Regression Discontinuity Design – Sharp

Motivating Example

Does winning an election make a politician more likely to win the next election? You might think so — incumbents have name recognition, fundraising advantages, and can deliver benefits to their districts. But testing this is hard. Politicians who win elections are different from those who lose: they may be more charismatic, better funded, or from more favorable districts. Simply comparing incumbents to non-incumbents conflates the effect of incumbency with all these pre-existing differences.

David Lee found an elegant solution(Lee, 2008). In a two-candidate race, the candidate who gets 50.1% of the vote wins, while the candidate who gets 49.9% loses. But these candidates are nearly identical — they ran in similar districts, spent similar amounts, and had similar qualities. The 0.2 percentage point difference in vote share that separates them is essentially random noise.

Lee's insight: by comparing candidates who barely won to candidates who barely lost, you get something close to a randomized experiment — but created naturally by the sharp cutoff rule that determines who wins. This approach is a regression discontinuity design.

The finding: barely winning an election increases the probability of winning the next election by about 45 percentage points. The incumbency advantage is real and enormous.

A. Overview

What RDD Does

A regression discontinuity design exploits a rule that assigns treatment based on whether a continuous variable (the running variable) falls above or below a known cutoff. Units just above and just below the cutoff are almost identical on average — the only systematic difference between them is whether they received treatment. By comparing outcomes on either side of the cutoff, you can estimate the causal effect of treatment.

Why It Works

The key intuition is that units very close to the cutoff are "as good as randomly assigned" to treatment. A student who scores 71 on an exam and receives a scholarship is not meaningfully different from a student who scores 69 and does not. Their tiny score difference is essentially noise — influenced by whether they slept well, guessed correctly on one question, or had a good day. By zooming in on this narrow window around the cutoff, you approximate the conditions of a randomized experiment.

Sharp vs. Fuzzy

In a sharp RDD, the treatment is a deterministic function of the running variable: everyone above the cutoff is treated, everyone below is not. There is no discretion, no exceptions, no partial compliance. Examples:

• Scholarship awarded if test score >= 80 (everyone above gets it, no one below does)
• Politician wins if vote share > 50%
• Firm is regulated if revenue exceeds a threshold

In a fuzzy RDD, crossing the cutoff changes the probability of treatment but does not determine it perfectly. See the Fuzzy RDD page for that case. The fuzzy design is closely related to instrumental variables, where the cutoff serves as the instrument for actual treatment.

When to Use RDD

• Treatment is assigned by a rule based on a continuous running variable and a known cutoff
• The running variable is observed and measured accurately
• There is no evidence that units manipulate the running variable to sort around the cutoff
• You have enough observations near the cutoff

When NOT to Use RDD

• The running variable is discrete with few values (e.g., age in whole years) — the "as good as random" argument is weaker
• There is strong evidence of manipulation (e.g., students who know the cutoff score target exactly that score)
• You want to estimate the effect for the entire population, not just for units near the cutoff
• The cutoff is not sharp (treatment leaks across the boundary) — consider fuzzy RDD instead

The Taxonomy Position

RDD is a design-based method. Its credibility comes from the institutional rule that generates the cutoff, not from a model of how confounders relate to outcomes. Unlike OLS, which requires the researcher to correctly specify all confounders, RDD leverages a specific institutional feature that produces quasi-random variation. Thistlethwaite and Campbell (1960) introduced the idea in the context of merit-based scholarships(Thistlethwaite & Campbell, 1960). Lee and Lemieux (2010) describe RDD as "the most credible and transparent non-experimental strategy for estimating causal effects." This characterization reflects RDD's strong internal validity when a sharp cutoff rule exists and manipulation is absent; it does not imply RDD is universally preferred, since the design is only applicable in settings with an institutional threshold and the resulting estimates are local to the cutoff.

Common Confusions

B. Identification

Assumption 1: Continuity of Potential Outcomes at the Cutoff

Plain language: Everything about units — their potential outcomes, their background characteristics, their unobserved traits — changes smoothly as the running variable crosses the cutoff. The only thing that jumps at the cutoff is treatment status. If you could graph every characteristic against the running variable, you would see smooth curves everywhere except in the treatment indicator, which jumps from 0 to 1.

Formally: $E[Y(0) | X = c]$ and $E[Y(1) | X = c]$ are continuous in $X$ at the cutoff $c$, where $Y(0)$ and $Y(1)$ are potential outcomes without and with treatment.

This continuity condition is the core identifying assumption. If it holds, any discontinuity in the outcome at the cutoff can be attributed to the treatment.

Assumption 2: No Manipulation (No Precise Sorting)

Plain language: Units cannot precisely control their running variable to sort around the cutoff. If students who would score 69 can strategically boost their score to 71 to get a scholarship, the students just above and below the cutoff are no longer comparable — the ones above are those who tried harder or had the resources to manipulate their score.

"No manipulation" does not mean units are unaware of the cutoff or do not try to cross it. It means they cannot precisely control the running variable. In elections, candidates try hard to win, but they cannot guarantee they will get exactly 50.1% of the vote. The residual uncertainty in the running variable is what creates the quasi-random assignment.

Assumption 3: Treatment Assignment is Sharp

Plain language: Treatment is a deterministic function of the running variable. Everyone above the cutoff is treated; no one below is treated. If there are exceptions — people above the cutoff who do not receive treatment, or people below who do — you have a fuzzy RDD, which requires different estimation methods (essentially, IV/2SLS at the cutoff).

