Fuzzy Regression Discontinuity Identification

Why Identification Is Fuzzy

In a sharp regression discontinuity design, crossing the cutoff determines treatment. In a fuzzy regression discontinuity design, crossing it changes the probability of receiving treatment, but does not determine treatment for every unit. For example, eligibility at a score threshold may make participation more likely while some eligible people decline and some ineligible people find another route into the program. The discontinuity in eligibility can then serve as a local source of variation in treatment receipt.

The central identification question is not simply whether outcomes differ on the two sides of the cutoff. It is whether that outcome discontinuity can be attributed to the treatment-receipt change induced by crossing the threshold. Under specific assumptions, the ratio of the two discontinuities identifies a local average treatment effect for people whose treatment status is changed by the threshold. It is local to the cutoff and to that group; it is not automatically the average effect for everyone eligible or everyone treated.

Definition: Let \(R\) be a running variable, \(c\) its cutoff, and \(Z=1\{R\ge c\}\) the threshold-based assignment indicator. Let \(D\in\{0,1\}\) indicate treatment received and \(Y\) be the outcome. A design is fuzzy when the probability of \(D=1\) has a discontinuity at \(R=c\), but \(D\) does not equal \(Z\) for every unit.

Potential Outcomes and the Local Ratio

For each unit, define \(D(1)\) and \(D(0)\) as the treatment receipt that would occur under assignment \(Z=1\) and \(Z=0\), respectively. Define \(Y(1)\) and \(Y(0)\) as the outcomes under treatment and no treatment. The observed values satisfy \(D=D(Z)\) and, under the exclusion condition below, \(Y=Y(D(Z))\). These definitions distinguish the threshold assignment from the treatment actually received.

The discontinuity in a conditional mean at the cutoff is its right-hand limit minus its left-hand limit. Define the outcome and treatment-probability discontinuities as follows:

\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].

The ratio \(\Delta_Y/\Delta_D\), when \(\Delta_D\ne 0\), is called the fuzzy regression discontinuity ratio or local Wald ratio. The identification result requires more than a nonzero denominator. In particular, crossing the threshold must not create another discontinuous route to the outcome, and the units just to either side must be comparable in their potential outcomes and treatment types.

Identification result: Suppose the potential outcomes and the proportions and outcome means of the treatment-response types vary continuously with \(R\) at \(c\); assignment affects \(Y\) only through \(D\); and there are no defiers, so \(D(1)\ge D(0)\) for units at the cutoff. If the first-stage discontinuity \(\Delta_D\) is nonzero, then \(\Delta_Y/\Delta_D\) identifies the limiting average treatment effect \(E[Y(1)-Y(0)\mid D(1)=1,D(0)=0,R=c]\) among compliers at the cutoff. The conditional mean at \(R=c\) is understood as a limit from nearby running-variable values.

The continuity condition is local. It requires that, absent the assignment change, the relevant potential-outcome means and proportions of response types would not themselves jump at the threshold. The no-manipulation and continuity checks from regression discontinuity analysis can provide evidence about the design, but cannot prove this condition. Exclusion means that crossing the threshold has no effect on the outcome except through treatment receipt. Monotonicity rules out defiers: units who would receive treatment under \(Z=0\) but not under \(Z=1\). Finally, a nonzero first stage ensures that assignment actually changes treatment receipt locally.

Why the Ratio Identifies a Complier Effect

The logic can be seen by separating units according to their potential treatment pair. Always-takers have \(D(1)=D(0)=1\); never-takers have \(D(1)=D(0)=0\); compliers have \(D(1)=1,D(0)=0\); and defiers have \(D(1)=0,D(0)=1\). Under monotonicity, the defier group is absent. At the cutoff, only compliers change their treatment receipt when assignment changes.

Let \(p_C\) be the limiting proportion of compliers at the cutoff. With no defiers, the change in treatment probability is \(p_C\): always-takers and never-takers contribute no change, while each complier contributes one unit of treatment-probability change. Under exclusion and continuity, the outcome change is the proportion of compliers multiplied by their average treatment effect. Thus:

\Delta_D=p_C,\qquad \Delta_Y=p_C E[Y(1)-Y(0)\mid \text{complier},R=c], \qquad \frac{\Delta_Y}{\Delta_D} =E[Y(1)-Y(0)\mid \text{complier},R=c].

