Identifying average causal effect in regression discontinuity design with auxiliary data

Pu Yang; Tingyu Li; Wenjie Hu; Xiao-Hua Zhou; Xinqin Feng

REVIEW 3 major objections 5 minor 42 references

Identifying average causal effect in regression discontinuity design with auxiliary data

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A sharp regression discontinuity design can identify whole-population average treatment effects when an auxiliary variable makes the running variable a surrogate and an auxiliary sample records both.

desk verdict Genuinely novel data-fusion identification for RD, but the treatment bridge function existence assumption is never verified and may fail for continuous U, making the simulations potentially vacuous. read the letter →

arxiv 2412.20840 v3 pith:JAIJTC6G submitted 2024-12-30 stat.ME

Xinqin Feng , Wenjie Hu , Pu Yang , Tingyu Li , Xiao-Hua Zhou This is my paper · ORCID

classification stat.ME MSC 62D2062G0562G20

keywords averagetreatmenteffectcausalinferencedatafusionextrapolationregressiondiscontinuitydesignbridgefunctionsdoublyrobustestimationauxiliary

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Regression discontinuity designs normally identify causal effects only at the threshold, because treatment is a deterministic function of the running variable and positivity fails away from it. This paper claims that the whole-population average treatment effect becomes identifiable if there exists an auxiliary variable $U$ such that, given $U$, the running variable $X$ carries no further association with the potential outcomes, and if an independent auxiliary sample records $(U,X)$ alongside the usual RD data. Under latent positivity and the existence of two bridge functions, the paper proves three equivalent identification formulas for $\tau_w = E[Y(w)]$ and gives estimators that mirror outcome regression, inverse probability weighting, and doubly robust estimation. If the claim is right, a researcher who can collect a separate sample of the running variable and its surrogate can estimate global average treatment effects from a regression discontinuity design, not just local effects at the cutoff.

What carries the argument

The load-bearing objects are the outcome bridge function $h_0(U,W)$ and the treatment bridge function $f_0(X,W)$, defined by the conditional moment equations $E[h_0(U,W) \mid X,W] = E[Y \mid X,W]$ and $E[f_0(X,W) \mid U,W] = 1/p_{W\mid U}(W \mid U)$. The treatment bridge function plays the role of an inverse propensity score on the latent $U$ scale, and the outcome bridge function plays the role of an outcome regression on the observed $X$ scale; chaining the two equations is what turns an expectation computable in the auxiliary sample into the global $\tau_w$. Proposition 1 recasts both defining equations as minimax population risks, and the paper estimates $h_0$ and $f_0$ by empirical minimax optimization over neural-network classes, then plugs them into three estimators: an outcome-regression analogue, an inverse-probability-weighted analogue, and a doubly robust analogue. The doubly robust estimator combines both bridge functions so that consistency survives if one of them is misspecified.

What would settle it

Simulate data in which $X$ has a direct effect on $Y$ that bypasses $U$ and $W$, then run the paper's estimator: the estimated $\tau_w$ will deviate from the true $E[Y(w)]$, exposing the latent-confounding assumption. In an observed-data setting where $U$ is actually available in the main sample, one can compute the standard fully adjusted doubly robust estimate and compare it with the proposed estimator that uses only the auxiliary $(U,X)$ sample; systematic disagreement beyond sampling error would indicate Assumption 1 or Assumption 2 is violated.

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Extended reading notes

Core claim

The central discovery is that the non-positivity of sharp RD can be converted into an unmeasured-confounding problem and then solved with a second, independent sample of the auxiliary variable and the running variable. Theorem 1 shows that, under latent confounding ($(X,W) \perp\!\!\perp (Y(0),Y(1)) \mid U$), exchangeability of $(U,X)$ across the main and auxiliary samples, latent positivity $0 < p_{W\mid U}(w \mid U) < 1$, and existence of a treatment bridge function $f_0$ satisfying $E[f_0(X,W) \mid U,W] = 1/p_{W\mid U}(W \mid U)$, the average potential outcome is identified as $$\tau_w = E[h_0(U,w)] = E[Y f_0(X,W)\mathbb{I}(W=w)] = E[Y f_0(X,W)\mathbb{I}(W=w) + h_0(U,w)(1 - f_0(X,W)\mathbb{I}(W=w))],$$ where $h_0$ is the outcome bridge function solving $E[h_0(U,W) \mid X,W] = E[Y \mid X,W]$. The identification does not require uniqueness of the bridge functions and does not require observing $U$ in the main regression discontinuity sample.

