1 The Locality Curse of RDD
The regression discontinuity design (RDD) buys its credibility at a price. Where a treatment switches on at a known threshold of a running variable a test score, an age, a vote share comparing units just above and just below the cutoff yields an almost experimental estimate of the treatment effect [Imbens and Lemieux, 2008, Calonico et al., 2014]. But the estimate pertains to a single, vanishingly thin slice of the population: those exactly at the cutoff. Formally, the sharp RDD with running variable X, cutoff c, and treatment D = 1{X ≥ c} identifies
and nothing more. A policymaker who wants to know the effect of a scholarship for students well above the eligibility threshold, or of a pension rule for people younger than the qualifying age, gets no answer. This is the "locality curse": RDD's internal validity is purchased by surrendering external validity everywhere except the margin.
The reason is a missing-data problem. For units above the cutoff we observe outcomes under treatment, so we can estimate E[Y(1)|X=x] for x > c; for units below it we observe untreated outcomes, identifying E[Y(0)|X=x] for x < c. At any given x we see only one potential outcome's mean. Extrapolation requires recovering the other one and the data alone cannot do it. Kwon and Deaner [2025] propose a transparent assumption that can: comonotonicity of the conditional potential outcome surfaces in covariates.
2 Setup: Bringing in Covariates
Suppose that alongside the running variable X each unit carries covariates W. Define the conditional mean surfaces, evaluated in a neighbourhood of the cutoff,
The treatment effect as a function of covariates is τ(w) = m₁(w) - m₀(w) The difficulty in extrapolating away from the margin is that, at running-variable values strictly above c, the data identify the treated surface but not the untreated one, and vice versa below. To learn τ(·) over a range of covariate or running-variable values rather than at the single point (1) we need a link between the two surfaces that lets the observed branch inform the unobserved one.
3 The Comonotonicity Restriction
Kwon and Deaner [2025]'s identifying assumption is that the treated and untreated conditional mean surfaces are comonotonic in the covariates: the covariate values associated with higher average untreated outcomes are also those associated with higher average treated outcomes. Stated as a rank condition, for any two covariate values w and w'
Equivalently, there exists a non-decreasing function ϕ such that m₁(w) = ϕ(m₀(w)): the treated surface is a monotone re-labelling of the untreated surface across covariate space. The two response functions need not be parallel, proportional, or linearly related only order-preserving. Units that have higher untreated potential outcomes (the "stronger" students, the "healthier" patients) also have higher treated potential outcomes; the treatment may compress, stretch, or shift the distribution, but it does not scramble the ranking.
This is a substantively interpretable assumption rather than a functional-form convenience. It encodes the belief that whatever latent quality the covariates proxy operates in the same direction on both potential outcomes a plausible claim in many settings (ability raises earnings with and without a programme; baseline health raises survival with and without a drug) and a checkable one in others.
How it identifies extrapolated effects. Comonotonicity is powerful because the two RDD branches together trace out both surfaces over the covariate distribution just on different sides of the cutoff. Above c we estimate m₁(·); below c we estimate m₀(·). The monotone link ϕ that aligns the two surfaces is pinned down where their covariate supports overlap, and once ϕ is known it transports the treated surface to running-variable regions where only untreated outcomes are observed (and conversely). The treatment effect τ(w) = ϕ(m₀(w)) - m₀(w) then becomes identified away from the margin, delivering the extrapolation that classical RDD cannot.
4 Estimation and Inference
Kwon and Deaner [2025] provide an estimator built on familiar local linear regression. The conditional mean surfaces in (2) are estimated by local linear smoothing of the outcome on the covariates near the cutoff, separately on each side; the monotone map ϕ is then recovered from the comonotone alignment of the fitted surfaces, with monotonicity imposed by rearrangement. Because the building blocks are local polynomials, the method inherits the well-developed bandwidth selection and bias-correction apparatus of modern RDD [Calonico et al., 2014], and standard errors follow from the asymptotics of the two-step estimator. With multiple covariates, the dimensionality of W raises the usual nonparametric concerns, and the procedure leans on the low effective dimension implied by the one-dimensional comonotone link.
5 Where It Sits, and What It Costs
Comonotonicity joins a small but growing toolkit for escaping the locality curse. Earlier approaches include using multiple cutoffs to trace effects across thresholds, bounding extrapolated effects under shape restrictions, and parametric extrapolation of the running-variable relationship. Relative to these, Kwon and Deaner [2025]'s contribution is an assumption that is simultaneously weak (it imposes only an ordering, not a shape) and interpretable (it has a clear behavioural reading), together with an estimator that practitioners can implement with off-the-shelf local regression.
The costs are real and should be stated plainly. Comonotonicity is fundamentally untestable away from the region where the surfaces overlap like every extrapolation assumption, it trades data for structure. It can fail when treatment reverses the covariate ordering (a programme that helps the disadvantaged most could induce a negative comonotone relationship, which the symmetric version of the assumption accommodates but which must be argued for). And richer covariate vectors strain the nonparametric estimation. Still, the broader significance is hard to overstate: RDD has spent three decades apologising for telling us only about the marginal unit. By naming a credible, interpretable condition under which the design speaks to units away from the threshold, the comonotonicity approach helps RDD graduate from a tool of internal validity into one that can, with care, inform policy beyond the cutoff.
References
- Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica, 82(6), 2295-2326.
- Imbens, G. W., and Lemieux, T. (2008). Regression discontinuity designs: a guide to practice. Journal of Econometrics, 142(2), 615-635.
- Kwon, S., and Deaner, B. (2025). Extrapolation in regression discontinuity design using comonotonicity. arXiv preprint arXiv:2507.00289.