Deriving The Fuzzy Regression Discontinuity Wald Ratio

From a Treatment Jump to an Outcome Effect

In a sharp regression discontinuity design, crossing the cutoff changes treatment status deterministically. In a fuzzy design, crossing the cutoff changes the probability of receiving treatment, but some units receive treatment on either side. The outcome may still jump at the threshold. The key question is how to interpret that jump when treatment receipt itself changes only partially.

The answer is to compare the outcome discontinuity with the treatment discontinuity. Both are defined by right- and left-hand limits as the running variable approaches the cutoff. Their ratio scales the outcome jump by the amount of treatment take-up induced at the threshold. Under additional assumptions, that ratio is a causal effect for units whose treatment receipt is changed by crossing the cutoff—not an effect for every unit near the threshold.

Setup and Conditions for Interpretation

Let $R_i$ be unit $i$'s running variable, $c$ the cutoff, and $Z_i=1(R_i\ge c)$ the indicator for being on the treatment-assignment side. Let $D_i\in\{0,1\}$ denote treatment receipt and $Y_i$ the outcome. In a fuzzy design, $Z_i$ is an instrument for $D_i$: assignment at the threshold affects the chance of treatment, but does not determine receipt for everyone.

Definition: The fuzzy regression discontinuity Wald ratio is $$ \tau_{\mathrm{FRD}}= \frac{\displaystyle \lim_{r\downarrow c}E[Y_i\mid R_i=r]-\lim_{r\uparrow c}E[Y_i\mid R_i=r]} {\displaystyle \lim_{r\downarrow c}E[D_i\mid R_i=r]-\lim_{r\uparrow c}E[D_i\mid R_i=r]}. $$ Here $r\downarrow c$ denotes approaching $c$ from above and $r\uparrow c$ denotes approaching from below. The denominator is the discontinuity in treatment receipt, also called the first-stage discontinuity.

The ratio has a causal interpretation only under design-specific conditions. State them before using the formula as an effect estimate.

Local support and a well-defined cutoff: observations occur on both sides arbitrarily close to $c$, and the limits used above exist. A positive density of the running variable near the cutoff is a common sufficient support condition.
Continuity at the cutoff: absent the assignment change, the relevant potential treatment and outcome quantities would vary continuously through $c$. More specifically, a sufficient condition is that the conditional joint distribution of potential treatment receipts and potential outcomes varies continuously as $R_i$ approaches $c$. This rules out a separate jump in the composition of units or their potential responses exactly at the threshold.
Exclusion: crossing the cutoff affects the outcome only through treatment receipt. In potential-outcome notation, write $D_i(z)$ for receipt under assignment status $z$, and $Y_i(d,z)$ for the outcome under receipt $d$ and assignment $z$. Exclusion requires $Y_i(d,1)=Y_i(d,0)=Y_i(d)$.
Monotonicity: no unit is made less likely to receive treatment by being assigned to the treatment side: $D_i(1)\ge D_i(0)$. This rules out defiers at the cutoff.
Nonzero first stage: the probability of treatment receipt must increase at the cutoff, so the denominator is positive rather than zero. A very small discontinuity also makes the ratio unstable and its causal interpretation empirically uninformative.

Consistency links observed and potential values: $D_i=D_i(Z_i)$ and $Y_i=Y_i(D_i)$. The setup also presumes that one unit's assignment and treatment do not alter another unit's outcome. Continuity conditions are local: they support comparing the limits at $c$, not extrapolating the same effect to units far from the cutoff.

Why the Ratio Is a Local Complier Effect

Define the principal strata by the pair $(D_i(0),D_i(1))$. Always-takers have $(1,1)$, never-takers have $(0,0)$, compliers have $(0,1)$, and defiers have $(1,0)$. Under monotonicity, the defier stratum is absent. A complier is therefore a unit whose treatment receipt changes from 0 to 1 when assignment changes from below to above the threshold.

Write each observed-side mean as a potential mean.
By consistency, above the cutoff the observed outcome is $Y_i(D_i(1))$, while below it the outcome is $Y_i(D_i(0))$. Continuity lets the right- and left-hand limits be compared at the cutoff population.

Express the outcome jump.
The outcome discontinuity is $E[Y_i(D_i(1))-Y_i(D_i(0))\mid R_i=c]$, where conditioning at $c$ represents the limiting population defined by continuity.

