Regression Discontinuity Design

Dr. R. Düsing · Osnabrück University
Running Example
XQualification test score (quasi-assignment variable, 0–100 points) cCutoff = 50 — anyone scoring ≥ 50 is admitted to the training program TProgram admission (T=1 if X ≥ c, otherwise T=0) YGross salary 2 years after program start (scaled 0–100, measurable for all applicants)
Does the training causally raise salary — or do higher-performing applicants simply earn more anyway? Just above and below the cutoff, applicants should differ only by chance: that's exactly what the RDD exploits.
① Scatterplot
② Regression Equation
③ Coefficients
▶ Extension — Polynomial Terms (Degree 2)
Polynomial RDD
If the true relationship between the assignment variable and the outcome is non-linear, this can bias the LATE estimate. Adding polynomial terms models the curve shape more flexibly:

ŷ = b₀ + b₁·(X−c) + b₂·T + b₃·(X−c)·T + b₄·(X−c)² + b₅·T·(X−c)² b₄ controls the curvature of the control-group line.
b₅ gives the curvature difference treatment−control (slope of the treatment curve = b₄ + b₅).

Caution: too-high polynomial degrees can lead to overfitting and artificially distort the LATE — especially at the edges of the data distribution.
b₄ (curvature, control) 0.000
b₅ (curvature diff.) 0.000
▶ Extension — Fuzzy RDD
Fuzzy RDD — When the Boundary "Blurs"
In a sharp RDD, treatment assignment follows the cutoff deterministically.
In a fuzzy RDD, there are two forms of non-compliance:

Non-compliance (treatment group): people above the cutoff don't take up the intervention.
Crossover (control group): people below the cutoff receive the intervention anyway.

The cutoff is then used as an instrument (Z) for actual treatment (T). The LATE results from the Wald estimator:

LATE = intent-to-treat effect / compliance rate Latent confounder (motivation): in this simulation, non-compliers decide based on their motivation — unmotivated treatment-group people decline the treatment, highly motivated control-group people actively seek it out. Since motivation also directly affects Y (γ), T is endogenous: naive OLS overestimates the LATE.
The Wald estimator uses G (assignment) as an instrument — G depends only on X, not on motivation, so the ITT is an unbiased starting point and LATE(Wald) ≈ b₂.
P(non-compliance | treated) 0.15
P(crossover | control) 0.10
Confounding γ 15
Concepts
What Is an RDD?
A regression discontinuity design uses a sharp cutoff c on a continuous assignment variable X: whoever is ≥ c is treated, whoever is below is not. Just above and below c, units differ only by chance — as in a local experiment. A jump in Y right at the cutoff is then the causal effect of the treatment.
Continuity Assumption
The central identification assumption: everything except treatment runs continuously at the cutoff. Without the program there would be no jump in Y at c — all other influencing factors change smoothly. If that's plausible, the jump is causally interpretable. (The RDD counterpart to the parallel-trends assumption in DiD.) If units deliberately manipulate their score to land just above c, the assumption is violated (McCrary density test).
LATE — the Local Effect at the Cutoff
The RDD estimates a Local Average Treatment Effect (LATE): the treatment effect exactly at the cutoff — the jump in Y at c. It only holds for units near the cutoff; people far below c differ systematically from those just below it, so the effect doesn't automatically generalize to them. High internal, limited external validity.
The Estimation Model
The LATE is estimated with Y = b₀ + b₁(X−c) + b₂·T + b₃·T(X−c). The assignment variable is centered at the cutoff (X−c) so that b₂ = the jump at c = LATE. b₁ is the slope on the left, b₃ the slope difference on the right. This gives both sides separate lines, and the vertical gap at the cutoff can be read directly as the effect.
Bandwidth & Polynomials — Pitfalls
A narrower window around c reduces bias (only comparable units) but increases variance (less data) — the classic bias-variance trade-off. High polynomial degrees seem flexible but produce overfitting and edge artifacts that distort the jump. Recommendation (Gelman & Imbens 2019): local-linear estimation in a narrow bandwidth instead of global high-degree polynomials.
Sharp vs. Fuzzy & Context
Sharp RDD: crossing the cutoff fully determines treatment (T = 1 ⇔ X ≥ c). Fuzzy RDD: the cutoff only increases the probability of treatment without forcing it — some just above c don't participate (no-shows), some just below are treated anyway (crossover). The treatment rate therefore jumps by less than 100% at c. A naive comparison of Y values thus underestimates the effect. The Wald estimator corrects this: it divides the jump in Y by the jump in the treatment rate, recovering the effect for the compliers (those who are actually moved to participate by the cutoff). RDD belongs to the quasi-experimental designs without randomization. → Difference-in-Differences · → Propensity Score Matching · → Causal Inference
Regression Discontinuity Design — Help
Example