Load-bearing premise

The load-bearing premise is Assumption 1: after conditioning on the auxiliary variable $U$, the running variable $X$ has no remaining association with the potential outcomes $Y(0),Y(1)$. If $U$ does not fully capture the link between $X$ and $Y$, the bridge functions solve different equations and the estimated average treatment effect is biased.

Editorial extensions

If this is right

Global average treatment effects, not just local effects at the cutoff, become identified and estimable in sharp RD whenever a suitable auxiliary variable and a separate $(U,X)$ sample are available.
The three estimators are consistent under growing sample sizes, with rates governed by localized Rademacher complexity; the doubly robust estimator is asymptotically normal with asymptotic variance split into a main-sample component and an auxiliary-sample component.
The doubly robust estimator is consistent if either the outcome bridge function or the treatment bridge function is consistently estimated, so misspecifying one of the two nuisance functions does not by itself destroy the inference.
A practitioner can take an existing RD study, collect only a new sample of the running variable and the auxiliary variable, and estimate the whole-population average effect without re-running the treatment study.
In the vitamin A and autism spectrum disorder application, the doubly robust estimate of the average SRS score is 90.87 under supplementation versus 98.02 under control, with a 95% bootstrap confidence interval for the average effect of (−12.48, 0.93), numerically favoring supplementation but not reaching statistical significance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A sensitivity analysis that reports how large a direct $X \to Y$ effect must be to move the estimated ATE by a given amount would make the latent-confounding assumption actionable; the paper does not provide one.
Because the treatment bridge function is guaranteed only under a completeness condition and a summability condition on the singular values of a conditional-expectation operator (Proposition 2), candidates for $U$ could be screened from the auxiliary sample by estimating how rapidly those singular values decay.
The same identification logic transfers to non-positivity problems outside RD: whenever a covariate's support prevents overlap between treated and control groups, an auxiliary sample recording the covariate and the non-overlapping variable may supply the missing joint distribution.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Thanks for the report. I read the paper carefully. The main claim—identification of the global ATE in sharp RD using an auxiliary (U,X) sample via bridge functions—is a genuine extension of Angrist-Rokkanen and proximal causal inference. The proof of Theorem 1 is clean, and the three estimators plus the doubly robust variant are a natural and useful translation of the classical causal inference trinity to this data structure. The real-data application is honest; they report a non-significant effect rather than overselling.

The soft spot is exactly the one the stress-test flags. Assumption 4—existence of a treatment bridge function—is the load-bearing premise. The paper states it as an assumption and gives a Proposition 2 with Picard-type sufficient conditions, but never checks those conditions in the simulations or the real data. For the continuous-U simulation (Setting 2), the equation is a Fredholm integral equation of the first kind; solvability is not automatic and may fail. If it fails, the identification formulas are vacuous, and the finite-sample performance may be regularization artifacts. This is not a minor technicality; it is the difference between estimating an identified parameter and estimating a projection artifact. The paper needs to either verify the condition (e.g., via completeness and Picard checks) or add a sensitivity analysis to show the estimates are stable when the bridge function is only approximately solvable.

Other weaknesses are secondary. Assumption 1 (latent confounding) is strong and untestable; a sensitivity analysis would help, especially for the vitamin A example where it is asserted from biological knowledge. The asymptotic proofs are sketches in places, e.g., the o_P(n^{-1/2}) remainder in Theorem 6 leans on an external result and some steps are compressed. No code or data is included, so the simulation results are not reproducible, and there are no baseline comparisons to existing RD extrapolation methods.