What RDD Identifies

Under these assumptions, the RDD estimates the local average treatment effect at the cutoff:

This quantity is the expected treatment effect for units whose running variable is exactly at the threshold.

C. Visual Intuition

Notice what happens as you increase the polynomial order above 2: the fit becomes erratic near the cutoff, and the estimate becomes unstable. This instability is why Gelman and Imbens (2019) recommend against global high-order polynomials. Stick with local linear (order 1) or local quadratic (order 2).

D. Mathematical Derivation

E. Implementation

The rdrobust Ecosystem

The standard approach for RDD estimation is the rdrobust package, developed by Cattaneo et al. (2020). It is available in Stata, R, and Python.

1library(rdrobust)
2library(rddensity)
3
4# Basic sharp RDD estimate
5rd <- rdrobust(y = df$y, x = df$x, c = 0)
6summary(rd)
7
8# RDD plot
9rdplot(y = df$y, x = df$x, c = 0, p = 1,
10     x.label = "Running Variable",
11     y.label = "Outcome",
12     title = "Sharp RDD Plot")
13
14# Manipulation test
15density_test <- rddensity(X = df$x, c = 0)
16summary(density_test)
17rdplotdensity(density_test, df$x)
18
19# With covariates
20rd_cov <- rdrobust(y = df$y, x = df$x, c = 0,
21                 covs = cbind(df$age, df$female))
22summary(rd_cov)
23
24# Sensitivity: vary bandwidth
25for (h in c(5, 10, 15, 20)) {
26cat("\nBandwidth =", h, "\n")
27print(summary(rdrobust(y = df$y, x = df$x, c = 0, h = h)))
28}

Understanding the rdrobust Output

The key numbers to report from rdrobust:

1. Conventional estimate: The local polynomial point estimate with conventional standard error
2. Bias-corrected estimate: The point estimate after bias correction
3. Robust confidence interval: Uses the bias-corrected estimate with a standard error that accounts for the additional variability introduced by the bias correction (preferred for inference)
4. Bandwidth: The data-driven optimal bandwidth used
5. Effective number of observations: How many observations fall within the bandwidth on each side

F. Diagnostics

F.1 McCrary Density Test (Manipulation Check)

An essential diagnostic for any RDD. If units can precisely sort around the cutoff, you will see a discontinuity in the density of the running variable at the cutoff. The McCrary test (and its modern successor rddensity) formally tests for this sorting behavior.

• If the density is smooth through the cutoff: no evidence of manipulation (good)
• If there is a jump in density: possible manipulation (serious threat to identification)

Example: If students know the scholarship cutoff is 80, and you see an excess of students scoring exactly 80-82 with a deficit at 78-79, that bunching is evidence of manipulation.

F.2 Covariate Balance (Placebo Tests)

Run the RDD on each pre-determined covariate (age, gender, prior achievement, etc.) as the outcome. If the running variable is as-good-as-randomly assigned near the cutoff, there should be no jump in any pre-determined covariate. This test is the RDD analog of the "balance table" in a randomized experiment.

F.3 Bandwidth Sensitivity

Re-estimate the treatment effect using a range of bandwidths (e.g., 0.5x, 0.75x, 1x, 1.25x, 1.5x, 2x the optimal bandwidth). If the estimate is stable across bandwidths, that stability is reassuring. If it swings wildly, the result may be fragile. For a broader discussion of robustness testing across specifications, see sensitivity analysis.

F.4 Placebo Cutoffs

Estimate the treatment effect at fake cutoffs where no treatment actually occurs (e.g., at the median of the running variable, or at percentiles far from the real cutoff). You should find no significant effects at these placebo cutoffs. If you do, something besides the treatment is causing discontinuities.

F.5 Donut Hole

Remove observations very close to the cutoff (e.g., within 1 unit) and re-estimate. If the result is driven entirely by the handful of observations right at the boundary, it may be fragile or reflect manipulation by units who scored exactly at the cutoff.

Interpreting Your Results

What to Report

1. The RDD plot with the fitted lines and data points
2. The point estimate, robust confidence interval, and p-value from rdrobust
3. The optimal bandwidth and the effective number of observations on each side
4. The McCrary density test results
5. Covariate balance tests at the cutoff
6. Sensitivity of the estimate to bandwidth choice

How to Interpret the Estimate

Common Misstatements to Avoid

• Avoid claiming the effect applies to all units — it is local to the cutoff
• Avoid using global high-order polynomials and presenting the approach as best practice
• Include the McCrary test
• Remember that the effective sample size is the number within the bandwidth, not the total sample

G. What Can Go Wrong

H. Practice

H.1 Concept Checks

H.2 Guided Exercise

H.3 Error Detective

H.5 You Are the Referee

I. Swap-In: When to Use Something Else

• Fuzzy RDD: When the cutoff induces a jump in the probability of treatment but not a deterministic shift — fuzzy RDD uses the threshold as an instrument for treatment receipt.
• Difference-in-differences: When there is no sharp threshold but a policy change creates treated and untreated groups with temporal variation.
• IV / 2SLS: When the source of exogenous variation is not a threshold on a running variable but a discrete instrument (e.g., lottery, draft number).
• Matching: When there is no threshold but rich observational data allow for matching on pre-treatment covariates.
• RDD with covariates: When the bandwidth is narrow and precision is low, adding pre-determined covariates can improve efficiency without threatening identification.

J. Reviewer Checklist