Why a Fuzzy Cutoff Can Act as an Instrument
Some programs use a measured score, age, or other running variable to determine eligibility at a threshold. When crossing the threshold changes the chance of receiving treatment but does not determine receipt perfectly, the design is called a fuzzy regression discontinuity (fuzzy RD). For example, eligibility may rise sharply at a score of 70, while some eligible people do not take up the program and some ineligible people obtain it another way.
The threshold creates a local change in treatment receipt that can be used as an instrument. The key calculation compares the change in outcome at the cutoff with the change in treatment receipt there. Its interpretation depends on the design assumptions: a ratio alone does not make the cutoff a valid instrument.
These are local discontinuities in conditional means, not comparisons of everyone above and below the threshold. Smooth changes in \(R\) may be associated with smooth changes in treatment and outcome; the design isolates the discontinuous changes at \(c\). The numerator and denominator must be estimated for the same cutoff, sample, bandwidth, and weighting scheme.
Assumptions and Local Interpretation
A useful interpretation of the ratio requires more than seeing a treatment jump. The running variable must not be precisely manipulated so that units just below and above the threshold differ discontinuously in their underlying characteristics. In the absence of the threshold-induced treatment change, expected potential outcomes should vary continuously through \(c\). Predetermined covariates should likewise have no unexplained jump. These conditions make observations very close to the cutoff comparable in expectation, though treatment receipt itself may jump.
The cutoff must affect the outcome through the treatment being studied, not through another policy or service that changes at the same threshold. The treatment jump must be nonzero. For the usual complier interpretation, crossing the cutoff must not make anyone less likely to receive treatment: in potential-treatment notation, \(D(1)\ge D(0)\). This is the local monotonicity condition. Consistency and no interference are also assumed: each unit’s observed outcome corresponds to its received treatment, and one unit’s treatment does not change another unit’s outcome.
The identification argument can be seen by imagining potential treatment \(D(z)\) under each eligibility state and potential outcome \(Y(d)\) under each treatment state for units at the cutoff. Continuity lets the right-minus-left limits identify the change induced by switching eligibility at \(c\). Under exclusion, the outcome under eligibility \(z\) is \(Y(D(z))\). Therefore:
Under monotonicity, units with \(D(1)-D(0)=1\) are compliers and units with \(D(1)-D(0)=0\) do not change treatment at the threshold. Thus \(\tau_D\) is the proportion of compliers at the cutoff, while \(\tau_Y\) is that proportion multiplied by their average treatment effect. Dividing gives the average treatment effect for compliers at the cutoff. With heterogeneous effects, this is not automatically the effect for everyone near the cutoff, all eligible people, or the full study population.
Worked Example: Calculating the Cutoff Wald Estimate
Suppose a hypothetical program uses an eligibility score cutoff. In a prespecified narrow window, the estimated treatment receipt rate is 0.68 just above the cutoff and 0.32 just below it. The estimated mean outcome is 72 points just above and 66 points just below. Assume the local continuity, exclusion, relevance, and monotonicity conditions are credible for this illustration.
The first-stage jump is \(0.68-0.32=0.36\), or 36 percentage points. The outcome jump is \(72-66=6\) points. The fuzzy RD estimate is:
Under the stated assumptions, the estimate is an average effect of receiving the program on compliers at the cutoff, estimated as 16.67 outcome points. It is not the effect of eligibility itself and does not establish that every participant gains 16.67 points. These summary estimates alone do not provide a standard error or confidence interval, so they do not establish statistical precision.
Estimating the Jumps in Practice
A common approach is to fit separate local linear regressions on each side of the cutoff, giving greater weight to observations nearer \(c\). For a chosen bandwidth \(h\), define \(x=R-c\), and use observations with \(|x|\le h\). On each side, fit a weighted line for \(D\) and another for \(Y\). The fitted intercept on each side estimates the conditional mean at \(x=0\); subtracting the left intercept from the right intercept gives the estimated jump. A local linear fit allows the smooth relationship between the running variable and each response to have a different slope on either side.
The regression is a way to estimate the limits, not a license to extrapolate from distant observations. Bandwidth choice trades precision against the risk that a simple local approximation is inadequate over a wide range. Report the bandwidth and model choices, and check whether reasonable alternatives give materially different jumps and ratios.
Worked Example: Jumps From Side-Specific Local Linear Fits
In a hypothetical analysis, local linear fits using the same bandwidth give these fitted values, where \(x=R-c\) is measured in score points:
At the cutoff, \(x=0\). The left and right treatment intercepts are 0.30 and 0.60, so \(\hat{\tau}_D=0.60-0.30=0.30\). The outcome intercepts are 50 and 54 points, so \(\hat{\tau}_Y=54-50=4\) points. Hence:
The slopes do not enter the jump calculation because the fitted lines are evaluated at the cutoff, where \(x=0\). If the design assumptions hold, this estimates a local complier effect. The fitted equations provide point estimates only; uncertainty must be calculated from the data and should account for estimating both jumps and their ratio.