Applicants with a qualification test score X ≥ 50 are admitted to the training program (T=1), everyone else is not (T=0). Salary Y (2 years later) is measurable for all applicants — regardless of whether they participated or not.

The problem: admitted and rejected applicants differ systematically in X, and hence in everything related to X. A naive group comparison would be biased. The RDD avoids this by evaluating only the local discontinuity at the cutoff.

RDD Logic & the Continuity Assumption

The core idea: directly to the left and right of the cutoff, applicants differ only by chance (they had almost the same score). Had the admission mechanism not intervened, both groups would show the same Y on average — the expected outcome curve would be continuous at the cutoff.

Continuity assumption: E[Y(0) | X] and E[Y(1) | X] are both continuous in X at c. Any observable jump in Y at the cutoff can then be causally attributed to the treatment.

Violation: applicants manipulate X to land just above c → selection bias

In the example: if applicants specifically practice before the test to hit exactly the cutoff, the groups just above/below c are no longer comparable — the continuity assumption would be violated.

Estimation Model & LATE

The model centers the running variable at the cutoff (X̃ = X − c) and adds a jump indicator variable T as well as, optionally, a slope difference (T·X̃):

Y = b₀ + b₁·X̃ + b₂·T + b₃·(T·X̃) + ε

b₀ = expected Y right at the cutoff (T=0 side)  |  b₁ = slope of the control group  |  b₂ = jump at the cutoff = LATE  |  b₃ = slope difference, treatment vs. control group

LATE = b₂ = lim[E(Y|X↓c)] − lim[E(Y|X↑c)]

In the example: b₂ = 15 means that program participants earn, on average, 15 scale points more salary at the cutoff than comparable non-participants — the local causal effect of the training.

Important: the LATE only holds for people near the cutoff (limited external validity). People far below the cutoff differ systematically from people just below it.

Scenarios A–D

A — No effect: b₂=0, b₃=0. The regression line is continuous at the cutoff — no jump, no difference in slopes. Visible: both lines meet seamlessly.

B — Jump (LATE): b₂≠0, b₃=0. A clear jump at the cutoff, but both sides have the same slope. The classic sharp-RDD pattern.

C — Kink (slope difference): b₂=0, b₃≠0. No level jump, but the slope changes. Useful when the treatment produces a continuous intensity difference (regression kink design).

D — Jump + kink: b₂≠0, b₃≠0. Both discontinuities at once. In the example this could mean: the training raises not only the salary level but also the salary increase per qualification point.

Polynomial Terms

A linear model assumes Y is linear in X on each side of the cutoff. If the true relationship is non-linear (curved), this can create an apparent discontinuity — or hide a real one.

The polynomial extension (degree 2) adds X̃² and T·X̃²:

Y = b₀ + b₁·X̃ + b₂·T + b₃·(T·X̃) + b₄·X̃² + b₅·(T·X̃²) + ε

In the example: if applicants with a very high score (X=90) would earn well even without the program (ceiling effect), a linear model would overestimate the LATE. The polynomial model captures this curvature.

Trade-off: higher polynomial degrees increase the variance of the estimate and can lead to overfitting near the cutoff. Degree 2 is usually sufficient — Imbens & Gelman (2019) recommend local linear estimation with an optimal bandwidth instead of high-degree polynomials.

Fuzzy RDD

In a sharp RDD, the treatment probability P(T=1|X) jumps from 0 to 1 at the cutoff — every applicant above the cutoff necessarily participates, none below do. Fuzzy RDD allows incomplete compliance: the jump in P(T=1|X) at the cutoff is real, but < 1.

