Robust Bias-Corrected Regression Discontinuity Inference

Why Bias Correction Changes Inference

A local-polynomial regression-discontinuity estimate is built from observations near a treatment cutoff. Restricting attention to nearby observations makes the comparison more local, but a finite bandwidth still averages over values on either side of the cutoff. When the conditional mean curves are curved, this averaging can create smoothing bias. The issue is not only whether to adjust the estimate: an adjustment is itself estimated, so inference must also reflect the uncertainty introduced by that adjustment.

Robust bias-corrected (RBC) inference addresses both parts. It estimates and subtracts the leading local-polynomial bias, then computes a standard error for the corrected estimator rather than reusing the standard error of the uncorrected one. With suitable smoothness and sampling conditions, this supports confidence intervals even when the estimation bandwidth is not deliberately made small enough to eliminate the leading bias.

Definition: In a sharp regression-discontinuity design, let \(X_i\) be the running variable, \(c\) the cutoff, and \(Y_i\) the outcome. Treatment is assigned by \(D_i=1\{X_i\ge c\}\). Write \(m_+(x)=E[Y_i\mid X_i=x]\) to the right of the cutoff and \(m_-(x)=E[Y_i\mid X_i=x]\) to the left. If these conditional mean functions have well-defined limits at \(c\), the cutoff effect is \(\tau=m_+(c)-m_-(c)\). Under the design's continuity-based identification conditions, this jump identifies the local average treatment effect at the cutoff.

The target is the discontinuity at \(c\), not a treatment effect averaged across the full range of \(X_i\). In practice, each side is approximated separately with a local polynomial. The polynomial order \(p\) and estimation bandwidth \(h\) determine the original estimate, which we denote by \(\hat{\tau}_{p,h}\). For example, \(p=1\) gives a local-linear fit. The earlier material on local polynomial estimation and bandwidth selection motivates these choices; here the focus is inference after a bandwidth has been chosen.

The Bias-Corrected Estimate

A local polynomial of order \(p\) generally has a leading smoothing bias of order \(h^{p+1}\), provided the conditional mean functions have enough derivatives near the cutoff. The precise leading term depends on the derivatives of those functions, the kernel, the polynomial order, and the distribution of observations around the cutoff. A bias estimate can be obtained by fitting a higher-order polynomial, commonly of order \(q=p+1\), and using its estimated derivatives to approximate that leading term.

Let \(\hat{B}_{p,h,q,b}\) denote this estimated leading bias. The bias-estimation fit may use a bandwidth \(b\), which need not equal \(h\). The bias-corrected estimate is

\hat{\tau}_{\mathrm{bc}} = \hat{\tau}_{p,h}-\hat{B}_{p,h,q,b}.

This notation suppresses details that depend on the kernel and implementation. The important features are that the correction estimates the leading bias of the original \(p\)-order estimator, and that the uncertainty in that correction is not treated as zero. Robust bias correction is not simply a claim that the adjusted point estimate must be closer to the truth in every sample. It is an inference procedure that adjusts the estimate and its sampling variability in a coordinated way.

Worked Example: Subtracting an Estimated Leading Bias

Suppose a hypothetical sharp RD analysis studies an outcome measured in test-score points. It uses a local-linear fit (\(p=1\)) with a selected bandwidth \(h\). In this illustrative analysis, 620 observations fall within that bandwidth, and the fitted jump is \(\hat{\tau}_{1,h}=0.42\) points. A quadratic fit (\(q=2\)) estimates the leading bias as \(\hat{B}_{1,h,2,b}=0.12\) points.

The bias-corrected estimate is

\hat{\tau}_{\mathrm{bc}} = 0.42-0.12 = 0.30\text{ test-score points}.

Thus, the corrected estimated discontinuity is 0.30 points. The value 0.12 is an estimate, not a known constant. It would be incorrect to retain the uncorrected estimate's standard error without checking how estimating this correction changes the variance. The next step is therefore to calculate uncertainty for the whole corrected estimator.

Why the Standard Error Must Also Be Corrected

The corrected estimate is a difference between two random quantities: the original local-polynomial estimate and its estimated bias. Its variance is

\operatorname{Var}(\hat{\tau}_{\mathrm{bc}}) = \operatorname{Var}(\hat{\tau}_{p,h}) + \operatorname{Var}(\hat{B}_{p,h,q,b}) - 2\operatorname{Cov}(\hat{\tau}_{p,h},\hat{B}_{p,h,q,b}).

