Why Correct Smoothing Bias?
A local polynomial regression discontinuity (RD) estimate fits separate outcome curves on either side of a cutoff and compares their fitted limits at that cutoff. The fit is local, not exact: observations away from the cutoff help determine the curve, and curvature can cause the fitted intercept to differ systematically from the true limit. This smoothing bias can matter even when the estimate’s sampling variability is small. Bias correction estimates the leading part of that discrepancy and subtracts it.
Let \(R\) be the running variable, \(c\) the cutoff, and \(Y\) the outcome. Write \(m_+(r)\) and \(m_-(r)\) for the conditional mean outcome functions on the right and left of the cutoff. Under the RD continuity and assignment conditions, the cutoff effect is the jump \(\tau=m_+(c)-m_-(c)\). The discussion below concerns the finite-sample approximation error in estimating this jump, not a change to its causal interpretation.
Where the Leading Bias Comes From
For polynomial order \(p\) and bandwidth \(h\), let \(u=(R-c)/h\), and let \(r_p(u)=(1,u,\ldots,u^p)'\). On each side \(s\in\{+,-\}\), fit a weighted polynomial by minimizing the sum of \(K(u_i)[Y_i-r_p(u_i)'\beta_s]^2\) over observations within bandwidth \(h\) of the cutoff on that side. Here \(K\) is a kernel weight function, and \(\beta_s\) is the vector of fitted coefficients. The first coefficient estimates the outcome limit on that side. Their difference, \(\hat\tau_p(h)\), estimates \(\tau\).
Suppose each side’s conditional mean has \(p+1\) continuous derivatives near \(c\), the running-variable density is positive there, and the kernel-weighted design matrices are nonsingular. A Taylor expansion gives the leading bias of each fitted intercept. Define \(I_s=[0,1]\) for the right side and \(I_s=[-1,0]\) for the left side when the kernel is supported on \([-1,1]\), and set \( \Gamma_s=\int_{I_s}K(u)r_p(u)r_p(u)'\,du \) and \( \Lambda_s=\int_{I_s}K(u)r_p(u)u^{p+1}\,du \). If \(e_1=(1,0,\ldots,0)'\), the leading bias term for the jump is
The factor \(h^{p+1}\) shows why bias generally declines as the bandwidth shrinks. It also shows why the derivative of order \(p+1\), which is not included in a degree-\(p\) fit, determines the leading bias. The two sides’ contributions must be differenced; curvature on the two sides can reinforce or offset one another.
For local linear estimation, \(p=1\). With a uniform kernel and the one-sided intervals above, \(e_1'\Gamma_s^{-1}\Lambda_s=-1/6\) on either side. Therefore
This expression is specific to the stated kernel and local linear fit. Different kernels or polynomial orders change the constants. The general expansion explains the structure of the bias; the correction must use constants corresponding to the actual estimator.
Worked Example: Calculating the Leading Local Linear Bias
Suppose a hypothetical analysis uses a uniform kernel and local linear fits with bandwidth \(h=2\) running-variable units. Assume the estimated second derivatives at the cutoff are \(m_+''(c)=0.8\) outcome units per squared running-variable unit and \(m_-''(c)=-0.4\) in the same units. The uncorrected estimated jump is \(5.2\) outcome units.
For these choices, the leading bias estimate is \( \hat B_1(2)=-(2^2/12)[0.8-(-0.4)] \). The derivative difference is \(1.2\), so \( \hat B_1(2)=-(4/12)(1.2)=-0.400 \) outcome units.
The uncorrected estimate’s leading bias is estimated to be negative. Subtracting it gives \( \hat\tau_{\mathrm{bc}}=5.2-(-0.400)=5.60 \) outcome units. This correction moves the estimate upward because the estimated smoothing bias was downward. It does not establish that the true bias is exactly \(-0.400\); it estimates only the leading term under the local approximation and derivative estimates.
Estimating and Subtracting the Bias
The derivative in the leading-bias formula is unknown. Estimate it with a polynomial of order \(q=p+1\), which includes the required derivative, using a bandwidth \(b\). If \(\hat\gamma_{s,q}\) is the coefficient vector from a fit in the scaled coordinate \(v=(R-c)/b\), then the estimated derivative of order \(q\) at the cutoff is \(q!\hat\gamma_{s,q}/b^q\). Substituting these derivatives into the leading-bias formula gives \(\hat B_p(h,b)\), the estimated bias of the original degree-\(p\) fit. The corrected estimate is
The original bandwidth \(h\) determines the estimate being corrected; \(b\) determines how the derivative used in the correction is estimated. They need not be the same. The higher-order fit is not simply a replacement estimate: its role here is to estimate the omitted derivative that governs the lower-order fit’s leading bias.
Worked Example: Obtaining a Correction From a Quadratic Fit
Continue with a hypothetical local linear RD estimate using the uniform-kernel bias constant. Let \(h=1.5\) running-variable units and let the quadratic derivative-estimation bandwidth be \(b=2\) units. Suppose the scaled-coordinate quadratic coefficients are \(\hat\gamma_{+,2}=1.2\) and \(\hat\gamma_{-,2}=0.4\), in outcome units. For a quadratic fit, the second derivative estimate is \(2!\hat\gamma_{s,2}/b^2\).
On the right, this gives \(2(1.2)/2^2=0.600\) outcome units per squared running-variable unit. On the left, it gives \(2(0.4)/2^2=0.200\). The estimated derivative difference is \(0.400\), so \( \hat B_1(1.5,2)=-(1.5^2/12)(0.400)=-0.0750 \) outcome units.
