Sharp Versus Fuzzy Regression Discontinuity

Regression discontinuity exploits a threshold rule — a test score of 70 that determines scholarship eligibility, an age cutoff that determines program access — to estimate a causal effect by comparing units just above and just below the cutoff. In the cleanest version, the sharp design, crossing the threshold perfectly determines treatment: score 70 and you get the scholarship, score 69 and you don't, with no exceptions.

Real policies are messier. A fuzzy RD arises when crossing the threshold only changes the probability of treatment rather than guaranteeing it — some students who score 69 still get a scholarship through a separate appeal, and some who score 71 decline it. The jump in outcomes at the cutoff is now a mix of the true treatment effect and the fact that treatment itself didn't jump all the way from 0% to 100%.

The fix rescales the outcome jump by the treatment jump. If test scores rise by 4 points at the cutoff, but the probability of actually receiving the scholarship only jumps from 20% to 80% (a 60-percentage-point jump, not 100), the fuzzy RD estimate divides the outcome jump by that treatment-probability jump: $4 / 0.60 \approx 6.7$ points as the effect on those actually induced to be treated by crossing the cutoff.

Ignoring the "fuzziness" and reporting the raw outcome jump as the treatment effect systematically understates it whenever the treatment probability doesn't jump all the way to 100%. In the example above, reporting the raw 4-point outcome jump as "the effect of the scholarship" would understate the true effect on students actually swayed by the cutoff by more than 40%, because most of those 4 points come from a group where only 60% of the people near the threshold actually changed treatment status.

This is exactly the same rescaling logic used in instrumental-variables estimation, where a variable that shifts treatment probability without directly affecting the outcome — here, the threshold itself — lets researchers recover a causal effect even when treatment isn't perfectly controlled by the researcher.