This argument also shows why the ratio is not the outcome discontinuity alone. The outcome jump combines the effect for threshold-induced treatment switchers with the fraction of local units who switch. Dividing by the treatment-probability jump removes that fraction under the assumptions. This is an identification argument, not a prescription for estimation; estimation and inference require additional choices about local fitting and uncertainty.

Worked Example: A Positive First Stage and Complier Effect

Suppose hypothetical local limits at a program eligibility cutoff show a mean test score of 70 points just below the cutoff and 74 points just above it. The treatment probabilities are 0.25 below and 0.65 above. Assume treatment is program participation and the outcome is a test score in points.

The outcome discontinuity is \(74-70=4\) points. The treatment-probability discontinuity is \(0.65-0.25=0.40\). Therefore the ratio is:

\frac{\Delta_Y}{\Delta_D} = \frac{74-70\text{ points}}{0.65-0.25} = \frac{4\text{ points}}{0.40} = 10\text{ points}.

The first stage is positive and nonzero, consistent with the threshold increasing participation. If continuity, exclusion, and no-defiers monotonicity hold, the result identifies an average treatment effect of 10 test-score points for local compliers: people near the cutoff who would participate when eligible and not participate when ineligible. It does not say that every participant gains 10 points, or that the program's average effect for all participants is 10 points.

Direction, Monotonicity, and Defiers

The coding of \(Z\) matters for describing monotonicity and the affected group. With \(Z=1\) defined as being on or above the cutoff, the usual assumption is \(D(1)\ge D(0)\): crossing to the \(Z=1\) side does not cause anyone to lose treatment. A positive \(\Delta_D\) is consistent with this direction, although a positive estimated jump does not establish individual-level monotonicity.

If treatment receipt instead falls as \(Z\) changes from 0 to 1, the usual no-defiers assumption is not appropriate for this coding. A distinct possible model is reverse monotonicity, \(D(1)\le D(0)\), under which there are no compliers and treatment switchers are defiers by the conventional definitions. If the other identification conditions hold, the ratio can then identify the average effect \(Y(1)-Y(0)\) for those defiers. They are people whose treatment is reduced by moving from \(Z=0\) to \(Z=1\), not compliers. A negative first stage alone is not enough to conclude reverse monotonicity; the assumption needs substantive justification.

Worked Example: A Negative First Stage Under Reverse Monotonicity

Consider a hypothetical threshold where \(Z=1\) still means being on or above the cutoff. The outcome mean is 83 points below the cutoff and 80 points above it, while treatment probabilities are 0.70 below and 0.50 above. Thus crossing to \(Z=1\) reduces treatment receipt.

The jumps and their ratio are:

\Delta_Y=80-83=-3\text{ points}, \qquad \Delta_D=0.50-0.70=-0.20, \qquad \frac{\Delta_Y}{\Delta_D} = \frac{-3\text{ points}}{-0.20} =15\text{ points}.

Suppose the substantive mechanism supports \(D(1)\le D(0)\), so that there are defiers but no compliers, and the continuity and exclusion conditions also hold. The treatment-probability jump is then minus the defier proportion, and the outcome jump is minus that proportion times the defiers' average treatment effect. Their ratio is 15 points, the average effect of treatment receipt for these defiers. It is not a conventional complier effect under the stated \(Z\) coding. Reversing the coding of \(Z\) would reverse both jumps and change the type label to compliers under that recoding, but would not change the ratio.

Failure Modes and Practical Interpretation

A ratio can be calculated even when it has no causal interpretation. The assumptions must be considered separately from the arithmetic. In particular, a threshold may change access to several services at once, treatment probability may have little or no jump, or the running variable may be manipulated so that units just above and below are not comparable. The ratio does not diagnose these problems by itself.

Worked Example: A Direct Assignment Effect Breaks Exclusion

Suppose a hypothetical program raises participation by 0.20 at a cutoff. Assume, for illustration, that participation itself raises the outcome by 20 points for the affected local units. If threshold assignment also directly raises the outcome by 2 points through a separate service, the outcome discontinuity is \(0.20(20)+2=6\) points.

The observed ratio is:

\frac{\Delta_Y}{\Delta_D} = \frac{6\text{ points}}{0.20} = 30\text{ points}.