Overall, this is a serious paper with an important idea, but the existence condition is not just a loose end; it is the hinge on which the whole identification argument turns. I would send it to review, but with the expectation that a referee pushes hard on Assumption 4. The paper is worth a reading group discussion precisely because the bridge-function existence question is subtle and often glossed over.

Referee Report

3 major / 5 minor

Summary. The paper proposes a framework for identifying the global average treatment effect in a sharp regression discontinuity design when treatment assignment is deterministic in the running variable and the usual positivity assumption fails. The key idea is to introduce a latent auxiliary variable U such that the potential outcomes are independent of the running variable and treatment given U (Assumption 1), and to suppose that an auxiliary dataset containing (U, X) is available jointly with the main RD dataset containing (X, W, Y). Under Assumptions 1-4, which additionally include exchangeability, latent positivity, and existence of a treatment bridge function, Theorem 1 identifies τw = E[Y(w)] through three equivalent formulae involving an outcome bridge function h0 and a treatment bridge function f0. The paper then develops minimax estimators for the bridge functions, proposes outcome-regression, inverse-probability-weighted, and doubly robust estimators of τw, and establishes convergence rates, consistency, and asymptotic normality under high-level conditions. Simulations and an application to vitamin A supplementation and autism severity illustrate the methods.

Significance. If the assumptions hold, the paper provides a practically valuable way to extrapolate RD causal effects away from the cutoff using a separate auxiliary sample, a setting that previous work addressed only under stronger data availability. The identification argument (Theorem 1) is cleanly derived and represents a useful extension of proximal causal inference ideas to regression discontinuity designs. The three estimators, especially the doubly robust one, are a natural and useful contribution, and the asymptotic analysis follows the minimax-learning template. The main weakness is that the key existence assumption for the treatment bridge function is not verified in the continuous-U simulation settings or in the real data application, so the empirical evidence and the practical guidance are not yet underpinned by the identification theorem. The paper's central theoretical result is sound conditional on its assumptions, but the applicability to the reported data is not established.

major comments (3)

[Section 5.2, Assumption 1] Assumption 4 (existence of the treatment bridge function f0 satisfying Eq. (2)) is the load-bearing condition for Theorem 1, but it is never verified in the continuous-U settings used in the simulations or in the real data application. The only existence example in Section 7.8 is for binary U, while Setting 2 (Table 1) has U ~ Uniform(0,1) and X|U ~ N(U-0.5,1), and the vitamin A application has U = retinoic acid, which is continuous. For sharp RD with continuous U, Eq. (2) with w=1 reads E[f0(X,1) | U, X>=c] = 1/P(X>=c|U), a Fredholm integral equation of the first kind. Latent positivity (Assumption 3) does not imply solvability; Proposition 2's Picard condition (condition (3)) is not checked for any of these settings. As a result, the simulation results in Tables 4-5 and the real-data estimates in Table 6 may be produced by the regularization and choice of function spaces even when no f0 exists, and Theorem 1 would not apply. Please either verify the existence condition for the simulation data-generating processes, provide a continuous-U existence example, or restrict the empirical claims to settings where the existence is guaranteed.
[Section 4.2, Theorem 6] The real-data conclusion in Section 5.2 depends on Assumption 1, which is asserted from biological reasoning about retinoic acid without any sensitivity analysis. The statement that serum retinol and SRS score are "likely to be independent conditional on retinoic acid level" is a plausibility argument, not a check. If U does not fully capture the association between X and the potential outcomes, the bridge-function equations identify quantities different from the target τw, and the reported estimates are biased. Since the main dataset has only 149 observations and the auxiliary dataset is separate, a sensitivity analysis (e.g., allowing a residual direct effect of X on Y of varying strength) would be needed to support the application; otherwise the results should be framed as conditional on an untestable assumption.
[Section 4.2, Theorem 6] The proof of the asymptotic decomposition for the doubly robust estimator is incomplete at a load-bearing point. In Section 7.7, the claim that the empirical-process remainder terms (ˆEm−E)[(ˆf−f0)I(W=w)Y] and (ˆEa−E)[(1−ˆf I(W=w))ˆh−(1−f0I(W=w))h0] are oP(n^{-1/2}) is justified by invoking the continuity of a Gaussian process at zero, citing Krätschmer and Urusov (2023). However, the argument that the sup over an L2-ball of the empirical process converges to zero under the Donsker and covering-number conditions is only sketched; the statement "converges to zero based on Corollary 1.2" is a substantial step. Since the asymptotic normality result and the variance formula depend on this remainder rate, a complete proof or a precise theorem reference with the verification of its conditions should be provided.