Diagnostics and Failure Modes
Diagnostics assess whether the design is plausible; they cannot prove its assumptions. Examine the distribution of \(R\) near the threshold for unusual bunching, gaps, or heaping. Such patterns may suggest precise sorting or data-recording problems. Check whether measured characteristics determined before eligibility, such as baseline health or prior outcomes, change discontinuously at \(c\). A jump in these characteristics raises concern that units on the two sides are not comparable. Also inspect placebo cutoffs and outcomes that should not respond to treatment.
A first-stage plot and estimate are essential. If treatment receipt barely changes at the cutoff, the ratio divides by a small number and can be highly unstable. A large point estimate in that setting is not evidence of a large effect by itself. Use inference appropriate for a potentially weak instrument; ordinary ratio-based confidence intervals can perform poorly when the denominator is near zero. The earlier material on weak-instrument inference is relevant here.
Worked Example: A Co-Occurring Policy Change Invalidates the Ratio
Suppose a hypothetical cutoff raises treatment receipt by 0.40. The observed outcome rises by 6 points, so the calculated ratio is \(6/0.40=15\) points per treatment. However, suppose a separate grant also begins at exactly the same cutoff and directly raises the outcome by 4 points, independently of the treatment being analyzed.
The exclusion condition fails: crossing the cutoff affects the outcome through both treatment and the grant. If the treatment’s true local effect were 5 points, its contribution to the outcome jump would be \(0.40\times5=2\) points. Adding the grant’s 4-point jump gives the observed 6-point jump. The ratio is therefore \(6/0.40=15\), even though the treatment effect in this illustration is 5 points.
The arithmetic is correct but the instrumental-variables interpretation is not. If another cutoff-triggered change is present, the outcome jump cannot be attributed solely to treatment. The analysis needs a more credible design or a carefully justified way to separate the effects; relabeling the ratio does not repair the violation.
How Fuzzy RD Fits Among Alternatives
| Approach | Source of comparison | Key distinction |
|---|---|---|
| Sharp RD | Units immediately on either side of the cutoff | Treatment is determined perfectly by cutoff status, so the treatment jump is 1 and the outcome jump is the RD effect. |
| Fuzzy RD | Units immediately on either side of the cutoff | Treatment changes imperfectly; the outcome jump is scaled by the treatment jump to estimate a local complier effect. |
| Broader IV analysis | Variation in an instrument across a study population | The source of instrument variation need not be a threshold, and the target population depends on the instrument and design. |
| Matching | Observed treatment and comparison units made similar on measured covariates | Matching addresses measured differences; it does not by itself use a cutoff-induced treatment jump as an instrument. |
Fuzzy RD’s strength is its local comparison near a known threshold, not automatic generalizability. Its conclusions can be sensitive to bandwidth and functional form, and may not transport to people far from the cutoff. If the treatment jump is weak, the ratio may be imprecise or its usual inference unreliable. If manipulation, discontinuous confounding, or exclusion violations are plausible, even a precisely computed ratio may not identify a treatment effect.
A useful reporting practice is to give the treatment jump, outcome jump, their ratio, units, bandwidth, and uncertainty, then state the assumptions and local target explicitly. Showing only the ratio conceals whether the instrument has a meaningful first stage and whether the outcome discontinuity is being interpreted appropriately.
Check Your Understanding
Work through each question, then compare your reasoning with the solutions.
- At a cutoff, treatment receipt rises from 0.25 to 0.55 and the outcome rises by 9 points. Calculate the fuzzy RD estimate and state its units.
- Why is the outcome jump alone not the fuzzy RD treatment-effect estimate when treatment receipt changes imperfectly?
- What does a nonzero jump in a predetermined baseline covariate suggest, and what does it not prove?
- If \(\hat{\tau}_D=0.02\) and \(\hat{\tau}_Y=1.0\) point, calculate the ratio and identify the main inferential concern.
- Under monotonicity, who are the compliers in a fuzzy RD design?
Solutions. (1) The first-stage jump is \(0.55-0.25=0.30\), so the ratio is \(9/0.30=30\) outcome points per treatment. Under the assumptions, it is a local complier effect at the cutoff. (2) The outcome jump reflects the effect of eligibility on the outcome, while only a fraction of units change treatment; dividing by the treatment jump scales the outcome change to the treatment change. (3) It raises concern about discontinuous composition or sorting, weakening the local comparability argument. It does not by itself prove manipulation or establish the direction or size of bias. (4) The ratio is \(1.0/0.02=50\) points per treatment, but the first stage is very small, so the ratio may be unstable and conventional inference unreliable. (5) Compliers are units whose treatment receipt changes from \(D(0)=0\) to \(D(1)=1\) when eligibility changes from below to above the threshold.