Estimating A Fuzzy Regression Discontinuity

From Two Local Jumps to One Effect

In a fuzzy regression discontinuity design, crossing the cutoff changes the probability of receiving treatment but does not determine treatment perfectly. Estimation therefore starts with two quantities: the discontinuity in the outcome and the discontinuity in treatment. Their ratio estimates the local effect of treatment for compliers at the cutoff, under the design’s identification assumptions.

The ratio is simple to calculate, but its uncertainty is not obtained by treating the two jumps as independent. Both are estimated using many of the same units near the cutoff, so their sampling errors can be correlated. A defensible standard error must account for that joint uncertainty.

In practice, the two discontinuities are estimated locally rather than by comparing all units above and below the cutoff. Define the centered running variable \(R_i=X_i-c\). For a chosen polynomial order \(p\), fit separate weighted polynomial regressions on the two sides, using observations within a bandwidth \(h\). A typical specification for an outcome \(Q_i\), where \(Q_i\) is either \(Y_i\) or \(D_i\), is

Each observation receives a kernel weight \(K(R_i/h)\). The fitted left limit at the cutoff is the left intercept, \(\hat\beta_{-,0}\), and the fitted right limit is the right intercept, \(\hat\beta_{+,0}\). The estimated jump is their difference, \(\hat\Delta_Q=\hat\beta_{+,0}-\hat\beta_{-,0}\). With local linear regressions, \(p=1\); the slopes allow the conditional mean to vary with the running variable on each side while the intercept difference estimates the cutoff jump.

Identification and Estimation Conditions

A local polynomial fit produces an estimated discontinuity, but the causal interpretation of the ratio requires more than a good fit. The identification argument for fuzzy regression discontinuity relies on continuity at the cutoff of the relevant potential-outcome and treatment-response functions in the absence of the threshold-induced change. Units must not precisely sort around the cutoff in a way that invalidates local comparability, and there must be no other discontinuous change at the threshold that affects the outcome.

For the ratio to identify a local complier effect, the threshold must change treatment receipt, the threshold must affect the outcome only through treatment (the exclusion restriction), and treatment response must be monotone with respect to the threshold assignment (no defiers). These conditions give the ratio its local causal interpretation; they do not follow merely from observing two jumps. The earlier material on fuzzy regression discontinuity identification motivates these assumptions. This tutorial focuses on estimating the jumps and their ratio.

For estimation, choose the cutoff, kernel, polynomial order, and bandwidth before interpreting the result. The bandwidth determines which observations contribute appreciably to the local fit and creates the usual bias-variance tradeoff. Separate bandwidth choices for the outcome and treatment regressions are possible, but they complicate the joint covariance calculation. A common bandwidth and kernel make the two fitted jumps easier to align and their dependence easier to estimate.

Joint Uncertainty and the Delta Method

Let \(\hat\Delta_Y\) and \(\hat\Delta_D\) be the estimated jumps, and let their estimated covariance matrix be

The covariance \(\widehat C_{YD}\) matters because outcome and treatment are measured on the same units and use overlapping local observations. It can be obtained from a joint estimation procedure or from the covariance of the two estimators’ influence functions, using an appropriate variance estimator for the sampling structure. Setting it to zero without justification is not generally valid.

A first-order Taylor expansion gives a standard large-sample approximation for the ratio. For \(g(a,b)=a/b\), the gradient is \((1/b,-a/b^2)\). Applying the variance formula for a differentiable function of a jointly estimated vector yields

The negative covariance term follows from multiplying the two gradient components, one of which is negative. Depending on the sign of the covariance, accounting for it can either decrease or increase the estimated variance. The approximation is most reliable when the treatment jump is sufficiently far from zero that the ratio behaves approximately linearly over the range of plausible jump estimates.

When the local-polynomial jumps are bias-corrected, use the bias-corrected outcome and treatment jumps in the ratio and use their joint covariance for inference. Bias correction changes both estimates and their sampling variation; calculating separate standard errors and then ignoring dependence does not produce a valid ratio standard error. Robust bias-corrected methods from the earlier material on regression-discontinuity inference provide a way to account for smoothing bias in the discontinuity estimates. The ratio’s inference must still propagate the joint uncertainty in both corrected jumps.

Inference When the Treatment Jump Is Uncertain

The delta-method interval can be misleading if the treatment jump is close to zero relative to its standard error. A ratio can vary sharply when its denominator is uncertain, so a seemingly moderate standard error is not guaranteed to describe the full uncertainty. A useful alternative in such cases is Fieller’s method, which inverts a test of a proposed ratio value \(\theta\). Under an approximately joint normal distribution for the two jump estimates, the corresponding confidence set consists of values satisfying

This inequality retains the covariance and does not require the ratio to be well approximated by a linear function around the estimates. If the treatment jump is weakly estimated, the resulting set may be very wide, unbounded, or disconnected. That is an informative warning: the data do not pin down a precise ratio. A conventional symmetric interval based on the delta method can obscure this behavior. Fieller inference does not repair a failure of the design assumptions or create a meaningful effect when the threshold barely changes treatment.

Diagnostics, Comparisons, and Limitations

A practical estimation sequence is to fit local regressions for both \(Y_i\) and \(D_i\), extract the two intercept differences, form their ratio, and then construct inference from the joint covariance matrix. Check that both regressions use observations with adequate support on each side and inspect whether the fitted functions behave plausibly close to the cutoff. Sparse support, strong curvature, or a large ratio driven by a small estimated treatment jump are reasons to investigate further, not reasons to report extra decimal places.

The estimate can also be compared with the main alternatives, while keeping the estimands distinct. A sharp regression-discontinuity estimate uses an outcome jump directly when crossing the cutoff determines treatment. In the fuzzy design, the outcome jump is scaled by the treatment jump because compliance is incomplete. A global regression across a broad range of the running variable may use more observations, but its result can depend heavily on functional-form extrapolation and need not estimate the same local discontinuity. Neither alternative removes the need to justify the design.

The fuzzy ratio is a local effect, not automatically an average effect for all treated people or for the full population. Its causal interpretation depends on continuity, exclusion, monotonicity, and a nonzero treatment discontinuity. Its precision depends on local sample support, bandwidth and polynomial choices, and the joint sampling behavior of the two estimated jumps. The estimate can be unstable when the denominator is imprecise, and no variance formula can compensate for sorting or another threshold change that violates identification.

• Mistake: “The standard error of the ratio can be found by dividing the outcome-jump standard error by the treatment jump.” This ignores uncertainty in the denominator and the covariance between jumps. Correction: use the joint delta-method variance, or an appropriate ratio-inference method when the denominator is uncertain.
• Mistake: “The treatment jump is positive, so the ratio is automatically causal.” A positive estimate does not establish continuity, exclusion, monotonicity, or adequate precision. Correction: distinguish the estimated ratio from the assumptions that identify it and report uncertainty.
• Mistake: “Similar ratios across two bandwidths prove the design is valid.” Similarity is a sensitivity check, not proof that there is no sorting, no other discontinuity, or no model bias. Correction: assess the design assumptions and local support as well as the numerical stability.