Bandwidth Bias-Variance Tradeoffs

Why Bandwidth Matters

A regression discontinuity (RD) estimate is local: it compares outcomes just below and just above a cutoff \(c\). In local polynomial estimation, the bandwidth \(h\) determines how far from \(c\) observations may contribute. A narrow window focuses on observations close to the cutoff, where the conditional mean is more plausibly approximated by a simple local trend. But it uses fewer observations and is therefore less precise. A wide window usually provides more information for estimating that trend, while relying on observations farther from the cutoff, where curvature can matter more.

The bandwidth is therefore not just a computational setting. It changes the balance between approximation error and sampling variability. This tutorial develops that balance for local linear RD estimation. The main point is that the leading bias depends on the curvature on each side and on that side’s design and weights. The constants for the two sides need not be equal.

Local Linear Estimation and Side-Specific Bias

Let \(Y\) be the outcome and \(R\) the running variable. Write \(m_+(r)=E[Y\mid R=r]\) for the conditional mean on the right of the cutoff and \(m_-(r)=E[Y\mid R=r]\) for the mean on the left. A local linear fit uses a separate weighted regression on each side, with an intercept and a slope in \(u=(R-c)/h\). The RD estimate is the right-side fitted intercept minus the left-side fitted intercept.

For the bias expression below, suppose each side’s conditional mean has a twice-continuously differentiable expansion near \(c\), the running-variable density has a positive finite limit at the cutoff, and the local weighted design matrices are nonsingular. Assume also finite conditional outcome variances near the cutoff. For the asymptotic approximations below, take \(h\to0\) and \(nh\to\infty\), with a fixed kernel and rescaled side supports, uniformly well-behaved density and conditional moments near \(c\), and nonsingular limiting weighted design matrices. The kernel and the side-specific supports must yield finite moments used in the formulas. These conditions justify the local Taylor expansion and the large-sample approximations; they do not by themselves establish the causal identification assumptions for an RD design.

For side \(s\in\{+,-\}\), let \(r(u)=(1,u)^\mathsf{T}\), and let \(\mathcal U_s\) denote the kernel-weighted support on that side. Define the moment matrix and curvature-moment vector by

The constants \(A_+\) and \(A_-\) summarize how the kernel, the local support, and the linear fit combine to weight curvature when estimating each side’s intercept. To see their role, expand the conditional mean on side \(s\): \(m_s(c+hu)=m_s(c)+m_s'(c)hu+\tfrac12m_s''(c)h^2u^2+o(h^2)\). The first two terms are represented by the local linear regression. The quadratic term is not, and its contribution to the fitted intercept is \(\tfrac12h^2 A_s m_s''(c)\). Consequently, the leading bias of the RD contrast is

This is a contrast of two side-specific biases, not automatically a common constant times the difference in curvatures. Even if the same kernel function is used on both sides, the side supports or the effective weighted designs can differ. A common \(A\) is valid only when an additional design condition makes \(A_+=A_-\), such as reflected side supports with a symmetric kernel.

How Bandwidth Changes Bias and Variance

The leading contrast bias is generally \(O(h^2)\); it is of order \(h^2\) when \(B=\tfrac12\{A_+m_+''(c)-A_-m_-''(c)\}\ne0\). Reducing \(h\) shrinks this approximation error quadratically in its leading term. At the same time, the expected number of observations in a window is proportional to \(nh\), where \(n\) is the total sample size and the density near the cutoff is positive. Under the regularity conditions above, the variance of each local intercept, and hence of the contrast, is of order \(1/(nh)\). A narrower window therefore tends to increase sampling variability.

More explicitly, if the conditional variances have finite positive limits \(\sigma_+^2(c)\) and \(\sigma_-^2(c)\), the leading variance has the form \(V/(nh)\), where \(V\) collects the density, kernel, side-specific moment matrices, and conditional variances. For example, a side’s contribution is proportional to \(\sigma_s^2(c)e_1^\mathsf{T}S_s^{-1}Q_sS_s^{-1}e_1/[f(c)]\), where \(Q_s=\int_{\mathcal U_s}K(u)^2r(u)r(u)^\mathsf{T}\,du\) and \(f(c)\) is the running-variable density at the cutoff. The two side contributions add to the variance of the difference when their observations are disjoint. Thus, for a fixed design and kernel, \(V\) is a positive constant that does not depend on \(h\) to leading order.