minor comments (5)

[Section 4.1, Theorem 5] The treatment rule is defined as W = I(X ≥ c), but in the vitamin A example the treatment is assigned when serum retinol concentration is below 1.05 µmol/L, i.e., W = I(X < c). The convention should be stated consistently, or the example should be aligned with the definition.
[Section 5.1, Tables] In Theorem 5, the conclusion says "we have ˆτ h w is consistent" but it should refer to ˆτ f w; this appears to be a typographical error that could confuse readers.
[Section 7.2] Table 6 has a column header "Treatment Control" that is ambiguous; the entries are estimates of τ1 and τ0, so the header should be "Treatment (w=1)" and "Control (w=0)". Also, the caption should explicitly state that the intervals are 95% confidence intervals.
[Section 5.1, Table 1] In the proof of Proposition 1, the notation h(U, X) appears where the definitions elsewhere use h(U, W); this inconsistency should be fixed to avoid confusion about the argument of the bridge function.
[Section 3, Eq. (2)] In Table 1, "U nif orm(0, 1)" contains a typo and should read "Uniform(0,1)".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the identification theorem is self-contained and the only self-citations appear in the real-data illustration, not in the derivation.

full rationale

The identification argument is self-contained. Theorem 1 (Section 3) defines the target as tau_w = E[Y(w)] and proves tau_w = E[h0(U,w)] = E[Y f0(X,W) I(W=w)] using only the conditional moment equations (1) and (2) together with Assumptions 1-4. Neither the outcome bridge function h0 nor the treatment bridge function f0 is defined in terms of tau_w, and the target parameter is never fitted directly; the estimators in Section 4 minimize empirical functionals based on the same pre-specified moment equations, so the resulting plug-in estimates are not predictions forced by fitting tau_w itself. The main unverified condition is Assumption 4, the existence of the treatment bridge function; the paper does not check the Picard/completeness condition of Proposition 2 in the simulations or in the vitamin A application. That is a substantive identification-condition gap or a robustness concern, but it is not circularity, because the existence assumption is not derived from, nor equivalent to, the target parameter. The only self-references are Feng et al. (2024a,b), which are used as sources of data and as prior analyses in the real-data example; they do not ground the identification theorem. I therefore find no circular step in the derivation chain.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The identification of τw rests on four domain assumptions (latent confounding, exchangeability, latent positivity, and existence of a treatment bridge function) that are stated but not empirically tested in the main sample; the application in Section 5.2 argues for latent confounding from biological knowledge. The estimation theory adds standard empirical-process regularity conditions. No new physical entity is introduced; the auxiliary variable U is a latent modeling construct that happens to be measured in the auxiliary dataset in the application.

free parameters (5)

λ and λ′ (regularization constants) = 1.0
Regularization constants in the minimax objectives (6) and (7), chosen by hand; the theory holds for any positive λ.
γ1 and γ2 (norm penalties) = 0.03
Penalty coefficients added to the inner optimization objective, set by hand in Section 7.9.
Basis dimensions d1, d2 = 10
Number of cosine basis functions for F′ and H′ in Section 7.9, chosen by hand.
Neural network hidden size = 10
Two-layer ReLU network with 10 hidden units for h and f, described in Section 7.9.
Learning rate and epochs = 0.05 or 0.1; 100 epochs
Optimization settings in Section 7.9, chosen by hand for the simulations.

