Regression Discontinuity: Sharp and Fuzzy Designs

Core Idea

Regression discontinuity designs exploit threshold rules in policy or eligibility. When treatment changes discontinuously at a cutoff (sharp RDD) or is affected by a running variable (fuzzy RDD), the discontinuity estimates local treatment effects. RDD requires no assumption about unconfoundedness away from the threshold.

Explainer

From your study of natural experiments and identification strategies, you know the core challenge: we want to compare treated and untreated units who are otherwise identical, but treatment is rarely assigned randomly in the real world. Natural experiments exploit situations where assignment is "as-if" random due to features of the institutional environment. Regression discontinuity design (RDD) is one of the most compelling natural experiments — and it works by exploiting a threshold rule that determines treatment.

The intuition is simple. Suppose a scholarship program admits students who score 70 or above on an exam and rejects everyone below. Students who score 69 and 71 are nearly identical in ability, motivation, and background — they're separated by a single point that may reflect measurement noise as much as true ability. Yet one group gets the scholarship and the other doesn't. RDD treats this cutoff as a locally randomized experiment: right around the threshold, the treated (71+) and control (69−) groups are comparable, and any difference in outcomes is plausibly caused by the treatment. This is a sharp RDD: everyone above the cutoff receives treatment, everyone below does not, producing a step-function in treatment probability at the threshold.

The running variable is the variable that determines treatment assignment — in this case, exam score. The key identifying assumption is continuity: in the absence of treatment, the outcome would change smoothly across the threshold. We check this by plotting the outcome against the running variable on both sides of the cutoff — if the relationship is smooth everywhere except at the threshold, and there is a visible jump exactly at the cutoff, that jump is our estimate of the local average treatment effect (LATE) at the threshold. "Local" is crucial: RDD only identifies the treatment effect for units near the cutoff, not for all units. Whether this generalizes depends on your research question.

Fuzzy RDD applies when the threshold doesn't perfectly determine treatment — it only makes treatment more likely. Maybe some students below the cutoff get the scholarship through an appeal process; some above it decline it. Now treatment probability jumps at the cutoff but doesn't jump from 0 to 1. Fuzzy RDD estimates the treatment effect using the threshold as an instrumental variable: the discontinuity in treatment probability instruments for actual treatment receipt, recovering a LATE for "compliers" (those whose treatment status actually changed at the threshold). Two key threats to validity that you always check: manipulation (did people game the running variable to land just above the cutoff?) — tested by examining whether the density of the running variable is smooth at the threshold (McCrary test) — and covariate smoothness (do observable pre-treatment characteristics jump at the threshold? If so, the as-good-as-random assumption is violated).