Fuzzy Regression Discontinuity as a Wald Estimator

Why a Fuzzy Cutoff Requires a Ratio

In a sharp regression discontinuity design, crossing the cutoff determines treatment. In a fuzzy design, crossing the cutoff changes the probability of receiving treatment, but some units on either side may not follow the assignment rule. The discontinuity in outcomes therefore combines the treatment effect with the size of the change in treatment receipt. Dividing the outcome jump by the treatment-probability jump removes that scaling and, under additional conditions, identifies an effect for units whose treatment is changed by crossing the cutoff.

The ratio is often called the fuzzy regression discontinuity estimand or the local Wald estimand. It is not automatically the average treatment effect for everyone near the cutoff. Its interpretation depends on which units respond to the cutoff and on assumptions about potential outcomes and treatment assignment.

Definition: Let $R$ be a running variable with cutoff $c$, and let $Z=1$ when $R\geq c$ and $Z=0$ when $R<c$. Let $D$ indicate treatment receipt and $Y$ be the outcome. Define the outcome discontinuity and first-stage discontinuity as $$ \Delta_Y=\lim_{r\to c^+}E[Y\mid R=r]-\lim_{r\to c^-}E[Y\mid R=r], \qquad \Delta_D=\lim_{r\to c^+}E[D\mid R=r]-\lim_{r\to c^-}E[D\mid R=r]. $$ When both limits exist and $\Delta_D\ne 0$, the fuzzy regression discontinuity Wald estimand is $\tau_{\mathrm{FRD}}=\Delta_Y/\Delta_D$.

Here $Z$ is the cutoff-based assignment indicator, whereas $D$ records treatment actually received. The discontinuities are differences between the conditional-mean limits from just above and just below the cutoff; they are not generally differences between raw averages over arbitrarily chosen groups. In applications, estimated local regressions or other local methods are used to estimate these limits. The ratio's causal interpretation is a separate identification question.

Potential Treatments, Types, and Required Conditions

Write $D(z)$ for the treatment status a unit would have under cutoff assignment $z$, and $Y(d)$ for the outcome it would have under treatment status $d$. The observed values satisfy $D=D(Z)$. Under the exclusion restriction, the observed outcome is $Y=Y(D(Z))$: assignment affects the outcome only by changing treatment receipt.

The pair $(D(0),D(1))$ defines each unit's treatment-response type. An always-taker has $(1,1)$; a never-taker has $(0,0)$; a complier has $(0,1)$; and a defier has $(1,0)$. These are potential-response categories, not groups identified by simply comparing observed treatment statuses on the two sides.

Identification conditions:

Continuity at the cutoff: The shares of the treatment-response types have common limits from both sides of $c$. In addition, for each type with positive limiting share, the type-specific conditional means $E[Y(d)\mid G=g,R=r]$, for $d=0,1$, have common limits as $r$ approaches $c$ from either side. Here $G$ denotes treatment-response type. This ensures the type composition and the relevant potential-outcome means can be compared using one common limiting population.
Exclusion: The cutoff assignment $Z$ has no effect on $Y$ except through actual treatment $D$. In potential-outcome notation, the outcome under assignment $z$ and treatment $d$ is $Y(d)$, not a separate $Y(z,d)$.
Monotonicity: $D(1)\geq D(0)$ for units in the local population. Thus there are no defiers: crossing from below to above the cutoff does not cause any unit to move from treatment to no treatment.
First-stage relevance: $\Delta_D\ne 0$. There must be a nonzero discontinuity in treatment receipt for the ratio to be defined as an estimand.

The continuity condition is deliberately about type-specific outcome means as well as type shares. Continuity of the marginal means $E[Y(0)\mid R=r]$ and $E[Y(1)\mid R=r]$ alone is not enough: those marginal means can remain smooth even if the outcomes of compliers and other types vary in offsetting ways. The derivation needs the type-specific means to have common limits, so that outcome contributions from always-takers and never-takers cancel across the cutoff.

Why the Ratio Is a Local Complier Effect

The identification result follows by decomposing each side's limiting outcome mean by treatment-response type. Let $p_g$ be the common limiting share of type $g$ at the cutoff, and let $\mu_{gd}$ be the common limiting mean of $Y(d)$ within type $g$. These quantities refer to the limiting population at $R=c$. Under monotonicity, the only types are always-takers, never-takers, and compliers.