But the treatment effect in this constructed example is 20 points, not 30. The extra 2-point direct assignment effect is incorrectly attributed to treatment receipt by the ratio. Exclusion fails, so the ratio does not identify the treatment effect. The observed discontinuities alone cannot reveal this direct pathway; it must be ruled out using knowledge of how the threshold and program operate.

Several diagnostics are useful, but none substitutes for the identifying assumptions. Examine whether the treatment probability has a clear, nonzero jump and whether the local support on both sides is adequate. Apply the standard regression discontinuity checks for sorting, predetermined covariate continuity, placebo cutoffs, and sensitivity to bandwidth and specification choices. A discontinuity in a predetermined covariate or evidence of precise sorting can undermine the continuity argument. Conversely, passing these checks does not prove exclusion or monotonicity, which are substantive claims about the assignment mechanism and treatment response.

A weak first stage is also a practical warning. When \(\Delta_D\) is close to zero, even modest uncertainty in the estimated discontinuities can make the ratio highly unstable. This tutorial concerns identification, not the inference procedures needed in that situation. The key interpretive discipline is to report the outcome jump, treatment-probability jump, and ratio separately, together with the assumptions and the local population to which the effect pertains.

Approach	What changes at the cutoff	Effect interpretation
Sharp regression discontinuity	Treatment receipt changes from 0 to 1 by assignment	Local treatment effect at the cutoff under the sharp-design assumptions
Fuzzy regression discontinuity	Treatment probability changes, but receipt is imperfect	Local average effect for threshold-induced treatment switchers under the fuzzy-design assumptions
Covariate adjustment without a discontinuous assignment mechanism	No threshold-based treatment-probability jump is required	Effect depends on the assumptions supporting that adjustment strategy; it is not identified by the cutoff ratio

The sharp design is a special case in which the treatment probability jumps by one, so dividing the outcome jump by that first stage leaves the outcome jump unchanged. In the fuzzy design, the ratio adjusts for incomplete take-up but narrows the target to local switchers. Unlike ordinary covariate adjustment, the design's identifying variation is the discontinuity at the threshold. Neither approach automatically supports generalization beyond its assumptions and target population.

Common Mistakes and Limitations

Calling all treated units compliers. Compliers are defined by the pair \(D(1)=1,D(0)=0\), not by their observed treatment status. The ratio concerns compliers at the cutoff under the specified assignment coding.
Ignoring the direction of the first stage. Under \(Z=1\) above the cutoff, a negative treatment jump conflicts with the usual no-defiers assumption. If reverse monotonicity is defensible, the switchers are defiers under the conventional definitions.
Reading a ratio as proof of causality. A numerical ratio does not establish continuity, exclusion, monotonicity, or a nonzero population first stage.
Overgeneralizing the effect. The identified effect is local to the cutoff and to the treatment switchers. It need not equal the average effect for all eligible people, all participants, or other thresholds.

The design is especially limited when the first stage is weak, when units can manipulate their running-variable values, or when assignment changes other determinants of the outcome. Treatment effects may also vary across people, so a local complier effect need not transport to a different population. These are limits on the causal claim, not matters that can be repaired by interpreting the ratio more broadly.

Check Your Understanding

Answer each question using the assignment coding and assumptions stated. Solutions follow the questions.

At a cutoff, the outcome jump is 6 points and the treatment-probability jump is 0.30. What is the ratio, and which group’s effect does it identify under the usual assumptions?
With \(Z=1\) above the cutoff, treatment probability falls from 0.60 to 0.40. Under reverse monotonicity, which response type changes treatment, and what is that type under the conventional definitions?
Why does a nonzero treatment-probability jump not, by itself, establish the ratio as a causal effect?
In a sharp design, treatment probability jumps from 0 to 1. How does the fuzzy ratio relate to the outcome jump?

Solutions. (1) The ratio is \(6/0.30=20\) points. With continuity, exclusion, and no defiers, it identifies the average effect for local compliers. (2) The switchers are defiers, with \(D(1)=0,D(0)=1\); under the reverse-monotonicity and other identification conditions, the ratio identifies their average treatment effect. (3) A direct assignment effect, discontinuity in potential outcomes, sorting, or unsupported monotonicity can invalidate the causal interpretation despite a nonzero jump. (4) The ratio is the outcome jump divided by 1, so it equals the outcome jump.