Fuzzy Regression Discontinuity

Definition

A Fuzzy Regression Discontinuity (FRD) design is a quasi-experimental identification strategy where a forcing variable $X$ crossing a threshold $T^*$ causes a discontinuous jump in the probability (but not a deterministic switch) of receiving treatment $W$ . The discontinuity-ratio estimator identifies the Local Average Treatment Effect (LATE) for the subpopulation of compliers — units who take treatment if and only if their forcing variable falls on the treated side of the threshold.

Key Ideas

Compliance types (Angrist-Imbens-Rubin 1996): compliers ( $W = \mathbf{1}\{X \le T^*\}$ ), always-takers ( $W = 1$ regardless), never-takers ( $W = 0$ regardless); defiers ruled out by monotonicity.
FRD estimand: $\tau^{\text{frd}} = \frac{\lim_{X\to T^{*+}} E[Y|X] - \lim_{X\to T^{*-}} E[Y|X]}{\lim_{X\to T^{*+}} E[W|X] - \lim_{X\to T^{*-}} E[W|X]} = \tau^{\text{LATE}}$ by Hahn-Todd-Van der Klaauw (HTV 2001).
Double locality: The estimate is local in two senses — valid only near the threshold in X, and only for compliers, not always-takers or never-takers.
Sharp RD is the limiting case where the jump in $E[W|X]$ at the threshold is exactly 1 (all units below threshold comply).
External validity (Bertanha-Imbens 2014, Assumption 5): $G_i \perp\!\!\!\perp (Y_i(0), Y_i(1)) \mid X_i$ — compliance type independent of potential outcomes given forcing variable. Under this condition the LATE generalises to the Average Treatment Effect (ATE) everywhere in the support of $X$ .
Graphical diagnostic: Under external validity, $E[Y^{\text{obs}} \mid W=w, X]$ is continuous in $X$ at $T^*$ for each $w \in \{0,1\}$ . Bertanha-Imbens (2014, Lemma 6) recommend plotting these two treatment-arm-specific regression functions — a jump in either provides evidence against external validity.

How It Works

Under standard regularity (Assumptions 1–4 of HTV 2001): random sampling, continuous conditional distributions of potential outcomes by compliance type, and non-degenerate treatment probability on both sides of $T^*$ , the FRD estimand equals LATE:

$\tau^{\text{LATE}} = E[Y(1) - Y(0) \mid G = \text{complier},\; X = T^*]$

Identification exploits the threshold as an instrument: units just below vs. just above $T^*$ are comparable in pre-treatment characteristics, but compliers below are induced into treatment. Local linear regression on each side estimates the two discontinuities.

The Hausman test ( $\tau^{\text{exo}} = \tau^{\text{frd}}$ ) and Angrist's (2004) test are both implied by but weaker than the external validity pair of restrictions; neither is invariant to monotone transformations of the outcome.

Why It Matters

FRD is one of the most credible quasi-experimental designs because identification rests on a single continuity assumption at the threshold. It is widely used to evaluate education programs, health policies, and regulatory cutoffs. Bertanha-Imbens (2014) extend the toolkit by providing a testable path to external validity — from the narrow LATE to the ATE — using only a standard nonparametric continuity check that any FRD analysis can produce at near-zero additional cost.

Open Questions