How does a fuzzy regression discontinuity differ from a sharp one, and what does it estimate?

## Sharp versus fuzzy Let `X` be the running variable, `c` the cutoff, and `D` the treatment actually received. - **Sharp**: `P(D = 1 | X)` jumps from 0 to 1 at `c`. Assignment and receipt are the same thing. - **Fuzzy**: `P(D = 1 | X)` jumps at `c` by some amount less than 1 — say from 0.15 to 0.70. Crossing the cutoff changes the odds of treatment, not the fact of it. Fuzziness arises whenever the rule is a strong nudge rather than a mechanism: administrators exercise discretion, eligible people decline, ineligible people obtain the treatment another way, or the rule is enforced with a lag. Maimonides' rule is the textbook case. The rule says a cohort should be split into two classes once enrolment exceeds 40 pupils. Reality is messier — schools split late, merge across years, or do not split at all — so enrolment crossing 40 raises the probability of a second (and hence smaller) class without determining it. ## The estimator Estimate two jumps at the cutoff and divide: `effect = (jump in E[Y] at c) / (jump in E[D] at c)` The numerator, the outcome jump alone, is the effect of *crossing the cutoff* — an intention-to-treat-style quantity that answers "what does the rule do?" The denominator rescales it by how much treatment the rule actually delivered. If crossing the threshold moves treatment probability by 0.5 and moves the outcome by 2 points, the implied effect on those who were moved is `2 / 0.5 = 4` points. Both jumps are estimated the same way as in a sharp design: a local fit on each side of the cutoff, within a bandwidth, read at the threshold. The denominator is estimated from a regression of the treatment indicator, not the outcome, on the running variable. ## What the ratio actually estimates Two assumptions carry it. 1. **Continuity**, as in the sharp case: average potential outcomes vary smoothly through the cutoff, so the only thing that breaks at `c` is the rule. 2. **Monotonicity at the cutoff**: crossing the threshold never moves anyone *away* from treatment. Nobody responds to becoming eligible by refusing something they would have taken while ineligible. Under these, the ratio identifies the average treatment effect for the subgroup whose treatment status is flipped by crossing the cutoff, and only near the cutoff. Two exclusions are baked in: units treated regardless of the rule contribute nothing, and units never treated regardless of the rule contribute nothing. The estimate speaks for the responsive middle, at the threshold. This is narrower than the sharp-design estimand, and saying so out loud is what separates a candidate who understands the design from one who memorised the formula. ## Where it goes wrong **A weak jump in treatment probability.** If crossing the cutoff moves treatment probability by only a few percentage points, the denominator is small and noisy, and dividing by it inflates both the estimate and its uncertainty dramatically. Always report the size of the treatment-probability jump alongside the effect; a plot of treatment rate against the running variable, with the same local fits, is the honest way to show it. If that jump is not visibly large, the design does not support a confident number. **Reporting the numerator as if it were the treatment effect.** The outcome jump alone is a legitimate and often preferable quantity — it answers what the policy does — but it is the effect of the rule, not of the treatment, and it is smaller in magnitude whenever compliance is partial. Label it correctly. **Monotonicity failures.** If a class-size rule causes some schools to consolidate rather than split, crossing the threshold pushes those units the wrong way, and the ratio no longer has a clean interpretation. Think about the institutional response before assuming monotonicity holds. **Sorting around the threshold.** Fuzziness and manipulation are different problems, and fuzziness makes manipulation easier to hide: if administrators can both choose enrolment near 40 and choose whether to split, the units on the two sides may differ for reasons unrelated to the rule. ## How to answer in the room Define both designs by what happens to the treatment probability at the cutoff, name a concrete fuzzy rule, write the ratio, and state the subgroup the ratio describes. Then volunteer the diagnostic: show the treatment-probability jump before showing the effect. That order — denominator first — is how practitioners actually present a fuzzy design, and interviewers notice.