Writing \(B=\tfrac12\{A_+m_+''(c)-A_-m_-''(c)\}\), the leading mean squared error (MSE)—the variance plus squared bias—is approximately

The first term grows with \(h\), while the second generally falls. This approximation explains why neither “always use the narrowest window” nor “use as many observations as possible” is a sound general rule. It also shows why curvature matters: larger \(|B|\) makes the bias cost of a given bandwidth more consequential. The value of \(B\) can be small because the curvatures are small, because the side constants differ in a way that offsets them, or both; it should not be assumed to be zero.

Curvature, Sample Size, and the Balance Point

The basic rate implied by the MSE approximation can be derived directly. Treat \(B\) and \(V\) as fixed positive constants and differentiate the leading MSE with respect to \(h\):

Setting this derivative to zero and solving for the positive bandwidth gives

This is the minimizer of the stated leading approximation when \(B\ne0\). It shows that more curvature (larger \(|B|\)) pushes the balance toward a smaller bandwidth, while greater variance or a smaller sample pushes it toward a wider bandwidth to retain enough observations. As \(n\) grows, this approximation gives a bandwidth proportional to \(n^{-1/5}\). The effective local sample size, proportional to \(nh\), still grows at rate \(n^{4/5}\). The bandwidth shrinks slowly with sample size because increasingly precise data permit a more local fit while retaining many observations.

This derivation is a guide to the structure of the tradeoff, not a complete recipe for choosing a bandwidth in practice. In particular, \(B\) depends on unknown curvature and side-specific design constants, while \(V\) depends on unknown variance and the design. When \(B=0\), the displayed leading bias term vanishes and this particular minimization does not apply; higher-order terms may then determine the tradeoff. In ordinary applications, one should not infer zero bias simply because an estimated contrast of curvatures happens to be near zero.

Interpreting Narrow and Wide RD Estimates

A narrow and a wide window may produce different point estimates, standard errors, and confidence intervals. The narrower estimate can be more sensitive to a few observations and typically has greater sampling uncertainty. The wider estimate may be more stable numerically but more exposed to departures from a linear approximation near the cutoff. Neither estimate is automatically more credible solely because its standard error is smaller or because it uses observations closer to the cutoff.

For example, suppose hypothetical local linear fits produce an RD contrast of 4.8 outcome units at \(h=5\) and 6.1 outcome units at \(h=10\). If the leading bias coefficient were estimated as \(B=-0.00833\) outcome units per squared running-variable unit, the corresponding leading bias approximations would be \(-0.00833(5^2)=-0.208\) and \(-0.00833(10^2)=-0.833\) outcome units. These calculations illustrate how the same curvature-related component scales with bandwidth; they do not establish which point estimate is closer to the causal effect. The estimates also vary because of sampling noise, and the bias coefficient itself is uncertain.

The bandwidth tradeoff matters because an RD estimate targets a local effect at the cutoff, while finite-sample estimation uses observations around it. The choice of window governs how that local target is approximated and how precisely it is estimated. A responsible comparison considers both the point estimates and their uncertainty, as well as whether a wider window may be relying on a less accurate local approximation. Bandwidth choice cannot repair a failure of the RD identification assumptions.

Common Pitfalls

• Using one bias constant without checking the designs. The same kernel name does not guarantee \(A_+=A_-\). Compare side supports and weighted moment matrices; use the separate constants unless symmetry or another condition justifies equality.
• Calling a smaller standard error evidence of less total error. A wide window can reduce variance while increasing bias. Standard errors describe sampling uncertainty under the fitted procedure; they do not automatically account for smoothing bias.
• Assuming curvature bias must cancel. Equal curvatures cancel from the leading contrast only if the side constants also match. Conversely, differing curvatures do not by themselves determine the sign of contrast bias; the constants matter too.
• Treating a bandwidth comparison as a causal validity test. Similar estimates across windows can be reassuring about sensitivity to this choice, but agreement does not prove continuity or the other assumptions needed for RD identification.
• Using the \(n^{-1/5}\) rate as a universal bandwidth rule. It follows from a particular local-linear leading-MSE approximation with nonzero leading bias. It is not a fixed numerical recommendation and does not replace estimating the relevant quantities.