Some of the most credible quasi-experimental estimates come from an elegant idea: when a treatment is assigned by a strict THRESHOLD on some continuous score, the units just above and just below the threshold are nearly identical — except that one side got the treatment. Comparing them yields a clean causal estimate. This is regression discontinuity, and it is everywhere in policy, because policies love thresholds.
The regression-discontinuity idea
A natural experiment at the threshold
Regression discontinuity (RD) applies when treatment is assigned by a cutoff on a continuous 'running variable' (also called the forcing or assignment variable): a scholarship for students scoring above a test threshold, a poverty programme for households below an eligibility score, a class-size rule that triggers at an enrolment count, a benefit at an age cutoff, an election won at 50% of the vote. The insight: units JUST ABOVE and JUST BELOW the cutoff are almost identical — a student scoring 69 and one scoring 71 (cutoff 70) are essentially the same in ability and everything else; which side of the line they fall on is essentially RANDOM (driven by the luck of a few exam questions). So comparing outcomes JUST above vs JUST below the cutoff is like a randomised experiment in the neighbourhood of the threshold: the JUMP (discontinuity) in the outcome AT the cutoff is the causal effect of the treatment. RD exploits the arbitrariness of a sharp threshold to create local randomisation — a 'natural experiment' that policies hand us whenever they use a cutoff.
Sharp and fuzzy RD
- Sharp RD — treatment is FULLY determined by the cutoff: everyone above gets it, no one below (a scholarship strictly given to those above the score). The jump in the outcome at the cutoff directly gives the treatment effect.
- Fuzzy RD — crossing the cutoff changes the PROBABILITY (not the certainty) of treatment: those above are MORE likely to be treated but not all are, and some below get treated anyway (the cutoff makes you eligible but take-up is imperfect). Here the cutoff is used as an INSTRUMENT for treatment (the IV logic of the non-compliance and matching modules): the jump in the outcome is scaled by the jump in the treatment probability to recover the effect on compliers (a LATE at the cutoff). Fuzzy RD is RD's analogue of the non-compliance/LATE story.
The local nature of RD
An effect only AT the cutoff
RD's great strength (credibility — local randomisation at the threshold) comes with a corresponding limitation: the estimate is LOCAL to the cutoff. It identifies the treatment effect only for units NEAR the threshold — those whose scores put them just above or below. It does NOT tell you the effect for units far from the cutoff (a student scoring 95, or 40, may respond very differently to the scholarship than one scoring 70). So RD has strong INTERNAL validity (a credible causal estimate at the cutoff) but inherently LIMITED external validity (the effect at the cutoff may not generalise to the rest of the distribution) — a specific instance of the external-validity issue (module 8). This matters for policy: an RD estimate answers 'what is the effect of the treatment for marginal units near the eligibility threshold?' — which is often the policy-relevant margin (the effect of EXPANDING eligibility slightly is exactly the effect at the current cutoff), but is not the effect for inframarginal recipients. Knowing that RD estimates a local effect at the cutoff is essential to interpreting and using it correctly.
The no-manipulation assumption
Can units sort around the cutoff?
RD's validity rests on units NOT being able to precisely MANIPULATE their position relative to the cutoff. If units can sort themselves just onto the favourable side (a student who knows the threshold getting their exam re-marked to 70; an official manipulating a poverty score to make a favoured household eligible; a firm misreporting to stay under a regulatory size cutoff), then those just above and just below are NO LONGER comparable — the ones who manipulated their way over are different (more motivated, better connected), reintroducing selection bias exactly at the cutoff where RD needs local randomness. The check: the McCrary density test — examine the DENSITY of the running variable around the cutoff. If there's no manipulation, the density should be SMOOTH through the cutoff; a JUMP/bunching (a pile-up of units just on the favourable side) signals manipulation and invalidates the design. A second check: covariates that should be unaffected by treatment should be CONTINUOUS (no jump) at the cutoff — a jump in a baseline characteristic at the threshold signals sorting. A credible RD study always shows the density test (no bunching) and covariate-continuity checks. The no-manipulation/no-bunching assumption is RD's analogue of the 'as good as random' requirement, and it's the first thing to scrutinise.
Exercise
A government gives a scholarship to all students who score 60 or above on a national exam (cutoff = 60), and wants to estimate the scholarship's effect on university completion. (1) Explain how a regression-discontinuity design would estimate this and why it's credible. (2) State whether this is sharp or fuzzy RD and what changes if take-up is imperfect. (3) Explain what the estimate does and doesn't tell the government (the local nature). (4) Explain the manipulation threat and how to check for it.