The covariance term matters because the two estimates are generally based on overlapping observations. The covariance can be positive or negative; omitting it does not have a generally correct direction. Nor is it enough to add the variances as if the estimates were independent. The robust bias-corrected standard error estimates the variability of the corrected estimator, including this covariance.

One way to see the calculation is to condition on the observed running-variable values. Local-polynomial estimates are linear in the outcomes when those values and the bandwidths are fixed. Write the original estimator as \(\hat{\tau}_{p,h}=\sum_i w_iY_i\), and the estimated bias as \(\hat{B}_{p,h,q,b}=\sum_i v_iY_i\), where the weights include the signs for the two sides of the cutoff. Then

\hat{\tau}_{\mathrm{bc}} = \sum_i (w_i-v_i)Y_i.

If the outcomes are conditionally independent given the running variables, with conditional variances \(\sigma_i^2=\operatorname{Var}(Y_i\mid X_i)\), the conditional variance of this linear estimator is

\operatorname{Var}(\hat{\tau}_{\mathrm{bc}}\mid X_1,\ldots,X_n) = \sum_i (w_i-v_i)^2\sigma_i^2.

This identity justifies the core of the robust variance calculation: use the combined corrected-estimator weights, not only the original-estimator weights. In applications the \(\sigma_i^2\) values are unknown, so they are estimated from residuals using a heteroskedasticity-robust procedure appropriate to the local-polynomial fit.

Worked Example: Accounting for Bias-Estimation Uncertainty

Continue with the hypothetical score analysis, where \(\hat{\tau}_{\mathrm{bc}}=0.30\) points. Suppose the variance estimates for the original jump and bias estimate are \(0.0064\) and \(0.0036\) points squared, respectively, and their estimated covariance is \(-0.00105\) points squared. These are illustrative variance components from the same fitted analysis.

The variance of the corrected estimate is

\widehat{\operatorname{Var}}(\hat{\tau}_{\mathrm{bc}}) = 0.0064+0.0036-2(-0.00105) = 0.0121\text{ points}^2.

Therefore,

SE_{\mathrm{bc}} = \sqrt{0.0121} = 0.110\text{ points}.

For a 95% normal-approximation interval, use \(1.96\) standard errors on either side of the corrected estimate:

0.30\pm 1.96(0.110) = 0.30\pm 0.2156 = [0.0844,\;0.5156]\text{ points}.

The negative estimated covariance increases the variance in this example, because subtracting twice a negative covariance adds a positive quantity. A calculation that ignored the covariance would give variance \(0.0100\) and standard error \(0.100\), understating the corrected estimator's uncertainty here.

Robust Bias-Corrected Confidence Intervals

Let \(SE_{\mathrm{bc}}\) be the standard error for the bias-corrected estimator. A two-sided, large-sample \(100(1-\alpha)\%\) RBC confidence interval is

\hat{\tau}_{\mathrm{bc}} \pm z_{1-\alpha/2}SE_{\mathrm{bc}},

where \(z_{1-\alpha/2}\) is the corresponding standard-normal quantile. The interval uses both the corrected estimate and its corrected standard error. Using \(\hat{\tau}_{\mathrm{bc}}\) with the uncorrected standard error, or using \(\hat{\tau}_{p,h}\) with the corrected standard error, combines components from different estimators and is not the RBC interval.

Under standard regularity conditions, the studentized corrected estimator is approximately standard normal. The conditions include a correctly specified sharp assignment rule and continuity-based identification at the cutoff; sufficient smoothness of the conditional mean functions on each side; a running-variable density that is positive and sufficiently regular near \(c\); enough observations close to the cutoff; and conditional outcome variation that is finite and nondegenerate. The bandwidths must shrink as the sample grows while retaining an increasing effective local sample, and the remaining bias after correction must be small relative to the corrected standard error. The variance estimator must also consistently estimate the variability of the corrected estimator.

A key practical benefit is that RBC inference can account for leading bias without requiring the estimation bandwidth to be undersmoothed solely to make that bias negligible. It does not remove every possible source of bias, nor does it guarantee accurate finite-sample coverage regardless of the data. The normal approximation and the estimated bias remainder still rely on the stated conditions.