If the uncorrected estimated jump is \(2.400\) outcome units, then \( \hat\tau_{\mathrm{bc}}=2.400-(-0.0750)=2.475 \) outcome units. The bandwidth \(b\) is accounted for when converting the scaled quadratic coefficient to a derivative; omitting the division by \(b^2\) would give a derivative in the wrong scale and a wrong correction.
Why the Correction Changes Uncertainty
The correction is estimated from the same outcomes and nearby observations as the original RD fit. Consequently, the corrected estimate is random not only because \(\hat\tau_p(h)\) varies across samples, but also because \(\hat B_p(h,b)\) varies. In general,
Thus, it is not generally correct to retain the original estimate’s standard error, nor to add the two variances while ignoring their covariance. The covariance term can be substantial because both quantities depend on overlapping data. In a local-polynomial formulation, the corrected estimate can be represented as a weighted sum of outcomes with adjusted observation weights. Its variance must reflect those adjusted weights and the outcome variability, including dependence between the original fit and the bias estimate.
Bias correction does not always increase the standard error. The estimated correction adds variability, but positive covariance with the original estimate reduces the variance of their difference; negative covariance increases it. A valid uncertainty calculation therefore uses the corrected estimator’s variance, rather than treating the estimated bias as a known constant. Robust bias-corrected inference develops this calculation and its inferential implications in detail.
Worked Example: Accounting for Covariance
Suppose, hypothetically, that the standard error of an uncorrected estimate is \(0.300\) outcome units, the standard error of its estimated bias is \(0.180\) outcome units, and their covariance is \(0.0360\) squared outcome units. The corrected estimate is the first quantity minus the second, so its estimated variance is \(0.300^2+0.180^2-2(0.0360)=0.0900+0.0324-0.0720=0.0504\) squared outcome units.
The corresponding standard error is \(\sqrt{0.0504}=0.2245\) outcome units. It is smaller than \(0.300\) in this example because the positive covariance is large enough to reduce the variability of the difference. If the covariance were zero, the variance would instead be \(0.1224\) and the standard error would be \(0.3499\). These calculations illustrate why neither “the correction always increases uncertainty” nor “the original standard error is still appropriate” is generally valid.
Assumptions, Diagnostics, and Alternatives
The correction relies on a local approximation, not on a guarantee that all bias has been removed. The conditional mean functions must be sufficiently smooth for the Taylor expansion to be informative, the observations near the cutoff must provide support on both sides, and the local polynomial designs must be estimable. The derivative estimates also need adequate local information. In asymptotic settings, \(h\) and \(b\) shrink as the sample grows while the effective numbers of observations in those neighborhoods grow; derivative estimation additionally requires enough data for the order-\(p+1\) fit. The exact bandwidth conditions depend on the estimator and design.
Useful diagnostics include inspecting the observations and fit near the cutoff, checking whether each side has enough distinct running-variable values to support the chosen polynomial, and examining how the estimated derivatives and correction change under reasonable bandwidth changes. A correction that changes dramatically with small bandwidth adjustments, is driven by a few observations, or has an unstable higher-order coefficient is a warning that the local derivative is poorly determined. Sparse support, severe curvature changes, or an inadequate smoothness approximation can make the leading-bias formula unreliable. A small corrected estimate is not itself evidence that these issues have been resolved.
Three approaches address smoothing bias differently. The uncorrected degree-\(p\) estimate is simplest, but its leading bias may be important for inference. Undersmoothing uses a smaller bandwidth to reduce bias relative to sampling variability, at the cost of using fewer observations and potentially increasing variance. A higher-order polynomial can reduce the order of approximation error, but may be more sensitive to the data and fit poorly near the boundary. Bias correction retains the original target fit and estimates its leading bias using a higher-order fit; its limitation is that the correction itself can be noisy and depends on derivative estimation and smoothness. These approaches are not interchangeable without considering their assumptions and uncertainty calculations.
The central point is that bias correction targets a particular leading term, not every source of error. Its value depends on whether the local polynomial approximation is credible and whether the derivative is estimated precisely enough. The correction and its uncertainty should therefore be considered together.
Check Your Understanding
Try each question, then compare your reasoning with the solutions.
- For a local linear fit, why does the second derivative enter the leading bias?
- With the uniform-kernel local linear formula, suppose \(h=1\), \(m_+''(c)=0.5\), and \(m_-''(c)=0.1\). What is the estimated leading bias?
- Why does a quadratic fit in the scaled coordinate require division by \(b^2\) to estimate a second derivative?
- If the original estimate and bias estimate have zero covariance, what is the variance of their difference?
- Give one diagnostic sign that a bias correction may be unreliable.
Solutions. (1) A degree-one fit omits the quadratic term in the Taylor expansion, whose coefficient is determined by the second derivative; its leading contribution is of order \(h^2\). (2) \( -(1^2/12)(0.5-0.1)=-0.0333 \) outcome units. (3) The fitted coefficient multiplies \(((R-c)/b)^2\); converting the coefficient to a derivative with respect to \(R\) introduces \(2!/b^2\). (4) It is the sum of the two variances, since \(\operatorname{Var}(A-B)=\operatorname{Var}(A)+\operatorname{Var}(B)\) when \(\operatorname{Cov}(A,B)=0\). (5) For example, an estimated derivative or correction that changes sharply with modest changes in \(b\) suggests unstable local derivative estimation.