Robust Bias-Corrected RD Inference

Why Bias Correction Changes Inference

A local polynomial RD estimate balances two sources of uncertainty: sampling variation and smoothing bias. With a finite bandwidth, a local polynomial fit can smooth across points that are not exactly at the cutoff, leaving a systematic difference between its estimated jump and the true jump. That bias matters for inference when it is large relative to the estimate’s standard error. A confidence interval centered on the uncorrected estimate can then miss its target more often than its nominal level suggests.

Robust bias-corrected (RBC) inference estimates and subtracts the leading smoothing bias. But the bias estimate is itself computed from outcomes in the local sample, so subtracting it also changes sampling variation. A valid interval must account for both facts: it centers on the bias-corrected estimate and uses the standard error of that corrected estimate.

The target here is the sharp RD jump. Its causal interpretation relies on the continuity-based identification conditions for sharp RD: the relevant potential-outcome means must be continuous at the cutoff, and treatment status must change there as specified by the design. Bias correction addresses a finite-bandwidth estimation problem; it does not repair a failure of those identification conditions.

Local Polynomial Estimates and Their Leading Bias

Write \(X_i=R_i-c\), and fit separate local polynomials on either side of zero, weighting observations according to their distance from the cutoff. For example, a local linear fit uses a polynomial of order \(p=1\), and its estimated jump is the difference between the two fitted intercepts. More generally, let \(\hat{\tau}_p(h)\) be the jump estimate from a polynomial of order \(p\) using bandwidth \(h\).

Under smoothness of the conditional mean on each side and regularity of the running-variable density and kernel, the leading smoothing bias has order \(h^{p+1}\). More precisely, for constants \(B_{p,+}\) and \(B_{p,-}\) determined by the polynomial order, kernel, and local design,

The constants need not be equal across the two sides. The bias estimate therefore uses the side-specific curvature or higher-order shape that contributes to the jump estimate, rather than assuming that the two regression functions have identical behavior.

A common construction estimates this leading term using local polynomials of order \(q=p+1\), often with a bandwidth \(b\). The higher-order fit estimates the relevant derivatives at the cutoff; these estimates are combined with the bias constants to give \(\hat{B}_p(h,b)\), an estimate of the leading bias of \(\hat{\tau}_p(h)\). The corrected estimate is

The bandwidth \(h\) controls the main RD fit, while \(b\) controls the fit used to estimate its bias. In standard RBC asymptotics, both shrink toward zero and their ratio stays bounded away from zero and infinity. In practice, the precise bias formula and constants depend on the chosen kernel, polynomial order, and implementation.

Why the Standard Error Must Also Be Corrected

To see the variance issue directly, express the local polynomial estimate and its estimated bias as weighted sums of the observed outcomes. Conditional on the running-variable values, write

The weights \(\ell_i\) and \(a_i\) are determined by the local fits, kernel, and bandwidths. Therefore the corrected estimate is also a weighted sum:

If observations are independent conditional on their running-variable values and have conditional error variances \(\sigma_i^2\), the conditional variance of that weighted sum is

In practice, a heteroskedasticity-robust estimate replaces the unknown variances with squared local-fit residuals. Equivalently, the corrected variance can be written using the variance of the original estimate, the variance of the bias estimate, and their covariance:

This expression explains the word robust in RBC inference: the standard error is computed for the estimator after bias correction, including the dependence between the estimate and its bias correction. It is not generally valid to subtract an estimated bias and then retain the uncorrected standard error as though the subtracted quantity were fixed and known.

Constructing the Robust Bias-Corrected Interval

Under the relevant regularity conditions, the corrected estimate, centered at the target and scaled by its corrected standard error, is asymptotically standard normal. For a two-sided confidence level \(1-\alpha\), an asymptotic RBC interval is

where \(z_{1-\alpha/2}\) is the corresponding standard normal quantile. This is an asymptotic interval: its coverage approaches the stated level as the sample size grows under the design and bandwidth conditions, rather than being guaranteed exactly in a finite sample.

Conditions for Reliable Coverage

Bias correction is useful only when the remaining approximation error is small enough for the normal approximation to work. A typical local-polynomial RBC argument requires independent observations for the variance calculation used here, a positive and sufficiently smooth density of the running variable near the cutoff, a suitable kernel, and enough smoothness of the conditional mean on each side. The local sample must grow while the bandwidth shrinks: informally, \(h\to0\) and \(nh\to\infty\). The bias-estimation bandwidth \(b\) must also shrink, with \(b/h\) kept at a nonzero finite scale in the standard formulation.

For a polynomial of order \(p\), the original leading bias is of order \(h^{p+1}\). Bias correction removes that leading term, leaving a smaller-order remainder under the required additional smoothness. The corrected standard error must include the variability of the estimated leading term; otherwise the reported interval is not calibrated to the corrected estimator. These conditions are asymptotic, so a very small effective local sample or a poor fit can still make an interval unreliable in a particular application.

There is also a distinction between bias correction and identification. RBC inference targets the same RD jump and does not make the continuity assumptions more plausible, eliminate sorting, or resolve a misspecified treatment rule. It addresses how local smoothing affects estimation and coverage when the RD design itself identifies the desired jump.

Common Mistakes

• Subtracting the bias estimate but keeping the original standard error. This treats an estimated quantity as fixed. The correction uses outcomes from the local sample, so its variance and covariance with the original estimate belong in the corrected standard error.
• Assuming bias correction eliminates all bias. It removes an estimated leading term under smoothness and bandwidth conditions. Higher-order remainder bias can remain.
• Confusing improved inference with stronger identification. RBC changes finite-bandwidth estimation and uncertainty, not the continuity assumptions that give the RD jump a causal interpretation.
• Describing an asymptotic interval as an exact probability statement. The nominal coverage is a repeated-sampling target under stated conditions, not a probability assigned to the fixed parameter after observing the interval.