\lim_{r\to c^-}E[Y\mid R=r] = p_A\mu_{A1}+p_N\mu_{N0}+p_C\mu_{C0},

\lim_{r\to c^+}E[Y\mid R=r] = p_A\mu_{A1}+p_N\mu_{N0}+p_C\mu_{C1}.

Below the cutoff, compliers have $D(0)=0$; above it, they have $D(1)=1$. Always-takers receive treatment on either side, and never-takers receive no treatment on either side. Exclusion therefore determines which potential outcome each type contributes to the observed outcome. Subtracting the two limits cancels the always-taker and never-taker contributions:

\Delta_Y=p_C(\mu_{C1}-\mu_{C0}).

The same type decomposition applied to treatment receipt gives a first stage of zero contribution from always-takers and never-takers, and a change from zero to one for each complier:

\lim_{r\to c^-}E[D\mid R=r]=p_A+p_C\cdot 0+p_N\cdot 0=p_A, \qquad \lim_{r\to c^+}E[D\mid R=r]=p_A+p_C\cdot 1+p_N\cdot 0=p_A+p_C.

Consequently, $\Delta_D=p_C$. When $p_C>0$, the ratio is

\tau_{\mathrm{FRD}} =\frac{\Delta_Y}{\Delta_D} =\frac{p_C(\mu_{C1}-\mu_{C0})}{p_C} =\mu_{C1}-\mu_{C0} =E[Y(1)-Y(0)\mid G=C,R=c].

This is the mean treatment effect for local compliers: units at the cutoff, in the limiting population, whose treatment status changes from untreated to treated when assignment changes from $Z=0$ to $Z=1$. The derivation requires the same limiting type shares and type-specific outcome means on both sides. Without that common limiting composition and those common means, the terms need not cancel in this way.

Worked Example: Calculating a Wald Discontinuity Ratio

Suppose a hypothetical program uses an eligibility-score cutoff. Local estimates of the conditional limits give an average outcome of 42 units just above the cutoff and 37.2 units just below it. The estimated treatment probabilities are 0.68 above and 0.38 below. Calculate the fuzzy regression discontinuity ratio and describe its interpretation under the identification conditions.

Outcome discontinuity. The estimated jump is

\widehat{\Delta}_Y=42-37.2=4.8\text{ outcome units}.

Treatment-probability discontinuity. The estimated first stage is

\widehat{\Delta}_D=0.68-0.38=0.30.

Wald ratio. Dividing the outcome jump by the treatment-probability jump gives

\widehat{\tau}_{\mathrm{FRD}} =\frac{4.8}{0.30}=16\text{ outcome units per treated unit}.

If continuity, exclusion, and monotonicity hold, and the first stage is nonzero, this ratio estimates the mean treatment effect for local compliers at the cutoff. It is not, without further assumptions, the average effect for all eligible people or even for every person near the cutoff. The numbers here are hypothetical local-limit estimates, not raw group averages.

Worked Example: Verifying the Type Decomposition

Consider a hypothetical limiting population with 20% always-takers, 55% never-takers, and 25% compliers. Suppose the limiting outcome means are as follows: for always-takers, $E[Y(1)\mid G=A,R=c]=30$; for never-takers, $E[Y(0)\mid G=N,R=c]=5$; and for compliers, $E[Y(0)\mid G=C,R=c]=10$ and $E[Y(1)\mid G=C,R=c]=18$. Assume type shares and these type-specific means are continuous at the cutoff, exclusion holds, and there are no defiers.

Below the cutoff, always-takers receive treatment, while never-takers and compliers do not. Thus the limiting outcome mean is

$$ 0.20(30)+0.55(5)+0.25(10)=6+2.75+2.50=11.25. $$

Above the cutoff, always-takers remain treated, never-takers remain untreated, and compliers become treated. The limiting outcome mean is

$$ 0.20(30)+0.55(5)+0.25(18)=6+2.75+4.50=13.25. $$

The outcome jump is $13.25-11.25=2.00$ units. The treatment probability is 0.20 below the cutoff and $0.20+0.25=0.45$ above, so the first-stage jump is $0.45-0.20=0.25$. Therefore,

\frac{\Delta_Y}{\Delta_D}=\frac{2.00}{0.25}=8.00\text{ outcome units}.

