Fuzzy RD Identification by a Wald Ratio

Why a Fuzzy Cutoff Needs a Ratio

In a sharp regression-discontinuity design, crossing the cutoff determines treatment perfectly. In a fuzzy design, eligibility or assignment changes at the cutoff, but actual treatment receipt does not change from zero to one for everyone. Some people on the eligible side may not receive treatment, while some on the other side may receive it anyway. The cutoff therefore creates an encouragement to receive treatment rather than a perfectly enforced treatment rule.

The outcome may still jump at the cutoff, but that jump is not, by itself, the treatment effect: only some people change treatment status. The key idea is to divide the outcome discontinuity by the discontinuity in treatment probability. This Wald ratio scales the observed outcome change by the size of the treatment change it induced.

Definition: Let $R$ be the running variable, $c$ the cutoff, $Z=1(R\geq c)$ the cutoff-based assignment, $D$ an indicator of treatment receipt, and $Y$ the outcome. Define the right-minus-left outcome discontinuity and treatment-probability discontinuity as $$ \Delta_Y=\lim_{r\downarrow c}E[Y\mid R=r]-\lim_{r\uparrow c}E[Y\mid R=r], \qquad \Delta_D=\lim_{r\downarrow c}E[D\mid R=r]-\lim_{r\uparrow c}E[D\mid R=r]. $$ If $\Delta_D\ne 0$, the fuzzy RD Wald ratio is $\tau_{\mathrm{FRD}}=\Delta_Y/\Delta_D$.

Here, $r\downarrow c$ denotes approaching the cutoff from above and $r\uparrow c$ denotes approaching from below. The signs in both differences use the same right-minus-left convention. A positive treatment jump and a positive outcome jump therefore produce a positive ratio. The outcome and treatment limits are local: they compare the two sides arbitrarily close to the cutoff, not broad groups far from it.

Potential States and the Identification Assumptions

To give the ratio a causal interpretation, distinguish cutoff assignment from treatment receipt. For each individual, let $D(1)$ and $D(0)$ be the treatment receipt that would occur under assignment above and below the cutoff, respectively. Let $Y(1)$ and $Y(0)$ be the outcomes under treatment and no treatment. The observed treatment is $D=D(Z)$. Under consistency and the exclusion restriction, the observed outcome is $Y=Y(D(Z))$: assignment affects the outcome only by changing treatment receipt.

The assumptions are local to the cutoff. They do not require treatment receipt to be random throughout the running-variable range. They do require the cutoff to act as a valid source of variation in treatment near $c$.

Continuity at the cutoff. The relevant conditional expectations of the potential treatment and outcome states have well-defined, matching limits as $R$ approaches $c$ from either side. In particular, the limits of the observed conditional means can be interpreted using the potential states of the population at the cutoff. This rules out an unrelated abrupt change in the composition or potential outcomes of individuals exactly at the threshold.
Exclusion restriction. Crossing the cutoff has no effect on $Y$ except through treatment receipt. If the cutoff also changes another service, information, or benefit that affects $Y$, the outcome discontinuity can include that additional effect.
Monotonicity. $D(1)\geq D(0)$ for individuals at the cutoff. The cutoff does not cause some individuals to lose treatment while causing others to gain it. This rules out “defiers” at the threshold.
Nonzero first stage. $\Delta_D\ne 0$. In practice, the cutoff must change the probability of treatment; without that change, the ratio is undefined and the cutoff supplies no treatment variation for this estimand.
Well-defined assignment and treatment. Assignment follows the cutoff rule, treatment and outcome are consistently defined, and one individual’s assignment does not alter another individual’s outcome in a way that invalidates the potential-state description.

Continuity is a substantive identifying condition, not a claim that the conditional means are flat near the cutoff. They may vary smoothly with $R$; what must not happen is a separate jump in their relevant potential-state behavior at $c$. The earlier material on sharp RD identification uses the same local-comparison idea, but the fuzzy design must additionally account for imperfect treatment take-up.