assumptions (5)

domain assumption Assumption 1 (Latent confounding): (X, W) ⊥ (Y(0), Y(1)) | U
Core identifying assumption in Section 2.1. It asserts the running variable and treatment are independent of potential outcomes given the auxiliary variable U. Untestable in the main sample and asserted from domain knowledge in the application.
domain assumption Assumption 2 (Exchangeability): F_m(U,X) = F_a(U,X)
Section 2.2. Requires the auxiliary sample to have the same joint distribution of (U,X) as the main sample. In the application the auxiliary sample comes from a different study (Feng et al. 2024b) with 378 observations; comparability is not formally checked.
domain assumption Assumption 3 (Latent positivity): 0 < p(W|U) < 1 almost surely
Section 3. Needed so that both treatment arms have support for every U, allowing the bridge functions to be well-defined. The paper notes it can be partially checked with the auxiliary sample.
domain assumption Assumption 4: existence of a treatment bridge function f0 in L2(X,W) satisfying E[f0|U,W] = 1/p(W|U)
Section 3, Definition 1 and Assumption 4. This is a completeness/solvability condition; Proposition 2 gives sufficient conditions (completeness and Picard conditions) but they are not verified in practice.
standard math Regularity conditions: boundedness of function classes, Glivenko-Cantelli/Donsker properties, critical radius conditions
Theorems 2-6 rely on these high-level complexity conditions from empirical process theory. The paper cites Bartlett et al. (2005), Wainwright (2019), Foster and Syrgkanis (2023).

invented entities (1)

Auxiliary variable U independent evidence
purpose: A latent variable such that conditioning on U eliminates the dependence of potential outcomes on the running variable, converting the RD non-positivity problem into an unmeasured confounding problem.
In the simulations and the application, U is measured (as retinoic acid in the auxiliary dataset), so it is not purely fictional. However, in the main dataset U is unobserved and its existence in the general framework is postulated.

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Pith. "Pith review of Identifying average causal effect in regression discontinuity design with auxiliary data." pith.science (2026). https://pith.science/paper/JAIJTC6G

@misc{pith2026241220840,
  author       = {Pith},
  title        = {Pith review of: Identifying average causal effect in regression discontinuity design with auxiliary data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JAIJTC6G}},
  note         = {Machine review of arXiv:2412.20840}
}

read the original abstract

Regression discontinuity designs are widely used when treatment assignment is determined by whether a running variable exceeds a predefined threshold. However, most research focuses on estimating local causal effects at the threshold, leaving the challenge of identifying treatment effects away from the cutoff largely unaddressed. The primary difficulty in this context is that the treatment assignment is deterministically defined by the running variable, violating the commonly assumed positivity assumption. In this paper, we introduce a novel framework for identifying the average causal effect in regression discontinuity designs. Our approach assumes the existence of an auxiliary variable for which the running variable can be seen as a surrogate, and an additional dataset that consists of the running variable and the auxiliary variable alongside the traditional regression discontinuity design setup. Under this framework, we propose three estimation methods for the ATE, which resembles the outcome regression, inverse propensity weighted and doubly robust estimators in classical causal inference literature. Asymptotically valid inference procedures are also provided. To demonstrate the practical application of our method, simulations are conducted to show the good performance of our methods; besides, we use the proposed methods to assess the causal effects of vitamin A supplementation on the severity of autism spectrum disorders in children, where a positive effect is found but with no statistical significance.

Figures

Figures reproduced from arXiv: 2412.20840 by the authors.

**Figure 2.** Performance of the three estimating procedures under setting 1 when one of the bridge [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Performance of the three estimating procedures under setting 2 when one of the bridge [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Histograms of retinoic acid for two subgroups defined by whether the serum retinol [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

Discussion (0). Continue with ORCID to comment.

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Reviewed August 10, 2026 · model on record in the stance chip above.

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