Pooling Multiple Regression Discontinuity Cutoffs

Why Pooling Cutoffs Requires a Target

A program may use different eligibility thresholds in different regions, years, or institutions. Each threshold can support a local regression-discontinuity comparison, but those comparisons need not estimate the same effect. Treatment effects, compliance, populations, and the density of observations near the threshold may all differ. Consequently, combining cutoff-specific estimates is not just a way to increase sample size: the pooling rule determines which combination of local effects the result represents.

This tutorial considers fuzzy regression discontinuity, where crossing a threshold changes the probability of receiving treatment rather than determining treatment perfectly. The same pooling distinctions apply to sharp designs, with the treatment probability jump equal to one at each cutoff. The introductory material on fuzzy regression discontinuity establishes how a single cutoff can identify a local effect; here the central issue is how multiple such local effects can be combined.

Identification Comes First at Every Cutoff

Pooling cannot repair a cutoff that does not identify a credible local effect. At each cutoff, the relevant assumptions must be plausible in the neighborhood used for estimation. In particular, the potential outcome and treatment-receipt relationships must be continuous through the threshold absent the eligibility change; individuals must not precisely manipulate their position around the threshold in a way that undermines the comparison; and the threshold must produce a nonzero discontinuity in treatment receipt. For the fuzzy-RD Wald ratio to have a local complier interpretation, eligibility must affect the outcome only through treatment and must not move treatment receipt in opposite directions for different individuals (monotonicity).

These assumptions are cutoff-specific. Evidence against sorting or continuity at one threshold is not cured by apparently clean diagnostics at another. Likewise, the first stage may be strong at one cutoff and weak at another. If all cutoff-specific effects are to be interpreted as effects for local compliers, the complier groups may differ across cutoffs. Their effects can be averaged, but that average is not automatically the effect for a single common group.

Before combining estimates, orient each cutoff consistently: “above” should represent eligibility at every cutoff. If one institution assigns eligibility below its threshold, redefine its indicator or reverse its side labels so the first-stage and outcome jumps use the same direction. Otherwise a negative jump may be a coding convention rather than evidence of a negative treatment effect.

Two Different Ways to Pool

One approach is to estimate each \(\tau_j\) and then take a weighted average. For prespecified nonnegative weights \(q_j\) that sum to one, the target is

The weights define the target: equal weights give each cutoff equal importance, while population-share weights target an average that gives more importance to cutoffs representing more of a specified population. Precision weights can be useful for summarizing estimates, but they generally target a precision-weighted combination, not necessarily a policy population. State the weights and their rationale rather than calling the result simply “the pooled effect.”

A different procedure pools the outcome and treatment jumps and takes their ratio. For nonnegative cutoff weights \(a_j\), this pooled Wald estimand is

provided the denominator is nonzero. This equality follows by substituting \(\Delta_{Yj}=\Delta_{Dj}\tau_j\) into the numerator. When all first-stage jumps are positive, the pooled Wald ratio is an average of the cutoff-specific effects with weights proportional to \(a_j\Delta_{Dj}\). Thus its effective weights depend on both the chosen pooling weights and the strength of the first stage. It is not generally the same as an average using \(a_j\) alone.

Normalization and the Meaning of a Common Neighborhood

For each cutoff, define a centered running variable \(X_{ij}=R_{ij}-c_j\). Centering makes the threshold zero, and positive rescaling, such as \(X_{ij}^{*}=X_{ij}/s_j\) for \(s_j>0\), also preserves which observations are above and below. In a correctly specified local linear model, this rescaling changes the slope’s units but not the fitted intercept at the threshold. However, it can change which observations enter a local fit and how they are weighted.

For example, a common bandwidth of \(0.5\) in \(X^*\) corresponds to a raw running-variable window of \(0.5s_j\) at cutoff \(j\). If scales differ, the same normalized bandwidth uses different raw windows. That may be a sensible choice when the scales have different meanings or spreads, but it is a substantive design decision, not a harmless formatting change. Conversely, using the same raw bandwidth at every cutoff may cover very different portions of the local score distributions.