This equals the complier effect $18-10=8$ units. The example shows exactly why the other types' outcome contributions disappear from the discontinuity: their treatment status does not change, and their type-specific potential-outcome means have the same limits on both sides.

What Can Break the Interpretation?

Each condition rules out a particular alternative explanation for the ratio. Continuity rules out a change at the cutoff in the relevant limiting composition or potential-outcome means. Exclusion rules out a direct effect of crossing the threshold that is not mediated by treatment receipt. Monotonicity rules out units who respond in the opposite direction from compliers. A nonzero first stage ensures that the cutoff actually changes treatment probability.

Worked Example: A Direct Assignment Effect Violates Exclusion

Suppose, hypothetically, crossing a cutoff changes treatment probability by 0.20. The true treatment effect is 5 outcome units for each unit, but crossing the threshold also directly raises the outcome by 1 unit, even among units whose treatment does not change. Assume for illustration that the other identification conditions hold.

The treatment-mediated outcome jump is $0.20(5)=1.00$ outcome unit. Adding the direct assignment jump gives $\Delta_Y=1.00+1.00=2.00$ units. The Wald ratio is therefore

\frac{\Delta_Y}{\Delta_D}=\frac{2.00}{0.20}=10.0\text{ outcome units}.

This ratio does not recover the treatment effect of 5 units. It also includes the direct effect of assignment, violating exclusion. A discontinuity in the outcome and a nonzero first stage do not by themselves establish the local-complier interpretation.

A further continuity pitfall is to check only the marginal potential-outcome means and the shares of treatment-response types. Those conditions alone do not guarantee that the complier-specific treatment effect has the same limit from both sides. For example, suppose half the limiting units are compliers and half are never-takers, with $Y(0)=0$ for everyone. Below the cutoff, let $Y(1)=0$ for compliers and 2 for never-takers; above it, let $Y(1)=2$ for compliers and 0 for never-takers. The type shares and marginal means of $Y(0)$ and $Y(1)$ are unchanged, yet the complier effect changes from 0 below to 2 above. The outcome jump is 1 and the treatment jump is 0.5, yielding a ratio of 2 that is not the complier effect at a common limiting population. This is why continuity of type-specific potential-outcome means is part of the required condition.

Interpretation and Common Mistakes

The local complier effect answers a focused policy question: for units at the cutoff whose treatment receipt is affected by crossing it, what is the mean difference between their treated and untreated potential outcomes? Its local nature is often an advantage because it targets units whose treatment status is actually shifted by the rule. But the effect need not generalize to other populations, other cutoff values, or people whose treatment status is unaffected by eligibility.

Calling the outcome jump the treatment effect. In a fuzzy design, the outcome jump is scaled by the size of the treatment-probability jump. Use $\Delta_Y/\Delta_D$, not $\Delta_Y$ alone.
Calling the ratio an average treatment effect for everyone. Under the stated assumptions, the ratio is a local complier effect at the cutoff, not automatically a population-wide effect.
Checking only marginal continuity. Smooth marginal potential-outcome means do not ensure smooth type-specific means. The relevant type shares and type-specific potential-outcome means need common limits.
Ignoring the exclusion restriction. If crossing the cutoff directly affects the outcome, the outcome jump includes more than the effect transmitted through treatment.
Using a ratio with no first stage. If $\Delta_D=0$, the Wald ratio is undefined; if it is close to zero, the ratio can be highly sensitive to estimation error.

Key takeaway: A fuzzy regression discontinuity Wald ratio identifies the mean effect for local compliers only when the cutoff changes treatment receipt, assignment affects outcomes only through treatment, defiers are absent, and the relevant treatment-type shares and type-specific potential-outcome means have common limits at the cutoff.

Check Your Understanding

Use the definitions and conditions above to answer the following questions.

A hypothetical design has an outcome discontinuity of 3.6 units and a treatment-probability discontinuity of 0.24. What is the Wald ratio, and which population does it represent under the identification conditions?
Why does monotonicity, together with the common limiting type shares, make the first-stage discontinuity equal to the complier share?
Why is continuity of type-specific potential-outcome means needed in addition to continuity of marginal means?
Give an example of a direct cutoff effect on the outcome that would violate exclusion, even if the treatment-probability jump were nonzero.
Why does a valid local-complier interpretation not establish that the same treatment effect applies to people far from the cutoff?