Use binary treatment and exclusion.
For each unit, $Y_i(D_i(1))-Y_i(D_i(0))=(D_i(1)-D_i(0))(Y_i(1)-Y_i(0))$. If receipt does not change, the difference is zero; if it changes, the difference is that unit's treatment effect.

Use monotonicity and divide by the treatment jump.
Under monotonicity, $D_i(1)-D_i(0)$ is 1 for compliers and 0 otherwise. The outcome jump is thus the complier share times the average treatment effect among compliers. The treatment jump equals that complier share, so their ratio is the complier average treatment effect.

In equations, let $C_i$ indicate that unit $i$ is a complier and let $\Delta_Y$ and $\Delta_D$ denote the outcome and treatment discontinuities. Then

\begin{aligned} \Delta_Y &=E[(D_i(1)-D_i(0))(Y_i(1)-Y_i(0))\mid R_i=c]\\ &=P(C_i=1\mid R_i=c)\, E[Y_i(1)-Y_i(0)\mid C_i=1,R_i=c],\\ \Delta_D &=E[D_i(1)-D_i(0)\mid R_i=c] =P(C_i=1\mid R_i=c). \end{aligned}

Consequently, if $\Delta_D>0$,

\tau_{\mathrm{FRD}} =\frac{\Delta_Y}{\Delta_D} =E[Y_i(1)-Y_i(0)\mid C_i=1,R_i=c].

Theorem: Under continuity at the cutoff, consistency, no interference, exclusion, monotonicity, and a positive treatment discontinuity, the fuzzy RD Wald ratio identifies the average treatment effect for compliers at the cutoff. It is a local average treatment effect (LATE), not generally the average effect for all units near the cutoff.

Worked Example: Calculating the Ratio From Cutoff Limits

Suppose a hypothetical job-training program uses an eligibility score cutoff. The limiting mean earnings are \$42,000 just above the cutoff and \$38,000 just below it. The limiting treatment-receipt proportions are 0.60 and 0.20, respectively. Assume the design conditions stated above hold, with no defiers and an assignment-induced increase in training.

The outcome discontinuity is $$ \Delta_Y=\$42{,}000-\$38{,}000=\$4{,}000. $$ The treatment discontinuity is $$ \Delta_D=0.60-0.20=0.40. $$ Therefore, $$ \tau_{\mathrm{FRD}}=\frac{\$4{,}000}{0.40}=\$10{,}000. $$

Under the assumptions, the estimated causal contrast represented by these population limits is an average increase of \$10,000 in earnings for cutoff compliers who receive training because they are on the eligible side. It is not the average effect for all applicants, nor does the ratio say that every complier gains exactly \$10,000. The limits and assumptions are hypothetical; in an applied analysis, the causal interpretation depends on whether the design conditions are credible.

Worked Example: Seeing the Complier Share in the Derivation

Consider a hypothetical cutoff population with 25% compliers, 10% always-takers, and 65% never-takers, and no defiers. Suppose crossing the cutoff changes treatment receipt only for compliers, whose average treatment effect on a test score is 8 points. Potential outcomes are continuous through the cutoff within these groups, and assignment has no direct effect on scores.

The treatment discontinuity is the complier proportion: $$ \Delta_D=0.25. $$ Always-takers receive treatment on both sides, and never-takers receive it on neither side, so neither group contributes an assignment-induced change in treatment. The outcome discontinuity is the complier share multiplied by their average effect: $$ \Delta_Y=0.25(8\text{ points})=2\text{ points}. $$ Thus, $$ \tau_{\mathrm{FRD}}=\frac{2\text{ points}}{0.25}=8\text{ points}. $$

The numerator is smaller than the complier effect because only one quarter of the cutoff population changes treatment status. Dividing by the treatment jump removes that scaling. This also shows why the denominator is essential: the outcome jump alone is not the treatment effect when treatment receipt changes only for some units.

Reading the Ratio and Comparing Designs

The sign of the Wald ratio follows the signs of its two discontinuities. With a positive first stage, a positive outcome jump gives a positive complier effect and a negative outcome jump gives a negative one. Units in the numerator and denominator must be kept clear: if $Y_i$ is measured in days and $D_i$ is an indicator, the ratio is in days per treatment, conventionally described as the effect of treatment in days.

A sharp RD is the special case in which treatment receipt changes from 0 to 1 at the cutoff for every unit. Its treatment discontinuity is 1, so the Wald ratio equals the outcome discontinuity. In a fuzzy RD, dividing by the treatment discontinuity adjusts for incomplete compliance. Neither design, by itself, identifies an average treatment effect for the whole population. Other instrumental-variable designs also use outcome and treatment changes to form a ratio, but the RD interpretation depends specifically on continuity of the relevant quantities at the threshold.

