Local Complier Effects in Fuzzy RD

Why the Fuzzy RD Ratio Is a Local Effect

In a fuzzy regression discontinuity design, crossing a cutoff changes the probability of receiving treatment, but it does not determine treatment perfectly. The discontinuity in the outcome therefore reflects both the effect of treatment and the fact that only some units change treatment status at the threshold. Dividing the outcome jump by the treatment jump isolates an effect for those units whose treatment is changed by crossing the cutoff: the compliers.

This interpretation is narrower than “the effect of treatment.” It does not generally describe everyone near the cutoff, everyone who takes treatment, or the full population. It describes a particular group at a particular location in the running variable’s distribution.

Definition: Let $R_i$ be unit $i$’s running variable, $c$ the cutoff, and $Z_i=1\{R_i\geq c\}$ the indicator for being assigned to the treatment-favoring side. Let $D_i(z)$ be the treatment status unit $i$ would have under assignment status $z$, and let $Y_i(d)$ be the outcome unit $i$ would have under treatment status $d$. A complier is a unit with $D_i(1)=1$ and $D_i(0)=0$: the unit receives treatment when assigned to the treatment-favoring side and does not receive it otherwise. The complier treatment effect is $Y_i(1)-Y_i(0)$.

The other principal types are always-takers, with $D_i(1)=D_i(0)=1$, and never-takers, with $D_i(1)=D_i(0)=0$. If crossing the threshold can instead make some units less likely to receive treatment, those units are called defiers. The standard local-complier interpretation assumes there are no defiers at the cutoff.

The Local Wald Ratio

Write the right-hand limit at the cutoff as $r\downarrow c$ and the left-hand limit as $r\uparrow c$. The fuzzy RD Wald estimand is the jump in the conditional outcome mean divided by the jump in the conditional treatment probability:

\tau_{\mathrm{FRD}} = \frac{ \lim_{r\downarrow c} E[Y_i\mid R_i=r] - \lim_{r\uparrow c} E[Y_i\mid R_i=r] }{ \lim_{r\downarrow c} E[D_i\mid R_i=r] - \lim_{r\uparrow c} E[D_i\mid R_i=r] }.

The denominator is the first-stage jump: how much the probability of receiving treatment changes at the cutoff. The numerator is the reduced-form outcome jump. The ratio is meaningful only when the first-stage jump is nonzero. With treatment more likely on the right, that jump is positive under the usual orientation.

For the ratio to identify a local complier effect, the threshold indicator must affect the outcome through treatment rather than through a separate direct channel. In potential-outcome terms, this is the exclusion restriction: the outcome under a given treatment status is not additionally changed by $Z_i$. The treatment response must be monotone at the cutoff, $D_i(1)\geq D_i(0)$, ruling out defiers. In addition, the relevant potential-outcome means and principal-stratum composition must vary continuously through the cutoff in the absence of the treatment-assignment change. These conditions allow the observed jumps to be attributed to assignment and treatment rather than to a separate discontinuity in who is observed or in their untreated outcomes. The precise continuity conditions are design assumptions, not consequences of computing the ratio.

Theorem: Under exclusion, monotonicity, a nonzero first-stage jump, and continuity at $c$ of the relevant potential-outcome means and principal-stratum shares, the fuzzy RD Wald estimand identifies the local average treatment effect for compliers at the cutoff: $$ \tau_{\mathrm{FRD}} = E\!\left[Y_i(1)-Y_i(0)\mid D_i(1)>D_i(0),\,R_i=c\right]. $$ When $R_i$ is continuous, the expression at $R_i=c$ denotes the limiting complier effect at the cutoff; it does not require a positive probability of observing a unit with exactly $R_i=c$.

Why the Ratio Selects Compliers

The interpretation follows from what each discontinuity counts. On the right-hand side, assignment status changes from $Z_i=0$ to $Z_i=1$. Always-takers receive treatment under either assignment, and never-takers receive it under neither, so their treatment status does not change. Under monotonicity, compliers are the only units whose treatment status changes.

Let $\pi_C(c)$ be the limiting share of compliers at the cutoff, and let $\tau_C(c)$ be their average treatment effect there. Continuity ensures that smooth changes in the distribution of types and their potential outcomes do not themselves create a jump. The first-stage jump is therefore the share of compliers, while the outcome jump is that share multiplied by their average effect:

\Delta_D(c)=\pi_C(c), \qquad \Delta_Y(c)=\pi_C(c)\tau_C(c).

Dividing the second expression by the first cancels the complier share. The ratio is an average effect among compliers, not a treatment effect for every unit contributing to the observations on either side.

\frac{\Delta_Y(c)}{\Delta_D(c)} = \frac{\pi_C(c)\tau_C(c)}{\pi_C(c)} = \tau_C(c), \qquad \pi_C(c)>0.

This calculation also explains why the first-stage jump matters for interpretation. If very few units change treatment status, the same outcome jump would imply a larger ratio, and estimates based on data would typically be less stable. The ratio remains a local estimand; dividing by the first stage does not broaden the group to which it applies.

Worked Example: Interpreting the Estimated Ratio

Suppose a hypothetical program uses an eligibility score with cutoff $c$. The outcome is a standardized test score measured in points. The estimated right- and left-hand limits of the outcome mean are 6.9 and 2.1 points, and the corresponding treatment probabilities are 0.55 and 0.25.

