Kernel Weights In Regression Discontinuity

Why Weight Observations Near the Cutoff?

A regression discontinuity (RD) estimate targets the difference between the outcome limits immediately to the left and right of a cutoff. Local polynomial estimation uses observations within a chosen neighborhood to approximate those limits. But observations inside that neighborhood need not contribute equally: an observation close to the cutoff may be more informative about the cutoff limit than one near the neighborhood’s edge. A kernel specifies how those observations are weighted.

Kernel choice is a choice about relative weights, not a replacement for the RD identification conditions. The running variable must be measured relative to the cutoff, and the outcome regression must be estimated separately on each side when fitting the usual sharp RD local linear specification. The continuity conditions underlying the RD interpretation remain necessary. A kernel cannot repair a failure of identification, a misspecified design, or a lack of observations near the cutoff.

The bandwidth \(h\) sets the neighborhood’s scale: with a compact-support kernel, it determines which observations can receive positive weight. The kernel’s shape determines how weights vary within that neighborhood. These choices are related, because changing \(h\) changes every normalized distance \(u_i\), but they are conceptually distinct.

Common Kernel Functions

Three widely used kernels are uniform (also called rectangular), triangular, and Epanechnikov. The definitions below use the conventional normalizations for which the kernel integrates to one. In weighted least squares, multiplying every weight by the same positive constant does not change the fitted coefficients, so other normalizations can yield the same regression estimates.

Here \(\mathbf{1}(\cdot)\) equals one when its condition is true and zero otherwise. The uniform kernel gives equal weight to every observation in the window. The triangular kernel gives the greatest weight at the cutoff and decreases linearly to zero at the edges. The Epanechnikov kernel also gives the greatest weight at the cutoff, but decreases according to a quadratic curve. All three assign zero weight outside the window.

For RD, kernel weights are commonly used in separate local polynomial regressions to the left and right of \(c\). In a local linear fit, each side estimates an intercept and a slope by weighted least squares. The RD estimate is the fitted right-side intercept minus the fitted left-side intercept. The intercepts represent the fitted outcome limits at \(X_i=c\); they are not generally the observed outcomes of units exactly at the cutoff.

In this expression, \(s_i=+\) indicates an observation on the right of the cutoff and \(s_i=-\) an observation on the left. The fitted value at the cutoff corresponds to \(u_i=0\), so the intercept \(a\) is the relevant side-specific limit. This formulation uses normalized distance \(u_i\) as the slope variable; using \(X_i-c\) instead changes the slope’s scale but not the fitted intercept at the cutoff.

How Kernel Weights Affect a Fit

The next calculation isolates the consequence of changing kernel shape. It uses a local constant weighted average, not the usual local linear RD estimator. A local constant fit does not include a slope, so it can respond more strongly to changing the relative weights. The example helps make the weighting mechanism visible; it should not be taken as a substitute for the local linear specification.

A useful algebraic fact is that a common rescaling of all weights within a fit does not affect the estimated coefficients. If every weight \(w_i\) is replaced by \(q w_i\), where \(q>0\), the weighted least-squares objective is multiplied by \(q\), leaving its minimizer unchanged:

This is why the conventional factor \(1/2\) in the uniform kernel, for example, does not affect the regression estimate. It is the pattern of relative weights that matters, not their common scale.

Why Kernel Choice Is Often Less Consequential Than Bandwidth

Within a fixed window, the kernels above differ in how they emphasize the center and edge. Changing the bandwidth, however, changes the window itself and therefore changes which observations can contribute at all. It also changes the normalized distance of every included observation. As a result, bandwidth changes can alter both the amount and the distribution of information used to estimate each cutoff limit.

For standard local linear RD estimation, common kernel choices often produce estimates with similar behavior when used with the same bandwidth and adequate data near the cutoff. This is a practical tendency, not a guarantee that estimates will be identical. Kernel shapes can differ in finite samples, and the impact can be substantial when observations are sparse, influential outcomes lie near the window edge, or the fitted regression is unstable. Theoretical precision can also depend on the kernel through constants, even when broad asymptotic properties are similar.

This example also clarifies why a difference between kernel estimates is not, by itself, evidence that one kernel is invalid. The estimates summarize different weighted fits to the available data. A pronounced change across reasonable kernels may instead be a warning to inspect support, influential observations, bandwidth, and model stability.

Diagnostics, Alternatives, and Limitations

After fitting a kernel-weighted RD model, inspect the data and the fit separately on each side. Check that there are observations close to the cutoff, that the weighted design matrix can identify the requested polynomial, and that the result is not being driven by one or two observations. A fit may be numerically computable but practically unstable if most effective information comes from very few units.

One descriptive diagnostic is the effective sample size of the nonnegative weights on a side:

It equals the number of observations when all positive weights are equal and decreases when weight is concentrated on a few observations. It is only a summary of weight concentration, not a substitute for checking the running-variable support or the regression’s uncertainty. Report the kernel and bandwidth so that the weighting rule is reproducible.

The uniform kernel is equivalent to giving equal weight to all observations inside the chosen window. It is a useful baseline, but observations near the window boundary can have substantial leverage. Triangular weighting downweights those observations and is a common choice for local linear RD. Epanechnikov weighting offers another smooth decline. No kernel eliminates the need to assess the fit itself.

A global polynomial fit is an alternative modeling strategy, not merely a different kernel: it uses a broader range of running-variable values and imposes a functional form over that range. Local kernel weighting instead emphasizes observations near the cutoff. A local constant fit is another alternative within the window, but unlike local linear regression it does not account for a local slope. These approaches answer the same cutoff question only under appropriate modeling and identification conditions; they should not be treated as interchangeable by default.

Common warning signs include very few positive-weight observations on one side, a nearly singular weighted regression design, a large shift when a single observation is removed, and sharply different estimates across kernels at the same bandwidth. Kernel choice is also constrained by the chosen support convention: for triangular and Epanechnikov kernels, an observation exactly at \(|u|=1\) receives zero weight, while the stated uniform convention gives it positive weight. With continuous running variables, exact boundary observations are typically uncommon, but the convention can matter in finite samples.

The practical lesson is measured: select and report a defensible bandwidth, use a standard kernel, and check whether conclusions are sensitive to reasonable alternatives. Kernel weighting changes how the local data contribute; it does not establish continuity, prevent manipulation, or guarantee an unbiased estimate. Those questions require separate design and diagnostic reasoning.