Compliance types: depending on assignment Z (Z=1: admitted; Z=0: not admitted) and actual participation T, people can behave differently:

In the example: some applicants with score X ≥ 50 don't take up the training (never-takers, e.g. due to lack of time). Then P(T=1|X) doesn't jump to 1 at the cutoff, but to, say, 0.85.

ITT — Intent-to-Treat

The intent-to-treat effect (ITT) measures how much the mere offer of treatment — the assignment Z — changes the outcome Y on average, regardless of whether the person actually participated:

ITT = E[Y | Z=1] − E[Y | Z=0] at the cutoff

In the RDD, Z corresponds to the indicator variable Z = 1[X ≥ c] — a person is "assigned" as soon as their score exceeds the cutoff.

Why is the ITT useful? Assignment Z is (at least near the cutoff) quasi-random — who lands just above or just below the cutoff differs only by chance. Hence the ITT is unbiased: it compares two groups that differ in expectation only by assignment, not by selection effects.

Why isn't the ITT the whole story? The ITT dilutes the true treatment effect because it also includes never-takers, who were assigned (Z=1) but never participated. Their outcome is "null" relative to the program — which pulls the ITT down. The more non-compliance, the more the ITT underestimates.

IV — Instrumental Variable

To get from the ITT to the true treatment effect, we need an instrument. An instrumental variable (IV) is a variable Z that satisfies three conditions:

  1. Relevance: Z affects the probability of actual participation T. In the example: those admitted (Z=1) participate with substantially higher probability. This is testable: check whether the jump in P(T=1|X) at the cutoff is significant (first-stage test).
  2. Exogeneity (independence): Z is not correlated with unobserved confounders U. In the RDD: whoever lands just above or just below the cutoff differs in expectation only by chance — the assignment is like a local randomization. This makes Z uncorrelated with everything else that affects salary (motivation, ability, background).
  3. Exclusion restriction: Z affects Y exclusively through T — there is no direct path from Z to Y. In the example: mere admission to the program must not have its own salary effect; only actual participation affects salary. If, say, the employer already treats admission itself as a signal of competence and raises salary because of it, this condition would be violated.

In the RDD, Z = 1[X ≥ c] is the natural instrument: it is relevant (jump in P(T=1|X)), locally exogenous (quasi-random assignment near c), and — under a plausible assumption — satisfies the exclusion restriction.

LATE(Wald) — Local Average Treatment Effect

The LATE (Local Average Treatment Effect), also called the Wald estimator, combines the ITT and the first stage:

LATE = ITT / first stage = jump in E[Y|X] / jump in E[T|X] at the cutoff

Intuition: the ITT is the effect of assignment Z on Y. But only a fraction of assigned people actually participate (the compliers). The jump in E[T|X] at the cutoff measures exactly this fraction — the first stage. Dividing the ITT by the first stage "corrects" for non-compliance: it yields the effect attributable to actual participation.

In the example (defaults: b₂=15, non-compliance=15%): ITT = 15 × 0.85 = 12.75k €, first stage = 0.85 → LATE = 12.75 / 0.85 = 15k € salary increase for compliers.

Why is the LATE better than naive OLS? A naive OLS comparison (participants vs. non-participants) is biased by self-selection: those who choose the program are often more motivated, more capable, or have better networks — independent of the program. This makes T endogenous (correlated with the error term ε). The IV/LATE approach sidesteps this problem by using only the variation in T caused by the quasi-random assignment Z — this variation is, by construction, uncorrelated with U.

The "local" in LATE has two meanings:

In the example: the LATE of 10k € holds for applicants who were just above the cutoff and would have accepted admission (compliers). Whether the program would be equally effective for never-takers or for applicants far above the cutoff cannot be inferred from this design.

References

Thistlethwaite, D. L. & Campbell, D. T. (1960). Regression-discontinuity analysis: An alternative to the ex post facto experiment. Journal of Educational Psychology, 51(6), 309–317.
Gelman, A. & Imbens, G. (2019). Why high-order polynomials should not be used in regression discontinuity designs. Journal of Business & Economic Statistics, 37(3), 447–456.