Worked Example: Interpreting an RBC Interval

Consider a second hypothetical sharp RD analysis of an outcome measured in waiting-time days. There are 1,200 observations in the analysis sample; 460 lie within the estimation bandwidth \(h=0.4\) running-variable units, and 810 lie within the larger bandwidth \(b=0.8\) used for the bias fit. A local-linear estimate and quadratic bias fit produce \(\hat{\tau}_{\mathrm{bc}}=0.28\) days, with \(SE_{\mathrm{bc}}=0.10\) days. Suppose the fits use observations on both sides of the cutoff and the design and smoothness assumptions are considered plausible for this illustration.

The 95% RBC interval is

0.28\pm 1.96(0.10) = 0.28\pm 0.196 = [0.084,\;0.476]\text{ days}.

Under the model and regularity conditions supporting the procedure, this interval is a confidence interval for the discontinuity in mean waiting time at the cutoff. Its frequentist interpretation is that the procedure, applied repeatedly to comparable samples, would cover the fixed cutoff effect about 95% of the time in the large-sample approximation. It is not a statement that the parameter has a 95% probability of lying in this particular interval. Because zero is outside the interval, the estimate is inconsistent with a zero discontinuity at the conventional two-sided 5% level under the same approximation.

When Coverage Can Still Fail

Bias correction improves inference only to the extent that the leading bias is well approximated and the corrected standard error is reliable. Coverage can remain poor in several settings:

Too few effective observations: A bandwidth may leave very sparse data near the cutoff, making both the local fit and its variance estimate unstable. A normal approximation can be inaccurate even when the formulas are implemented correctly.
Insufficient smoothness or rapidly changing curvature: If the conditional mean functions do not have the derivatives required by the approximation, or their higher-order behavior changes sharply near the cutoff, the estimated leading bias may not capture the remaining bias.
Bandwidths that do not support the approximation: An excessively wide bandwidth can draw in observations from regions where the local polynomial is a poor approximation. An extremely narrow bandwidth can leave too little information. Bias correction does not make any bandwidth choice automatically valid.
Irregular data or variance estimation: Strong dependence, extreme observations, or an unreliable estimate of local outcome variability can undermine the standard error. The independent-observation variance formula above requires conditional independence; dependent data require inference that reflects the dependence structure.
Failure of the RD design's identification conditions: A precise interval does not repair a discontinuity in potential outcomes unrelated to treatment or an invalid assignment rule. Statistical precision and causal identification are distinct questions.

These limitations make it useful to examine the data and the design, and to assess whether the conclusion is sensitive to reasonable choices of polynomial order and bandwidth. Such checks do not replace the assumptions behind RBC inference; they help reveal when a nominally precise interval depends heavily on fragile local approximations.

Common Mistakes

Subtracting a bias estimate but keeping the old standard error. The bias estimate is random and typically correlated with the original estimate. Use a standard error for the combined corrected estimator.
Ignoring covariance. The variance of a difference includes \(-2\operatorname{Cov}(\hat{\tau},\hat{B})\). Assuming independence without justification can overstate or understate uncertainty.
Calling the correction exact. RBC estimates and removes the leading bias term. Higher-order remainder bias and finite-sample error can remain.
Interpreting a confidence interval as a probability statement about a fixed parameter. The confidence level describes the long-run coverage of the interval procedure under its assumptions, not a probability assigned to the parameter after observing the data.
Equating a narrow interval with credible causal identification. The interval quantifies sampling uncertainty under the design and smoothness conditions; it cannot establish that those conditions hold.

Key takeaway: Robust bias-corrected inference uses a higher-order local-polynomial fit to estimate and subtract leading smoothing bias, then estimates uncertainty using the corrected estimator's full variability, including covariance between the original estimate and the bias estimate. Its coverage remains conditional on adequate data, smoothness, valid identification, and a reliable variance approximation.

Check Your Understanding

Use the definitions and calculations above to answer the following questions.

For a local-linear RD estimate, what order of local polynomial is commonly used to estimate the leading bias, and why is that fit not itself the final treatment-effect estimate?
If the original estimate has variance \(0.010\), the bias estimate has variance \(0.004\), and their covariance is \(0.001\), calculate the variance of their difference.
Why can the covariance between the original estimate and the estimated bias not generally be omitted?
A corrected jump estimate is 0.20 outcome units with RBC standard error 0.15 units. Calculate its approximate 95% confidence interval and state what it implies about a zero discontinuity.
Name two conditions under which an RBC interval may have poor coverage even when the bias-correction formula is calculated correctly.