Why the Wald Ratio Identifies a Local Effect

Write the discontinuity using the potential states at the cutoff. Continuity lets the right-side limit represent the mean under $Z=1$ and the left-side limit represent the mean under $Z=0$, for the same limiting population. Therefore the treatment-probability jump is

\Delta_D=E[D(1)-D(0)\mid R=c].

Under exclusion, the outcome corresponding to assignment $z$ is $Y(D(z))$. The outcome jump is thus

\Delta_Y =E\!\left[Y(D(1))-Y(D(0))\mid R=c\right].

Subtracting the two possible outcomes under assignment gives an individual-level identity. If $D(1)=D(0)$, assignment changes neither treatment status nor outcome under exclusion, so the difference is zero. If $D(1)=1$ and $D(0)=0$, assignment changes treatment from zero to one, so the difference is $Y(1)-Y(0)$. Monotonicity rules out the reverse switch. Consequently,

Y(D(1))-Y(D(0)) =\bigl(D(1)-D(0)\bigr)\bigl(Y(1)-Y(0)\bigr).

Taking conditional expectations and using monotonicity, $D(1)-D(0)$ equals one for individuals whose treatment status changes from untreated to treated at the cutoff, and zero for everyone else. Let $C$ denote that group, the cutoff compliers. Then

\Delta_D=P(C\mid R=c), \qquad \Delta_Y=E[Y(1)-Y(0)\mid C,R=c]P(C\mid R=c).

Dividing the second equality by the first yields the identification result:

Theorem: Under local continuity, exclusion, monotonicity, consistency, and a nonzero treatment-probability discontinuity, the fuzzy RD Wald ratio identifies the average treatment effect for cutoff compliers: $$ \frac{\Delta_Y}{\Delta_D}=E[Y(1)-Y(0)\mid C,R=c]. $$ This is an average effect for individuals near the cutoff whose treatment receipt is changed by crossing it. It is not automatically the average effect for all individuals at the cutoff or for a wider population.

The derivation also explains why the first stage matters. The outcome jump aggregates treatment effects only for people whose treatment changes; the denominator is the share of the local population whose treatment changes. Their ratio is the average effect among that group. With heterogeneous effects, this need not equal an effect for people whose treatment status is unaffected by the cutoff.

Worked Examples

Worked Example: Calculating a Fuzzy RD Wald Ratio

Suppose a hypothetical program has a cutoff at $c=50$. In a local analysis, the estimated limiting mean outcome is 10.0 units just below the cutoff and 12.4 units just above it. The corresponding treatment probabilities are 0.28 below and 0.68 above. Assume the identifying conditions are credible and the treatment jump is positive.

First calculate each discontinuity using right minus left:

\Delta_Y=12.4-10.0=2.4\text{ outcome units}, \qquad \Delta_D=0.68-0.28=0.40.

Then divide the outcome jump by the treatment-probability jump:

\widehat{\tau}_{\mathrm{FRD}} =\frac{2.4}{0.40}=6.0\text{ outcome units}.

Under the assumptions, the estimate says that treatment increases the outcome by an average of 6.0 units for cutoff compliers. It does not say that the observed outcome rose by 6.0 units at the cutoff; the observed jump was 2.4 units. The ratio adjusts that jump for the fact that treatment probability rose by 0.40 rather than by one.

Worked Example: Seeing the Ratio in Treatment-Response Groups

Consider a hypothetical population at the cutoff. Suppose 30% are compliers, 20% would receive treatment under either assignment, and 50% would receive no treatment under either assignment. Assume the cutoff has no direct effect on the outcome, and the average treatment effect among compliers is 4 outcome units. The treatment effect for the other two groups can differ, but their treatment status does not change when assignment changes.

For compliers, $D(1)-D(0)=1$; for the always-treated and never-treated groups it is zero. Thus the treatment-probability discontinuity is

\Delta_D=(0.30)(1)+(0.20)(0)+(0.50)(0)=0.30.