The outcome jump is $6.9-2.1=4.8$ points. The treatment-probability jump is $0.55-0.25=0.30$, or 30 percentage points. The Wald ratio is

\widehat{\tau}_{\mathrm{FRD}} = \frac{6.9-2.1}{0.55-0.25} = \frac{4.8}{0.30} = 16 \text{ points}.

If the exclusion, monotonicity, and continuity conditions are credible, the estimate means that treatment raises the test score by an average of 16 points for compliers at the cutoff. It does not mean that treatment raises every participant’s score by 16 points, nor that the average effect for all eligible people is 16 points. The outcome jump is in test-score points, so the ratio is also in test-score points; the probability scale in the denominator is dimensionless.

Worked Example: Seeing the Complier Share Cancel

Consider a hypothetical setting in which the limiting population at the cutoff is 40% compliers, 20% always-takers, and 40% never-takers. Assume no defiers, no direct effect of assignment, and smooth potential-outcome means across the threshold. Suppose the average treatment effect for the cutoff compliers is 10 units of the outcome.

Moving from the left to the right side changes the treatment status of the 40% who are compliers. The always-takers and never-takers do not change status. The first-stage jump is therefore $0.40$. The outcome jump is the complier share times the complier effect:

\Delta_D(c)=0.40, \qquad \Delta_Y(c)=0.40(10)=4.

The ratio is $4/0.40=10$ outcome units. This example illustrates the logic of the identification argument: the outcome discontinuity is diluted because only a fraction of units change treatment, and dividing by that fraction recovers the average effect for the group that does change treatment.

Why the Effect Is Local, Not Population-Wide

“Local” has two related meanings here. First, the effect pertains to units at or arbitrarily close to the cutoff, represented by limits from either side. Second, within that local group, the effect is averaged over compliers rather than over all units. The ratio does not reveal the effect for always-takers or never-takers, whose treatment status does not respond to crossing the cutoff.

There is no general reason for cutoff compliers to have the same treatment effects as people far from the threshold. They may differ in eligibility, severity, resources, or other characteristics. Likewise, even near the cutoff, the average effect for compliers need not equal the average effect for all units. Moving from a local complier effect to a population average requires additional information and assumptions; it is not achieved by relabeling the Wald ratio.

Worked Example: A Local Effect Can Differ From the Population Average

Suppose a hypothetical population contains 100 people. Twenty are compliers close to an RD cutoff, and their average treatment effect is 12 outcome units. The other 80 people are outside this local complier group, and their average treatment effect is 2 units. For simplicity, assume those are the only two groups in this population calculation.

The population average treatment effect is the group-size-weighted average:

\frac{20(12)+80(2)}{100} = \frac{240+160}{100} = 4 \text{ outcome units}.

The fuzzy RD ratio at the cutoff identifies 12 outcome units for the local compliers, not the population average of 4. The discrepancy is not a contradiction: the two quantities average effects over different groups. The RD discontinuity is generated by assignment changes at the cutoff and has no direct information about effects for people whose treatment status is not changed there.

Assumptions and Interpretation Checks

The numerical ratio alone does not establish a causal effect. Its local-complier interpretation depends on substantive design conditions as well as a nonzero treatment jump. In particular, check the following before interpreting it:

Relevance: the probability of treatment actually changes at the cutoff. A zero first-stage jump leaves the ratio undefined.
Exclusion: crossing the threshold does not change the outcome through a route other than treatment.
Monotonicity: assignment to the treatment-favoring side does not cause any unit to lose treatment, so there are no defiers at the cutoff.
Continuity and comparability: absent the assignment-induced treatment change, the relevant potential outcomes and type composition do not jump at the threshold.
Consistent treatment and outcome definitions: the treatment being shifted by the cutoff and the outcome being compared are well-defined on both sides.

These conditions are not interchangeable. For example, a positive first stage does not rule out defiers, and a smooth observed outcome away from the cutoff does not by itself prove the exclusion restriction. The assumptions justify interpreting the ratio; the arithmetic does not test all of them.

Common Mistake: Calling It the Effect for Everyone

A common but incorrect conclusion is: “The ratio is 16, so the program increases the outcome by 16 units for every person who receives it.” This changes both the population and the type of effect. The ratio averages causal effects among units whose treatment status responds to threshold assignment, and it is local to the cutoff. Treatment recipients can include always-takers as well as compliers, while people far from the cutoff may have different effects.

A better report names the target group and conditions: “Under the stated exclusion, monotonicity, and continuity assumptions, the fuzzy RD estimate is an average treatment effect of 16 outcome units for compliers at the cutoff.” If those assumptions are uncertain, describe the ratio as an estimate whose causal interpretation depends on them rather than as an unconditional fact.

Key takeaway: The fuzzy RD Wald ratio divides the outcome discontinuity by the treatment-probability discontinuity. Under the identifying assumptions, this removes the dilution from units whose treatment does not change and yields an average treatment effect for cutoff compliers—not a universal or population-wide effect.

Check Your Understanding

Answer each question by identifying both the quantity and the group to which it refers.

At a cutoff, the outcome jumps by 3 points and treatment probability jumps by 0.25. What is the Wald ratio, and what is its interpretation under the identifying assumptions?
Why do always-takers and never-takers not contribute to the first-stage jump?
What role does monotonicity play in the local-complier interpretation?
Why does the existence of a nonzero first-stage jump not make the ratio a population-average treatment effect?
If crossing the cutoff directly changes the outcome even when treatment is held fixed, which identifying condition fails, and why is the ratio’s causal interpretation then threatened?