Pooling Effects Across Multiple RD Cutoffs

Why Pooling Requires a Target

A regression-discontinuity (RD) analysis estimates an effect at a particular threshold, for the units whose running-variable values are near that threshold. If several sites or programs use different thresholds, their local effects need not be the same, nor need they describe the same population. Combining estimates can be useful, but the result answers a well-defined question only when the combination rule—and therefore its target—is made explicit.

The central distinction is between estimating effects at several cutoffs and defining one summary of those effects. A pooled estimate does not automatically become the average treatment effect for everyone in the sites. It is generally a weighted summary of cutoff-specific local effects. Interpreting it more broadly requires additional assumptions about how those local effects relate to effects away from the thresholds.

Definition: Let $j=1,\ldots,J$ index RD cutoffs. Write $\tau_j$ for the causal effect identified locally at cutoff $j$, under the assumptions of the relevant RD design. For weights $w_j\geq 0$ satisfying $\sum_{j=1}^J w_j=1$, the pooled estimand is $$ \tau_{\mathrm{pool}}=\sum_{j=1}^J w_j\tau_j. $$ The weights define the target summary. Unless further assumptions are justified, this is a weighted average of cutoff-specific local effects, not an average treatment effect for all units in the sites.

What Each Cutoff-Specific Effect Represents

Let $R$ be the running variable, $c_j$ the cutoff at site $j$, $D$ treatment receipt, and $Y$ the outcome. In a sharp RD design, treatment changes deterministically at the threshold. With $Y(d)$ denoting the potential outcome under treatment status $d$, the local effect is the difference between the limiting mean potential outcomes on the two sides of $c_j$:

\tau_j^{\mathrm{sharp}} = \lim_{r\to c_j^+} E[Y\mid R=r] - \lim_{r\to c_j^-} E[Y\mid R=r].

This jump identifies a causal effect at the threshold when, in particular, the relevant conditional means of potential outcomes are continuous there and units cannot precisely manipulate which side of the cutoff they occupy. Those assumptions are assessed cutoff by cutoff; evidence supporting them at one site does not establish them at another.

In a fuzzy RD design, crossing the threshold changes the probability of treatment rather than assigning treatment perfectly. Define the reduced-form outcome jump $\Delta Y_j$ and the treatment-probability jump $\Delta D_j$ using right-hand limit minus left-hand limit. When the treatment jump is nonzero, the local Wald estimand is:

\tau_j^{\mathrm{fuzzy}}=\frac{\Delta Y_j}{\Delta D_j}.

Under the fuzzy-RD identification conditions—including continuity at the threshold, a nonzero first stage, and the exclusion and monotonicity conditions—the ratio identifies a local average treatment effect for compliers at cutoff $j$. Thus, even when every cutoff-specific estimate is causal, the effects may apply to different local complier groups.

Worked Example: A Weighted Summary of Two Sharp-RD Effects

Suppose two hypothetical programs use different eligibility thresholds. A sharp RD analysis estimates an effect of 2 months of additional employment at cutoff 1 and 5 months at cutoff 2. An agency wants a summary that selects cutoff 1's local effect with probability 0.70 and cutoff 2's local effect with probability 0.30. This specifies weights $w_1=0.70$ and $w_2=0.30$.

The pooled estimand and its estimate are:

\tau_{\mathrm{pool}} =0.70(2\text{ months})+0.30(5\text{ months}) =1.40+1.50 =2.90\text{ months}.

The interpretation is a weighted average of the two cutoff-specific local effects, using the stated 70%/30% target weights. It is not automatically the average effect for all people in the two programs. For example, knowing that one site has 1,000 residents and another has 100 residents does not establish that their local RD effects equal their full-site average treatment effects. To make that broader claim, one would need additional evidence or assumptions linking each cutoff-local effect to effects for the people away from its threshold.

Choosing Weights: Target First, Precision Second

Weights can represent different questions. Equal weights describe the average of the $J$ cutoff-specific effects when each cutoff is given the same importance. Weights based on the sizes of prespecified local target groups can describe a mixture that selects a cutoff-local effect in proportion to those group sizes. Such weights are defensible only if those groups and the intended selection rule are clearly defined. Weights based on full site population sizes instead define a population-size-weighted summary of local effects; they do not, by themselves, turn those effects into full-population effects.

Inverse-variance or other precision weights are often useful for estimating a common effect under a model that assumes the cutoff effects are equal. But if effects differ, a precision-weighted average is still a weighted estimand—with weights reflecting the estimation procedure, not necessarily a meaningful population. Report the weights and their rationale, and distinguish a target chosen for interpretation from weights chosen to improve precision.