Only the compliers contribute to the outcome discontinuity under exclusion. Their share is 0.30 and their average treatment effect is 4, so

\Delta_Y=(0.30)(4)+(0.20)(0)+(0.50)(0)=1.20\text{ outcome units}.

The Wald ratio is

\frac{\Delta_Y}{\Delta_D}=\frac{1.20}{0.30}=4.0\text{ outcome units}.

The ratio recovers the stipulated complier effect. The outcome discontinuity is smaller because only 30% of individuals change treatment status at the cutoff.

Worked Example: Why Exclusion Is Needed

Suppose a hypothetical cutoff increases treatment probability by 0.25. Treatment raises the outcome by 2 units for every person whose treatment changes, but crossing the cutoff also directly raises the outcome by 0.5 units, independently of treatment. This direct effect violates exclusion.

The outcome jump now contains both components:

\Delta_Y=(0.25)(2)+0.5=1.0\text{ outcome unit}.

The calculated ratio is

\frac{\Delta_Y}{\Delta_D}=\frac{1.0}{0.25}=4.0\text{ outcome units}.

The treatment effect in the example is 2 units, not 4. The ratio includes the direct 0.5-unit cutoff effect, amplified by division by the 0.25 first stage. Arithmetic alone cannot reveal this failure: the causal interpretation depends on the exclusion restriction.

Interpreting and Reporting the Result

A fuzzy RD result should report both discontinuities as well as their ratio. The outcome jump gives the size and direction of the local outcome change; the treatment jump shows how strongly the cutoff changes treatment receipt. Reporting only the ratio hides whether it is scaled by a substantial first stage or a very small one.

Keep the units straight. If $Y$ is measured in days and $D$ is a probability, then $\Delta_Y$ is in days, $\Delta_D$ is dimensionless, and the ratio is in days. If the outcome is binary, the ratio is in probability units, often expressed as percentage points after conversion. The ratio can be negative when the outcome and treatment jumps have opposite signs; its interpretation also depends on the direction of the first stage and on treatment coding.

Identification is not the same as estimation. In data, the limits are approximated from observations near the cutoff, so the calculated ratio is an estimate and has sampling uncertainty. The formulas here describe the causal estimand and its assumptions, not a particular choice of regression, bandwidth, or inferential procedure.

Common Mistakes

Calling the outcome jump the treatment effect. In a fuzzy design, the outcome jump reflects treatment changes for only part of the local population. Divide by the treatment-probability jump to obtain the Wald estimand.
Reversing the sides in one discontinuity. Both $\Delta_Y$ and $\Delta_D$ must use the same right-minus-left convention. Reversing only one changes the ratio’s sign.
Interpreting the ratio as an effect for everyone. The ratio identifies the average effect for cutoff compliers under the assumptions, not a population-wide average effect.
Treating a nonzero first stage as sufficient. A treatment jump is necessary, but exclusion, continuity, and monotonicity are also needed for the stated interpretation.
Assuming smooth trends repair a direct cutoff effect. Smoothness in the running variable does not eliminate a separate discontinuity in another outcome-relevant policy or service at the threshold.

Key takeaway: In a fuzzy RD design, the outcome discontinuity is divided by the treatment-probability discontinuity. Under continuity, exclusion, monotonicity, and a nonzero first stage, this Wald ratio identifies the average treatment effect for individuals near the cutoff whose treatment status changes because of the cutoff.

Check Your Understanding

Use the definitions and assumptions above to answer the following.

Just above a cutoff, the outcome limit is 18 units and treatment probability is 0.75. Just below, the corresponding limits are 15 units and 0.25. Calculate both discontinuities and the Wald ratio, with units.
Why does the outcome jump generally differ from the complier treatment effect in a fuzzy design?
In potential-treatment notation, what does monotonicity rule out, and why does it matter for interpreting the ratio as an average effect?
Suppose crossing the threshold also provides a counseling service that affects the outcome. Which assumption is threatened, and what part of the ratio’s interpretation becomes unreliable?
If the treatment-probability discontinuity is zero, what happens to the Wald ratio, and what does that say about identification from this cutoff?