Regression discontinuity designs

When to use it

Programmes ration access with rules, and many of those rules are thresholds on a continuous score: households below a proxy-means-test score receive a cash transfer; students above an exam mark enter a scholarship scheme; proposals above a review score win the grant; children born before a date enter school a year earlier. Wherever such a rule exists and is actually enforced, a regression discontinuity design is available — often retrospectively, from administrative records alone [1, 3].

The insight is that the units immediately either side of the cutoff are, for practical purposes, the same population. A household scoring one point below the poverty threshold is not meaningfully poorer than one scoring one point above; which side of the line each lands on is close to chance. Near the cutoff, the assignment rule behaves like a local lottery, and the comparison inherits much of the credibility of a randomised experiment without anyone having randomised anything [1, 2].

Reach for RDD before matching whenever a genuine cutoff exists: the assignment rule is known and verifiable, whereas the selection-on-observables assumption behind matching never is.

Sharp and fuzzy designs

Two variants, distinguished by what the cutoff actually does [1, 4]:

• Sharp RDD. Crossing the threshold fully determines treatment: everyone below the poverty-score cutoff receives the transfer, no one above it does. The jump in outcomes at the cutoff is the treatment effect for units at the threshold.
• Fuzzy RDD. Crossing the threshold changes the probability of treatment without fixing it — some eligible households never enrol; some ineligible ones are admitted on appeal. Here the cutoff is used as an instrument for actual take-up: the jump in outcomes is scaled by the jump in participation, and the estimate applies to the units whose participation the threshold changed. Fuzzy is the common case in development practice, where enforcement is imperfect.

Three terms recur in every write-up: the running variable (the score), the cutoff (the threshold), and the bandwidth (the window of scores either side of the cutoff actually used in estimation).

Estimation: fit locally, not globally

The estimand is the difference between two regression functions at a single point — the cutoff. Modern practice, systematised in Cattaneo, Idrobo and Titiunik’s workflow, estimates it by local polynomial regression: fit a low-order polynomial (local linear is the default) to the observations within a bandwidth on each side of the cutoff, weighting observations by their distance from it, and take the gap between the two fits at the threshold [2].

Three practical rules follow:

1. Choose the bandwidth by criterion, not convenience. The bandwidth trades bias (wide windows drag in units unlike those at the cutoff) against variance (narrow windows leave few observations). Data-driven, mean-squared-error-optimal bandwidth selectors are standard, and results should be shown to be stable across a range of bandwidths around the selected one [2].
2. Resist high-order global polynomials. Fitting a single high-degree polynomial across all the data lets observations far from the cutoff drive the estimated jump. The local, low-order fit is the defensible default [2].
3. Use inference that accounts for the fitting. Conventional confidence intervals ignore the bias of the local fit near the boundary; robust bias-corrected inference is the current standard and is implemented in the widely used rdrobust packages accompanying the Cattaneo–Idrobo–Titiunik workflow [2].

Always show the plot. A binned scatter of the outcome against the running variable, with the fits overlaid, is the single most informative exhibit in an RDD report — a jump the eye cannot see in the raw data deserves suspicion [1, 2].

Validity: the checks that make or break the design

RDD’s identifying assumption is continuity: in the absence of treatment, the outcome’s relationship to the running variable would pass smoothly through the cutoff. Two families of falsification tests probe it [1, 2]:

• Manipulation of the running variable. If people can nudge their score across the threshold — an assessor rounds a household’s poverty score down as a favour; students appeal marks just below the pass line — the units either side of the cutoff are no longer comparable, because the sorted ones differ in motivation, connections or need. The signature is bunching: a spike in the density of the running variable just on the favourable side of the cutoff. Formal density tests exist and belong in every RDD write-up, alongside an institutional account of who computed the score, who saw the threshold, and whether appeals were possible [1, 2].
• Covariate continuity. Baseline characteristics that treatment cannot have caused — household size, age, prior outcomes — must show no jump at the cutoff. Run the same RDD estimation with each baseline covariate as the outcome; significant jumps indicate sorting or a confounded threshold. Placebo cutoffs (testing for jumps at score values where nothing happens) complete the battery [2].

One further institutional check costs nothing: confirm the cutoff was actually applied. Administrative thresholds are waived, regionalised and quietly revised more often than programme documents admit; the enforced rule, not the written one, defines the design [3, 4].

Interpreting the estimate honestly

The RDD estimate is a local effect: the impact of the programme on units at (or, in fuzzy designs, on compliers at) the cutoff [1, 4]. A cash transfer’s effect on households just at the poverty line says nothing direct about its effect on the destitute far below it. This is not a flaw to hide but a scope condition to state — and it is sometimes exactly the policy question, since decisions about expanding or contracting a programme operate precisely at the margin the cutoff defines [1, 3].

Sharp and fuzzy regression discontinuity compared

	Sharp RDD	Fuzzy RDD
Cutoff determines	Treatment itself	Probability of treatment
Common in practice when	Rules are strictly enforced	Enrolment is incomplete or appeals exist
Estimate applies to	Units at the cutoff	Compliers at the cutoff
Extra data needed	Scores, treatment, outcomes	Plus actual take-up either side

Where RDD appears in development practice

The design’s natural habitat is any programme rationed by a score. Proxy-means-tested transfers assign benefits below a poverty-index threshold; school-entry age rules create birthday cutoffs; scholarship and grant schemes rank applicants and fund above a line; performance-based financing pays facilities that clear a quality score [3]. In each case the evaluation can often be run from records the programme already holds — scores, treatment status and outcomes — making RDD one of the cheapest credible designs available, provided the score data survive with enough precision and the cutoff was enforced [3, 4]. Where the threshold itself is being set or revised, the baselines and targets page in the indicators cluster covers the measurement side of that decision.

Checklist before you commit to RDD

• A continuous running variable and an enforced cutoff exist, and both are documented.
• Score data are available at fine enough resolution, with enough observations near the cutoff to support a local fit.
• A density test shows no bunching at the threshold, and the institutional story supports the absence of manipulation.
• Baseline covariates are continuous at the cutoff; placebo cutoffs show no effects.
• Estimation is local polynomial with a criterion-chosen bandwidth, robust inference, and sensitivity shown across bandwidths.
• The report presents the binned scatter plot and states plainly that the effect is local to the cutoff population.