Donut RD and Threshold-Adjacent Observations

Why Exclude Observations Near the Cutoff?

Regression discontinuity (RD) designs use a running variable and a cutoff to compare units just below and just above a treatment threshold. The credibility of that comparison depends on behavior near the cutoff: in the absence of treatment, outcome patterns should not jump there for other reasons. But the observations closest to the threshold can be unusually problematic. A recorded score might be rounded or mismeasured near the eligibility boundary, or units might be able to manipulate a score precisely enough to cross it.

A donut RD removes a specified neighborhood of the cutoff and estimates the discontinuity from observations farther away on either side. This can reduce reliance on observations suspected of being unreliable. It also creates a gap in the data precisely where ordinary RD uses local comparisons. With a fixed gap, continuity by itself does not tell us what the outcome limits at the cutoff are. The fitted cutoff values are extrapolations from outside the gap, so interpreting them causally requires additional assumptions.

The hole is a design choice, not a routine correction. Its size should correspond to a concrete concern, such as known rounding or a specific manipulation process. Removing observations simply because they influence the estimate is not a sound justification: it risks choosing the sample to produce a preferred result. A donut also does not establish that the remaining observations are free of manipulation or other threats to the design.

Estimating a Sharp Donut RD

In a sharp RD, treatment assignment is determined by the threshold: \(D=1(X\geq 0)\). Write \(m_-(x)\) and \(m_+(x)\) for the conditional mean outcome functions on the left and right of the cutoff. If the cutoff limits are identified, the sharp RD estimand is the right-hand limit minus the left-hand limit:

A common donut specification fits a separate local linear regression on each side using only observations outside the hole and within bandwidth \(h\). For clarity, with unweighted least squares these fits are

The estimated discontinuity is \(\hat{\tau}_{\mathrm{donut}}=\hat{a}_+-\hat{a}_-\). If kernel weights are used, the same two regressions are weighted according to the chosen kernel and bandwidth; the fitted values at \(X=0\) remain the two intercepts. The intercepts are not observed outcomes at the cutoff. Each is the value predicted by its side-specific fitted line at zero.

What a Fixed Hole Changes About Identification

In a standard RD with observations arbitrarily close to the cutoff, continuity of the relevant conditional mean functions lets the limits be learned from observations approaching zero from each side. A fixed donut hole changes the support: the left-side observations stop at \(-\delta\), and the right-side observations begin at \(+\delta\). Even with a very large sample outside the hole, the data do not reveal the function values inside it.

Continuity alone does not fill this gap. One can construct distinct continuous functions that agree with all observed values outside the hole but take different values at zero. Thus the two cutoff limits, and their difference, are not nonparametrically identified from the donut sample merely by assuming continuity. Estimating local linear regressions supplies a way to extrapolate, but it does not make the extrapolation assumption-free.

The local linear model is an approximation, not a consequence of the RD design. Its credibility depends on the shape of the outcome relationship over the interval from the retained observations to the cutoff. If that relationship bends sharply inside the hole, the two fitted lines may predict inaccurate cutoff values even when they fit the observed data well. A smaller hole reduces the distance extrapolated but may retain observations affected by the original concern; a larger hole removes more such observations but makes the extrapolation longer and usually reduces the sample size.

Fuzzy RD: The Ratio Requires Credible Extrapolation

In a fuzzy RD, crossing the cutoff changes the probability of receiving treatment rather than determining treatment perfectly. Let \(T\) denote treatment received and \(Z=1(X\geq0)\) denote threshold eligibility. In an ordinary fuzzy RD, the ratio of the discontinuity in the outcome to the discontinuity in treatment probability can identify a local average treatment effect for compliers at the cutoff under the usual fuzzy RD conditions: continuity of the relevant potential outcome and treatment functions at the cutoff, a nonzero first stage, exclusion of a direct effect of eligibility on the outcome, and monotonicity of treatment take-up with eligibility.

With a fixed donut, the issue is not removed by taking the ratio. Both cutoff discontinuities are based on extrapolated intercepts. The usual fuzzy RD assumptions alone do not identify those limits from data outside the hole. In addition, assumptions must make the extrapolated outcome and treatment-probability intercepts credible—for example, correctly specified local linear models across the hole. Only under these extra extrapolation assumptions, together with the usual fuzzy RD assumptions, can the ratio be interpreted as a local causal effect for compliers at the cutoff.

Choosing and Reporting the Donut

The exclusion radius should be stated in the units of the running variable, along with the bandwidth and the rule for whether observations exactly at the hole boundary are retained. The choice should be linked to evidence about the suspected problem—for example, a known rounding interval—not selected solely after inspecting which estimate is most attractive. If the running variable is discrete, removing the nearest support points may create a particularly large gap, so the amount of extrapolation should be described directly.

A careful report distinguishes three claims: which observations were excluded, what the fitted model estimates, and what assumptions support a causal interpretation. It is useful to show how conclusions depend on the chosen hole, but stability across choices does not prove identification, and instability is a warning that results depend on modeling decisions. Donut estimation can address a localized concern only if the concern is genuinely concentrated in the removed region and the remaining data and extrapolation model are credible.

Common Mistakes

• Assuming continuity recovers the missing cutoff values. With a fixed hole, continuity alone does not identify the limits from observations bounded away from zero. State the added extrapolation restriction explicitly.
• Calling the fitted intercepts observed cutoff means. They are predictions from side-specific regressions, not averages of units observed at the cutoff.
• Claiming a fuzzy donut ratio is automatically a complier effect. The usual fuzzy RD assumptions do not by themselves justify the extrapolated intercepts; additional assumptions are needed for both numerator and denominator.
• Treating exclusion as proof that manipulation is absent. A donut removes selected observations; it does not show that sorting is absent among the retained units or that the exclusion has repaired every source of bias.
• Ignoring the cost of a larger hole. A wider exclusion can remove more potentially problematic observations, but it also increases extrapolation distance and often decreases precision.