The way data are pooled also matters. A model that forces one common discontinuity imposes a homogeneity restriction: it treats the cutoff-specific effects as equal on the model's outcome scale. A model allowing separate discontinuities can estimate the $\tau_j$'s first, then combine them using declared weights. A single pooled regression that does not allow cutoffs to differ may conceal heterogeneity, and its implicit weighting can depend on the model, sample sizes, and variances.

Worked Example: Precision Weights Are Not Automatically Target Weights

Suppose two hypothetical cutoff-specific sharp-RD estimates, both measured in additional months of employment, are $\hat{\tau}_1=2.0$ with standard error 1.0 month and $\hat{\tau}_2=5.0$ with standard error 2.0 months. For an inverse-variance summary, the unnormalized weights are $1/SE_j^2$: 1.00 and 0.25. The normalized weights are therefore $w_1=1/1.25=0.80$ and $w_2=0.25/1.25=0.20$.

The precision-weighted estimate is:

\hat{\tau}_{\mathrm{pool}} =0.80(2.0)+0.20(5.0) =1.60+1.00 =2.60\text{ months}.

This calculation is arithmetically valid, but its interpretation depends on the goal. If a common effect is assumed, it is an efficient summary under the corresponding model. If effects differ, 2.60 months is the average with 80% weight on cutoff 1 and 20% on cutoff 2. It does not represent an equal-cutoff average, which would be $(2.0+5.0)/2=3.50$ months, or a population-targeted average unless the 80%/20% weights were independently justified for that target.

Pooling Fuzzy-RD Effects

A fuzzy RD provides both an outcome jump and a treatment jump. There are two distinct operations: average the cutoff-specific Wald ratios, or form a ratio of pooled jumps. These generally produce different results. The ratio of pooled jumps weights the cutoff-specific effects in proportion to the contribution of each cutoff to the pooled first stage.

Result: Let $a_j\geq 0$, with $\sum_j a_j=1$, be weights applied to the reduced-form and treatment jumps, and suppose $\Delta D_j>0$ for every included cutoff. Then $$ \frac{\sum_j a_j\Delta Y_j}{\sum_j a_j\Delta D_j} = \sum_j \left( \frac{a_j\Delta D_j}{\sum_k a_k\Delta D_k} \right) \frac{\Delta Y_j}{\Delta D_j}. $$ Thus the ratio of weighted jumps is a weighted average of the cutoff-specific fuzzy-RD Wald effects, with effective weights proportional to $a_j\Delta D_j$. These weights are not generally the same as the original $a_j$.

The identity follows by substituting $\Delta Y_j=\Delta D_j(\Delta Y_j/\Delta D_j)$ into the pooled numerator:

\frac{\sum_j a_j\Delta Y_j}{\sum_j a_j\Delta D_j} = \frac{\sum_j a_j\Delta D_j(\Delta Y_j/\Delta D_j)} {\sum_k a_k\Delta D_k} = \sum_j \frac{a_j\Delta D_j}{\sum_k a_k\Delta D_k} \frac{\Delta Y_j}{\Delta D_j}.

This algebra clarifies the target but does not remove the need for the fuzzy-RD assumptions at each cutoff. Nor does the ratio necessarily target an equal-cutoff average of local complier effects. If first stages have different sizes, the cutoffs contribute differently to the pooled ratio.

Worked Example: Equal Jump Weights, Unequal Fuzzy-RD Effects

Consider two hypothetical fuzzy cutoffs. At cutoff 1, the treatment probability jumps by $\Delta D_1=0.80$, and the outcome jumps by $\Delta Y_1=4.80$ outcome units. At cutoff 2, the treatment probability jumps by $\Delta D_2=0.40$, and the outcome jumps by $\Delta Y_2=4.00$ outcome units. The cutoff-specific Wald estimates are:

\hat{\tau}_1=\frac{4.80}{0.80}=6.00, \qquad \hat{\tau}_2=\frac{4.00}{0.40}=10.0 \quad\text{outcome units per treatment}.

An equal-cutoff average of the two Wald estimates is $(6.00+10.0)/2=8.00$ outcome units per treatment. By contrast, using equal weights $a_1=a_2=0.50$ on the jumps gives a ratio of pooled jumps:

\frac{0.50(4.80)+0.50(4.00)} {0.50(0.80)+0.50(0.40)} = \frac{4.40}{0.60} =7.333 \quad\text{outcome units per treatment}.

The effective weights on the Wald estimates are $0.50(0.80)/0.60=2/3$ and $0.50(0.40)/0.60=1/3$. Consequently, $(2/3)(6.00)+(1/3)(10.0)=7.333$, as the identity predicts. The ratio of pooled jumps puts more weight on cutoff 1 because its treatment jump is larger. The 8.00 and 7.333 estimates answer different pooling questions.

