of the most widely taught examples of the fuzzy RDD approach.

Cunat, V., Gine, M., & Guadalupe, M. (2012). The Vote Is Cast: The Effect of Corporate Governance on Shareholder Value.

Journal of FinanceDOI: 10.1111/j.1540-6261.2012.01776.x

Cunat, Gine, and Guadalupe use a fuzzy RDD around the majority threshold in shareholder governance proposals to estimate the causal effect of governance provisions on firm value. This paper is a leading example of fuzzy RDD applied to corporate governance and finance.

Van der Klaauw, W. (2002). Estimating the Effect of Financial Aid Offers on College Enrollment: A Regression-Discontinuity Approach.

International Economic ReviewDOI: 10.1111/1468-2354.t01-1-00055

Van der Klaauw applies a fuzzy RDD to study how financial aid offers affect college enrollment decisions, exploiting discontinuities in an aid assignment rule where eligibility changes at GPA thresholds but compliance is imperfect. This paper is one of the earliest and most influential applications of fuzzy RDD.

Survey (4)

Angrist, J. D., & Pischke, J.-S. (2009). Mostly Harmless Econometrics: An Empiricist's Companion.

Princeton University PressDOI: 10.1515/9781400829828

Angrist and Pischke write one of the most influential modern textbooks on applied econometrics, organizing the field around a design-based approach to causal inference. The book provides essential treatments of instrumental variables, difference-in-differences, and regression discontinuity, each grounded in the potential outcomes framework. It remains the standard reference for graduate students learning to evaluate and implement identification strategies.

Cattaneo, M. D., Idrobo, N., & Titiunik, R. (2020). A Practical Introduction to Regression Discontinuity Designs: Foundations.

Cambridge University PressDOI: 10.1017/9781108684606

Cattaneo, Idrobo, and Titiunik provide a practical and accessible guide to implementing regression discontinuity designs, covering both sharp and fuzzy cases with worked examples and code. Part of the Cambridge Elements series, it provides step-by-step guidance on bandwidth selection, estimation, and inference using the rdrobust toolkit.

Cunningham, S. (2021). Causal Inference: The Mixtape.

Yale University PressDOI: 10.12987/9780300255881 Replication

Cunningham provides an accessible textbook with an excellent DiD chapter that walks through the intuition, the math, and the code (in Stata and R). Freely available online at mixtape.scunning.com, it is a valuable companion for students who want worked examples alongside formal treatment.

Lee, D. S., & Lemieux, T. (2010). Regression Discontinuity Designs in Economics.

Journal of Economic LiteratureDOI: 10.1257/jel.48.2.281

Lee and Lemieux write the standard survey of RDD methods in economics, covering both sharp and fuzzy designs, validity tests, and extensions. This paper is the standard reference for understanding the econometric theory and practical implementation of RDD.

One-Line Implementation

Download Full Analysis Code

Motivating Example: Maimonides' Rule and Class Size#

AOverview#

Sharp RDD (Quick Review)#

Fuzzy RDD#

The Key Insight: Fuzzy RDD = IV at the Cutoff#

Common Confusions#

BIdentification#

The Three Requirements#

Monotonicity#

Formal Estimand#

CVisual Intuition#

DMathematical Derivation#

EImplementation#

FDiagnostics#

First-Stage Strength#

Manipulation Test#

Covariate Balance at the Cutoff#

Bandwidth Sensitivity#

Interpreting Your Results#

GWhat Can Go Wrong#

Weak First Stage at the Cutoff

Manipulation of the Running Variable

Using a Global High-Order Polynomial Instead of Local Regression

HPractice#

Paper Summary

Key Table

Authors' Identification Claim

ISwap-In: When to Use Something Else#

JReviewer Checklist#

Critical Reading Checklist

Paper Library

Foundational (8)

Application (3)

Survey (4)

Tags

Motivating Example: Maimonides' Rule and Class Size

AOverview

Sharp RDD (Quick Review)

Fuzzy RDD

The Key Insight: Fuzzy RDD = IV at the Cutoff

Common Confusions

BIdentification

The Three Requirements

Monotonicity

Formal Estimand

CVisual Intuition

DMathematical Derivation

EImplementation

FDiagnostics

First-Stage Strength

Manipulation Test

Covariate Balance at the Cutoff

Bandwidth Sensitivity

Interpreting Your Results

GWhat Can Go Wrong

HPractice

ISwap-In: When to Use Something Else

JReviewer Checklist