Worked Example: A Negative Effect in a Hypothetical Design

Suppose a hypothetical clinical service becomes more likely at a referral-score cutoff. Just above the cutoff, the limiting treatment probability is 0.70; just below, it is 0.30. The limiting mean hospital stay is 5.4 days above and 7.0 days below. Assume assignment has no direct effect on hospital stay and the other continuity and monotonicity conditions hold.

The treatment jump is $$ \Delta_D=0.70-0.30=0.40, $$ and the outcome jump is $$ \Delta_Y=5.4-7.0=-1.6\text{ days}. $$ The ratio is $$ \tau_{\mathrm{FRD}}=\frac{-1.6\text{ days}}{0.40}=-4.0\text{ days}. $$

For compliers at the cutoff, the interpretation is an average reduction of 4.0 days in hospital stay due to receiving the service, under the assumptions. It does not mean the service reduces every patient's stay by four days. A negative estimate is compatible with a positive treatment first stage: treatment receipt increases while the outcome decreases.

Diagnostics, Failure Modes, and Limits

The ratio is only as credible as its design conditions. The earlier material on testing for sorting at the threshold, predetermined covariate continuity checks, placebo cutoffs and outcomes, and bandwidth sensitivity and support diagnostics provides relevant ways to probe the local design. Such checks can reveal warning signs, but passing them cannot prove exclusion, monotonicity, or continuity of every unobserved potential outcome.

A negligible treatment jump: if $\Delta_D$ is near zero, the ratio divides by a small number. Small changes in either estimated discontinuity can then produce large changes in the ratio. A zero jump makes the ratio undefined and provides no cutoff-based treatment contrast.
Sorting or changing composition: if units can precisely manipulate which side of the threshold they occupy, the populations just above and below may differ for reasons other than assignment. A discontinuity in predetermined characteristics is a warning that the continuity comparison may fail.
Direct assignment effects: if crossing the threshold changes access to information, services, or other benefits besides $D_i$, exclusion fails. The outcome jump then contains more than the effect of treatment receipt.
Defiers: if some units become less likely to receive treatment on the treatment side, monotonicity fails. The ratio can become a signed weighted average rather than the complier average effect derived above.
Insufficient support or a discrete running variable: if there are no observations arbitrarily close on one side, or the running variable has too few distinct values, the stated limits may not be supported by the data. A smooth-looking fitted curve cannot create missing local support.

A frequent mistake is to call the ratio “the treatment effect at the cutoff for everyone.” The derivation shows why that is too broad: the denominator is the share of compliers, and only their treatment effects contribute to the numerator under the assumptions. Another mistake is to treat a nonzero outcome jump as sufficient evidence of a treatment effect. Without a nonzero treatment jump and exclusion, that jump does not have the stated treatment-effect interpretation.

The Wald ratio is an identification result, not a guarantee that a particular estimate is precise or stable. Its local complier effect may differ from effects for always-takers, never-takers, or units far from the cutoff. It also does not identify the complier group for each individual: principal-stratum membership is defined by potential treatment receipts, of which only one is observed for any unit. Estimation and uncertainty quantification require additional statistical work beyond this population-limit derivation.

Check Your Understanding

Work through each question, then compare your reasoning with the brief solutions.

At a cutoff, the outcome limit increases by 3 units and treatment receipt increases by 0.25. What is the Wald ratio, and who does it describe under the required assumptions?
Why does monotonicity make the treatment discontinuity equal to the complier share?
Suppose the outcome discontinuity includes a direct assignment effect of 2 units, in addition to the outcome change caused by treatment. Does the usual ratio still identify the complier average treatment effect? Explain.
In a sharp RD, treatment receipt jumps from 0 below the cutoff to 1 above it. What does the fuzzy RD ratio reduce to?

Solutions. (1) The ratio is $3/0.25=12$ outcome units per treatment. Under the assumptions, it is the average effect for compliers at the cutoff, not for everyone near it. (2) With binary treatment and no defiers, $D_i(1)-D_i(0)$ equals 1 for compliers and 0 for everyone else. Its expectation is therefore the complier proportion. (3) No. The outcome jump contains the direct assignment effect, violating exclusion, so dividing by the treatment jump does not isolate the treatment effect. (4) The denominator is $1-0=1$, so the ratio equals the outcome discontinuity.