What is Regression Discontinuity Design?
Regression Discontinuity Design is a quasi-experimental design that identifies a local causal effect at a known cutoff when treatment assignment or treatment probability changes discontinuously while untreated and treated potential outcomes remain continuous in the running variable.
Quick Facts
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How It Works
Identification comes from continuity at the cutoff
In a sharp RD, the treatment effect is the difference between the right- and left-hand limits of the conditional outcome at cutoff c. The design requires no other cause of the outcome to jump at that same threshold, a stable assignment rule, consistent treatment, and no interference. Imbens and Lemieux review the design, graphical analysis, local estimation, bandwidth choice, and validity checks.
Local polynomial and bandwidth choices control bias and variance
Estimate separate local linear or low-order polynomial regressions on each side, usually weighting observations nearer the cutoff more heavily. Narrow bandwidths reduce smooth-function bias but increase variance; wide bandwidths add precision while demanding stronger approximation. Report robust bias-corrected intervals and sensitivity to defensible bandwidths, kernels, polynomial orders, donut exclusions, and covariate adjustments. Avoid global high-order polynomials.
Manipulation and coincident changes threaten the design
Inspect the running-variable density, heaping, sorting incentives, predetermined covariate continuity, treatment-probability jump, placebo outcomes, and placebo cutoffs. A density discontinuity is diagnostic rather than a complete validity verdict, and smooth density does not exclude hidden policy changes at the threshold. Document the assignment process and limit conclusions to the supported local population.
Key Characteristics
- Uses a known threshold in a running or assignment variable
- Estimates a local effect from limits on both sides of the cutoff
- Distinguishes sharp assignment from fuzzy treatment compliance
- Balances local approximation bias against sampling variance
- Requires continuity, no precise sorting, and no coincident interventions
- Has strong local interpretation but limited automatic external validity
Common Use Cases
- Evaluating eligibility rules based on an exam or risk-score threshold
- Measuring effects of age-based or income-based policy cutoffs
- Estimating a local effect when treatment compliance is incomplete
- Auditing whether users manipulate a product eligibility score
- Comparing results across bandwidths and placebo thresholds
Example
Loading code...Frequently Asked Questions
What is the difference between sharp and fuzzy Regression Discontinuity?
In sharp RD, crossing the cutoff deterministically changes treatment. In fuzzy RD, it changes treatment probability, so cutoff eligibility acts as an instrument and the ratio of outcome and treatment jumps identifies a local complier effect under additional IV assumptions.
Why is bandwidth selection important in RD?
A narrow bandwidth compares more similar units and reduces smooth-function approximation bias but uses fewer observations. A wider bandwidth improves precision while requiring the regression shape to remain credible farther from the cutoff. Data-driven selection and sensitivity checks should accompany robust intervals.
Does a density test prove there is no manipulation?
No. A discontinuity in running-variable density can reveal sorting, but a smooth density does not prove units lacked influence or that no other process changed at the cutoff. Combine density evidence with institutional knowledge, heaping checks, covariate continuity, and assignment audits.
Can an RD estimate be generalized away from the cutoff?
Not automatically. The primary estimand is local to units near the threshold, and a fuzzy design is further local to compliers. Extrapolation needs additional structural assumptions, data at other cutoffs, or a transport design and should be reported separately.
Why should high-order global polynomials be avoided in RD?
They can oscillate, overweight distant observations, produce unstable boundary estimates, and give poor confidence-interval coverage. Separate local linear or quadratic fits with transparent bandwidth and robust bias correction are generally easier to diagnose and defend.