Uncertainty for a Pooled Estimate

Uncertainty must reflect the estimator actually reported. For a weighted average with fixed weights \(q_j\), its variance is \(\sum_j\sum_k q_jq_k\operatorname{Cov}(\hat\tau_j,\hat\tau_k)\). If estimates use disjoint, independent samples, the cross-covariances may be zero; if observations overlap or share shocks, they should not be omitted without justification. A confidence interval based on this standard error describes sampling uncertainty for the specified weighted estimand, subject to the validity of the cutoff estimates and variance procedure.

For the pooled Wald ratio, uncertainty is not obtained by averaging the standard errors of the cutoff-specific ratios. Let \(\hat N=\sum_j a_j\widehat{\Delta}_{Yj}\), \(\hat F=\sum_j a_j\widehat{\Delta}_{Dj}\), and \(\hat\tau_{\mathrm W}=\hat N/\hat F\). A first-order delta-method approximation uses the variability of \(\hat N-\hat\tau_{\mathrm W}\hat F\):

This expression includes covariance between outcome and first-stage jumps, as well as any dependence across cutoffs. When the denominator is small or poorly estimated, a conventional ratio interval can be unreliable; weak first-stage diagnostics and an inference method appropriate for uncertain denominators are then important. A large sample at one cutoff does not by itself guarantee a strong pooled first stage if the weighted treatment jumps are small or offsetting.

Diagnostics, Alternatives, and Limitations

A transparent analysis should show cutoff-specific estimates before presenting a pooled summary. Review each cutoff’s first-stage magnitude and uncertainty, outcome jump, effective sample size, bandwidth and support, and the sorting, covariate-continuity, and placebo diagnostics relevant to the design. Inspect the contribution \(a_j\widehat{\Delta}_{Dj}/\sum_k a_k\widehat{\Delta}_{Dk}\) for a pooled Wald ratio. A cutoff with an unusually large contribution can dominate the result even if the stated \(a_j\) weights appear balanced. Re-estimating after omitting one cutoff is a useful influence check, not a substitute for a prespecified target.

The main alternatives answer different questions. Reporting separate cutoff-specific estimates preserves heterogeneity and avoids imposing a common effect, though it may be less concise. A weighted average is appropriate when its weights match a clearly defined target. A pooled Wald ratio may be natural when combining weighted reduced-form and first-stage jumps, but it implicitly weights treatment effects by first-stage strength as well as by \(a_j\). A stacked regression with cutoff-specific treatment jumps can retain heterogeneity; a model that constrains those jumps or effects to be common gains precision only by adding a substantive homogeneity restriction. When that restriction is doubtful, report heterogeneity rather than hiding it in a single coefficient.

Pooling also has limits. Cutoff-specific local effects generally apply to different marginal populations, so even a carefully weighted average may not represent an effect for a broad population far from the thresholds. Different outcome definitions, treatment versions, eligibility rules, or follow-up periods can make the effects incomparable. Normalization can align the numerical scale of running variables but cannot make underlying populations or interventions identical. If treatment jumps have opposite signs after eligibility orientation, the simple positive-weight interpretation of the pooled Wald ratio fails and the design and coding require investigation.

A Common Pooling Mistake

A common mistake is to say, “We weighted the cutoffs by sample size, so the pooled Wald ratio is the sample-size-weighted average treatment effect.” That conclusion ignores the first-stage jumps. The ratio’s effect weights are proportional to sample-size weights multiplied by treatment-probability jumps. In the worked example, sample-size weights of 100 and 300 gave an effect average of 5.50, while the pooled Wald ratio was 5.20 because the corresponding effective weights were 0.40 and 0.60. The correction is to calculate and report the implied weights—or directly state that the estimand is the ratio of weighted jumps.