Bandwidth Selection for Regression Discontinuity

Why Bandwidth Selection Matters

In a regression-discontinuity (RD) design, the estimated effect at the cutoff is built from observations near that cutoff. The bandwidth determines how near: a narrow window uses fewer, more local observations, while a wider window includes more observations at the cost of relying on a less local approximation to the outcome relationship. Choosing the bandwidth therefore changes both the precision and the potential bias of the estimate.

There is no single bandwidth that is best for every purpose. A bandwidth chosen to minimize the estimated point estimator's mean squared error (MSE) need not give a confidence interval with coverage close to its nominal level. This distinction matters in applications: reporting a precise-looking interval does not resolve the problem if the estimate has appreciable bias relative to its standard error.

Setup and Definitions

Let \(X_i\) be the running variable centered so that the cutoff is zero, and let \(Y_i\) be the outcome for unit \(i\). In a sharp RD design, treatment changes from zero to one at the cutoff. The target is the difference between the two conditional outcome limits at zero. Write \(m_+(x)\) and \(m_-(x)\) for the conditional mean functions on the right and left, respectively; the cutoff effect is \(\tau=m_+(0)-m_-(0)\). Identification requires the relevant untreated and treated conditional mean limits to be well defined and continuous at the cutoff, as established by the continuity-based RD framework.

For estimation, fit a polynomial separately on each side using observations with \(|X_i|\) no greater than a chosen bandwidth \(h\), and weight them according to a kernel \(K\). The fitted intercepts at zero are subtracted to estimate \(\tau\). The polynomial degree \(p\) determines how flexible each local fit is: \(p=1\) is local linear and \(p=2\) is local quadratic. The kernel determines how weights vary within the window; \(h\) determines the window's scale.

The standard bias–variance approximations below require a positive, smooth density of \(X_i\) near zero; conditional outcome means that are sufficiently smooth on each side; finite, nonzero conditional outcome variances at the cutoff; and a sampling structure for which the variance estimator and limit approximation are appropriate. For a degree-\(p\) fit, suppose the first omitted local-polynomial term is nonzero, so the leading bias is of order \(h^{p+1}\). If this term vanishes, the leading order and optimal rate can differ. Dependence, clustering, or other sampling features must be reflected in the variance calculation rather than ignored.

How the MSE-Optimal Bandwidth Arises

For a fixed kernel and degree \(p\), the leading bias of the cutoff estimate is typically \(B h^{p+1}\), where \(B\) collects curvature or higher-derivative terms and kernel constants. Its variance is typically \(V/(nh)\), where \(V\) collects variance, density, and kernel constants. Thus a useful first-order approximation is

This expression is not an exact finite-sample risk formula. It makes the tradeoff explicit: increasing \(h\) raises the squared-bias term, while usually reducing the variance term because more observations contribute.

Differentiate the approximation with respect to \(h\), set the derivative to zero, and solve for the positive stationary point:

The approximation decreases toward this point from the variance-dominated region and increases beyond it as squared bias grows, so it gives the MSE-minimizing bandwidth within this approximation. In particular, with a local-linear fit (\(p=1\)), \(h_{\mathrm{MSE}}\) is proportional to \(n^{-1/5}\). This rate describes how the bandwidth shrinks with sample size under the stated smoothness and leading-bias conditions; it does not mean the optimal numerical bandwidth is the same in every setting.

Point-Estimation Error Is Not Interval Coverage

A conventional confidence interval for the cutoff effect is often formed as the estimate plus or minus a normal critical value times its standard error. Its usual nominal-coverage justification requires the estimator's bias to be negligible relative to its standard error, or to be handled by an appropriate inference method. Merely estimating the standard error does not remove bias.

Under the approximations above, the standard error has order \((nh)^{-1/2}\), while the leading bias has order \(h^{p+1}\). Their ratio has order

At the MSE-optimal rate \(h\propto n^{-1/(2p+3)}\), this ratio is generally of constant order rather than tending to zero. Thus, with an ordinary uncorrected interval, bias can remain consequential for coverage even as the sample grows. The interval may then cover the target less often or more often than its stated nominal level. A bandwidth that is attractive for point-estimate MSE is not automatically appropriate for conventional inference.

One basic way to make the leading bias small relative to the standard error is undersmoothing: choose \(h\) so that \(h=o(n^{-1/(2p+3)})\), under the same leading-order conditions. The bias-to-standard-error ratio then tends to zero, while the smaller window increases variance. Undersmoothing is a principle, not a guarantee that any arbitrarily small bandwidth works well in a finite sample: too few observations can make estimates unstable. More generally, bandwidths designed to control coverage error balance finite-sample bias, variance, and approximation error. Their exact choice depends on the inferential procedure and assumptions. In common settings, a conventional coverage-oriented bandwidth is smaller than the point-estimation MSE choice, but that is not a universal ordering.

Kernel and Polynomial Order

The kernel and polynomial order affect the constants in both bias and variance, and therefore affect the numerical bandwidth selected. A uniform kernel assigns equal weight inside its support and zero weight outside. A triangular kernel gives greater weight to observations close to the cutoff and decreases linearly to zero at the bandwidth boundary. With the same \(h\), both use observations within the same distance, but they do not generally produce the same estimate or the same bias and variance constants.

Likewise, changing the degree changes the local approximation. A local-linear fit has leading bias typically of order \(h^2\); a local-quadratic fit has leading bias typically of order \(h^3\). The higher-degree fit can represent more curvature, but estimating additional coefficients can increase variance and can make the result more sensitive to the data near the window boundary. These statements concern the usual leading orders under smoothness conditions; the actual constants depend on the outcome functions, density, variance, and kernel. Consequently, neither “use the narrowest window” nor “use the highest polynomial” is a general selection rule.

Practical Selection and Common Pitfalls

A disciplined analysis makes the selection target explicit before reporting a bandwidth. If the principal target is the point estimate, an MSE-oriented selector is relevant. If the principal target is a conventional confidence interval, inspect whether the bandwidth and inferential method address leading bias relative to the standard error. Do not silently treat one selection criterion as a substitute for the other.

The bandwidth is measured in the units of the running variable. If that variable is rescaled, the numerical value of \(h\) changes correspondingly, even though the substantive window need not change. The kernel also matters: bandwidth comparisons are most interpretable when the kernel, polynomial degree, and other estimation choices are held fixed. Researchers commonly report sensitivity to reasonable nearby bandwidths and alternative local specifications, while avoiding a search for the specification that happens to produce a preferred result.

• State the target. Say whether the bandwidth is chosen for point-estimation MSE or for inference and coverage considerations.
• Keep choices distinct. Report the bandwidth, kernel, polynomial degree, and interval procedure; none is a substitute for the others.
• Check the local approximation. A large window may include regions where a low-degree polynomial poorly represents the conditional mean; a very small window may leave substantial sampling variability.
• Interpret intervals cautiously. A nominal level describes a repeated-sampling coverage goal under its assumptions, not a probability that a fixed parameter lies in the particular computed interval.

Common mistake: “The MSE selector chose this bandwidth, so the usual 95% confidence interval must have 95% coverage.” The reasoning fails because the MSE criterion trades bias against variance for the point estimate, while conventional interval coverage also depends on bias being negligible relative to the standard error or otherwise accounted for. The correction is to evaluate bandwidth choice and inference together, using a method whose coverage justification matches the bias behavior of the estimator.