Standard Errors and Clustering in RD

Why Standard Errors Matter in RD

An RD estimate is local: it summarizes the outcome discontinuity at a cutoff using observations in a chosen neighborhood. Its standard error quantifies sampling uncertainty in that estimate. A variance estimator that ignores unequal outcome variability or dependence within assignment units can make the standard error too small or too large, changing the apparent precision of the RD result.

These are separate issues. Heteroskedasticity means the conditional outcome variance can differ across observations, even when observations are independent. Clustering means outcomes for observations in the same group may be dependent. RD analyses often need a variance estimator robust to both.

Local RD Regression and Its Variance

Let \(R_i\) be the running variable for observational unit \(i\), let \(c\) be the cutoff, and define \(D_i=1[R_i\ge c]\). A local polynomial regression fits a polynomial in \(R_i-c\) on each side of the cutoff, using observations near the cutoff and kernel weights. In the sharp RD case, the coefficient on \(D_i\), under a specification that allows the two sides’ slopes to differ, estimates the local outcome jump. The bandwidth and polynomial order define the fitted estimator; they do not determine which variance estimator is appropriate.

For clarity, write the weighted regression in matrix form as \(Y=X\beta+u\), where \(Y\) is the outcome vector, \(X\) contains the local polynomial terms and side indicators, and \(W\) is the diagonal matrix of kernel weights. The weighted least-squares estimate is \(\hat\beta=(X'WX)^{-1}X'WY\). The RD estimate \(\hat\tau\) is the component of \(\hat\beta\) corresponding to the cutoff jump. This regression description applies to the local polynomial fit; inference must also reflect any bias correction used.

The conventional homoskedastic covariance formula imposes a common error variance. If that restriction is not credible, a heteroskedasticity-robust sandwich estimator replaces it with a sum of observation-level residual contributions. For a weighted fit, let \(\hat u_i\) be the residual and \(x_i\) the row of \(X\). The HC0 covariance estimator is

The standard error for \(\hat\tau\) is the square root of the corresponding diagonal element. The squared weights appear because the weighted regression’s observation-level score is \(w_i x_i'\hat u_i\). Other HC versions add finite-sample adjustments; the exact adjustment should be reported rather than treated as interchangeable in small samples.

For robust bias-corrected RD inference, the standard error must be for the bias-corrected estimator, not simply the ordinary standard error from the original local fit. Bias correction changes the estimator’s uncertainty. A reported robust bias-corrected interval should therefore use the variance calculation and critical value appropriate to that corrected estimator.

Clustering by the Assignment Unit

In many RD applications, the assignment rule operates at a group level even though outcomes are recorded for individuals. For example, a school’s running-variable value may determine whether all students in that school are eligible for a program. Students in the same school can share unobserved conditions, so their outcomes may be correlated. Treating all students as independent would then count within-school information as if it were independent evidence.

The cluster-robust sandwich estimator aggregates the regression score within each cluster before forming the “meat” of the sandwich. Let \(g=1,\ldots,G\) index clusters, and let \(X_g\), \(W_g\), and \(\hat u_g\) denote the corresponding rows, weights, and residuals. Define the cluster score \(S_g=X_g'W_g\hat u_g\). A basic cluster-robust covariance estimator is

A finite-sample multiplier is often applied to this estimator. Its form depends on the software and specification, so analysts should identify the adjustment used. The essential feature is aggregation: within-cluster residual contributions can reinforce or offset one another before the cluster scores are squared. When outcomes are positively correlated within clusters, the resulting standard error is often larger than the observation-level robust standard error, but that is not guaranteed in every data set.

The cluster definition should follow the source of dependence and the assignment process, not be chosen to obtain a desired standard error. The number of independent clusters matters more for the quality of the usual cluster-robust approximation than the total number of individual observations alone. With few clusters, conventional cluster-robust standard errors and normal critical values can be unreliable; suitable small-sample corrections or methods designed for few clusters may be needed. Clustering does not repair a misspecified RD model, inadequate bandwidth choice, or unaddressed bias.

Confidence Intervals and Corrected RD Estimates

A confidence interval combines an estimate, its standard error, and a critical value appropriate to the inferential procedure. In a large-sample normal approximation, the interval has the form \(\hat\tau\pm z_{1-\alpha/2}SE(\hat\tau)\). With cluster-based inference, the critical value or degrees-of-freedom adjustment may differ, especially when the number of clusters is limited. An interval for a bias-corrected estimate must pair that estimate with its bias-corrected standard error.

Choosing and Reporting the Variance Estimator

A practical choice begins with the design and data structure. If observations are independent but conditional variances may differ, use a heteroskedasticity-robust estimator. If observations may be dependent within assignment units, cluster at the level that captures that dependence; a cluster-robust estimator also permits heteroskedasticity across observations and clusters. If both features are present, clustering is generally the relevant dependence adjustment, not a reason to assume equal variance.

For fuzzy RD, the estimated effect is a ratio involving discontinuities in treatment receipt and outcome. Its uncertainty must account for the variance of both estimated discontinuities and their covariance; when assignment units induce dependence, that covariance calculation must also respect clustering. The same general principle applies: the standard error must correspond to the estimator actually reported.

Common Mistakes and Why They Matter

• Using ordinary standard errors because the sample is large. A large number of individuals does not make within-cluster dependence disappear. If there are few independent assignment units, the effective information for inference may be much closer to the number of clusters than the number of records.
• Clustering only as a formality. The cluster level should reflect plausible dependence and the assignment mechanism. An arbitrary grouping does not guarantee valid inference.
• Assuming a robust standard error fixes bias. Robust variance estimators address uncertainty under their dependence assumptions; they do not remove local polynomial approximation bias. Use the appropriate bias-corrected procedure when needed.
• Reading a wider interval as a smaller estimated effect. Clustering changes the estimated uncertainty, not necessarily the point estimate. It can change whether an interval excludes zero without changing the estimated jump.
• Using a cluster-robust standard error with an unsuitable critical value. A normal approximation may be poor with few clusters. State how the critical value and any finite-sample correction were chosen.