Different Outcome Scales and Different Local Populations

Effects can only be added meaningfully when their outcome units are commensurate. If one site reports an outcome in millimeters and another in centimeters, convert one estimate before pooling. If outcomes are different constructs or instruments, a numerical average may have no coherent interpretation even when both are recorded in points. A standardized effect can sometimes facilitate comparison, but it changes the estimand: it is expressed in standard-deviation units, and the standard deviation used for each site must be stated.

Scale alignment does not make the local populations identical. A weighted average across a threshold for younger applicants and a threshold for older applicants remains a mixture of effects for those two cutoff-local groups. If the intended target is a common population, the analysis needs a defensible transport or standardization strategy and assumptions supporting it; merely pooling the estimates does not supply that justification.

Worked Example: Converting Scales Before Pooling

Suppose two hypothetical RD analyses estimate reductions in the same physical quantity. Cutoff 1 reports a reduction of 4.0 mmHg, while cutoff 2 reports a reduction of 0.50 kPa. Use $1\text{ kPa}=7.50\text{ mmHg}$ to express both on the first site's scale. The second effect is $0.50(7.50)=3.75$ mmHg of reduction. If the intended mixture gives weights 0.60 and 0.40 to the two cutoff-local effects, the pooled reduction is:

0.60(4.0\text{ mmHg})+0.40(3.75\text{ mmHg}) =2.40+1.50 =3.90\text{ mmHg}.

This is a 3.90 mmHg weighted average of the two threshold-local reductions, under the stated 60%/40% mixture. It is not automatically the average reduction for all people at the two sites. The conversion resolves the units; it does not resolve differences in the cutoff-local populations or justify generalizing beyond them.

Uncertainty for a Pooled Estimate

For estimates $\hat{\tau}_1,\ldots,\hat{\tau}_J$ and fixed, prespecified weights, the pooled estimate is $\hat{\tau}_{\mathrm{pool}}=\sum_j w_j\hat{\tau}_j$. Its estimated variance is $w^\mathsf{T}\widehat{\Sigma}w$, where $\widehat{\Sigma}$ is the covariance matrix of the cutoff-specific estimates. If the estimates are independent, this reduces to $\sum_j w_j^2SE_j^2$; if they share individuals, sites, or other sources of sampling error, covariance terms must be included. A confidence interval can then be formed using an appropriate critical value and standard error, with inference accounting for the RD estimation procedure.

For example, in the precision-weighted example above, if the two estimates are independent and the weights are treated as fixed, the pooled standard error is:

SE(\hat{\tau}_{\mathrm{pool}}) = \sqrt{0.80^2(1.0^2)+0.20^2(2.0^2)} = \sqrt{0.64+0.16} = 0.8944\text{ months}.

This calculation uses the independence assumption; it would need covariance terms otherwise. In practice, weights estimated from the data and bias-corrected RD procedures may require inference that reflects those features rather than treating the weights as fixed. A narrow interval quantifies sampling uncertainty for the specified pooled estimand; it does not establish that the weights represent a desired population or that RD assumptions hold at every cutoff.

Common Pooling Mistakes

Calling a weighted average of local effects a full-population effect. Site-size weights alone do not show that effects at the cutoffs equal effects for everyone in those sites. Describe the result as a weighted average of cutoff-specific local effects unless broader transport assumptions are stated and justified.
Assuming a pooled regression has no target implications. A common discontinuity model imposes effect homogeneity on its scale; a model allowing cutoff-specific effects followed by explicit weighting makes heterogeneity and target weights more visible.
Confusing a ratio of pooled jumps with an average of Wald ratios. The ratio of pooled fuzzy-RD jumps generally uses effective weights proportional to the treatment jumps. State which quantity is being estimated.
Averaging incompatible outcome units. Convert units or define a substantively meaningful standardized scale before combining estimates, and say what the resulting unit represents.
Using precision weights as if they were population shares. They reflect uncertainty or a model, not necessarily a target population. Report their purpose and the estimand they induce.

Key takeaway: A pooled RD effect is defined by its cutoff-specific effects, outcome scale, and weights. State all three, distinguish a local-effect mixture from a broader population effect, and verify the RD identification assumptions separately at each cutoff.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following.

Two sharp-RD effects are 3 and 7 units. What estimand results from weights 0.25 and 0.75, and what does it represent?
Why do weights proportional to site population sizes not, by themselves, identify the average treatment effect for all individuals in those sites?
Two fuzzy-RD cutoffs have equal weights on their outcome and treatment jumps but different treatment jumps. Why might the resulting ratio differ from the equal-cutoff average of their Wald estimates?
What additional issue must be resolved before averaging a cutoff-specific effect in centimeters with one reported in inches?
If two cutoff-specific estimates are correlated, what information is needed to calculate the standard error of their weighted average?