On Extrapolation of Treatment Effects in Multiple-Cutoff Regression Discontinuity Designs

Yuta Okamoto; Yuuki Ozaki

REVIEW 2 major objections 5 minor 55 references

On Extrapolation of Treatment Effects in Multiple-Cutoff Regression Discontinuity Designs

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows when the standard constant-bias assumption for extrapolating treatment effects across multiple regression-discontinuity cutoffs fails, and offers bounds that stay valid.

desk verdict A genuinely useful partial-identification result for multi-cutoff RD, with an overreaching microfoundation claim that needs a fix. read the letter →

arxiv 2412.04265 v3 pith:BI7OYLQW submitted 2024-12-05 econ.EM

Yuta Okamoto , Yuuki Ozaki This is my paper · ORCID

classification econ.EM

keywords multiple-cutoffregressiondiscontinuityextrapolationpartialidentificationconstantbiasassumptionmonotonicitydominanceuniforminferencedecisionmodel

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

This paper asks when treatment effects estimated by regression discontinuity at one cutoff can be credibly extrapolated to other values of the running variable when several cutoffs exist. The authors show that the standard constant-bias assumption—that the gap between two groups' untreated outcome functions is flat—is justified only when the running variable cannot be manipulated by agents and cutoffs are assigned as if randomly. When the running variable is partially manipulable, such as a test score, a rational-agent model predicts that cutoff shifts change effort, so the assumption generically fails and extrapolations can be biased. As a complement, the paper derives bounds on the extrapolated effect for the lower-cutoff group at any point between the two cutoffs, using monotonicity of the untreated outcome functions and a dominance condition, and shows these bounds are pointwise sharp. It then develops estimation and uniform inference procedures and illustrates them on two financial-aid programs.

What carries the argument

The identification machinery is the pair of bounds $\underline{\nabla}_l(\bar{x}) = \mu_{1,l}(\bar{x}) - \mu_{0,h}(\bar{x})$ and $\overline{\nabla}_l(\bar{x}) = \mu_{1,l}(\bar{x}) - \mu_{0,l}(l)$, delivered by monotonicity of the lower-cutoff group's control outcome function (upper bound) and dominance of the higher-cutoff group's control outcome (lower bound). The theoretical machinery for the failure result is a rational-agent model in which effort $e_i$ shifts both the running variable and the control outcome; a cutoff shift changes effort whenever the idiosyncratic score-shock density $f_{\eta^s}$ is not periodic with period $h-l$. For inference, the paper uses local linear estimation with an equivalent-kernel expansion, robust bias correction, and a multiplier bootstrap with Mammen weights to obtain uniform confidence bands.

What would settle it

Estimate the control-outcome function $\mu_{0,l}(x)$ for $x \in (l,h)$ using auxiliary data or a design that removes treatment for the lower-cutoff group; a non-monotone estimate invalidates the upper bound and the sharpness claim. For the decision-model failure result, estimate the density of score shocks across the two cutoff groups: an absence of $h-l$ periodicity indicates, by Proposition 2, that optimal effort responds to the cutoff and the constant-bias assumption fails.

Watch

Extended reading notes

Core claim

The paper's central claim is twofold. First, a microeconomic decision model shows that the constant-bias assumption—that $\mu_{0,h}(x) - \mu_{0,l}(x)$ is flat over $(l,h)$—is justified when the running variable is non-manipulable and cutoff assignment is as-if random, but fails generically when the running variable is partially manipulable, even when the groups are identical in ability and beliefs. Second, under continuity, monotonicity of $\mu_{0,c}$, and dominance $\mu_{0,l} \le \mu_{0,h}$, the extrapolated treatment effect for the lower-cutoff group at any $\bar{x} \in (l,h)$ is pointwise sharply bounded by $\underline{\nabla}_l(\bar{x}) = \mu_{1,l}(\bar{x}) - \mu_{0,h}(\bar{x})$ and $\overline{\nabla}_l(\bar{x}) = \mu_{1,l}(\bar{x}) - \mu_{0,l}(l)$.

Load-bearing premise

The load-bearing premise is that the lower-cutoff group's untreated outcome function $\mu_{0,l}(x)$ is monotone over the extrapolation interval, a restriction that is never observed and cannot be tested with regression-discontinuity data; if it fails, the upper bound collapses.

Editorial extensions

If this is right

When the running variable is non-manipulable and the groups are comparable in unobservables, the constant-bias extrapolation is justified.
When the running variable is partially manipulable, the constant-bias extrapolation can be substantially biased, and the proposed bounds remain valid without it.
The bounds are pointwise sharp, so no tighter interval can be obtained at a single point under the maintained assumptions.
Adding intermediate cutoff groups narrows the lower bound, so richer multi-cutoff designs give more informative identified sets.
The same bounding logic extends to one-sided fuzzy designs, covering settings in which eligible individuals may not take up the treatment.

Reading between the lines

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

A practical diagnostic emerges from the decision model: estimate the density of the idiosyncratic shock to the running variable, and if it shows no periodicity of length $h-l$, the constant-bias assumption is likely violated.
Because the upper bound rests on an untestable monotonicity condition, an external validation study that observes control outcomes for lower-cutoff units just above their cutoff would provide a direct check on the headline bounds.
The pointwise-sharp-but-not-uniformly-sharp distinction suggests that developing uniform-sharp bounds under slightly stronger shape restrictions would be a useful next step.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Two things to know. First, the main contribution is solid: under monotonicity of the control outcome and dominance across cutoffs, the paper derives pointwise sharp bounds on extrapolated treatment effects, and backs them with a uniform inference procedure. That is a clean, implementable result that should be useful to applied researchers. The authors also honestly disclose that the bounds are not uniformly sharp, which is the right call. Second, the microfoundation analysis of the constant bias assumption is directionally interesting but the headline 'if and only if' in Proposition 2 is stronger than the proof supports.

The proof of Proposition 2 shows that if optimal effort is cutoff-invariant, then fηs(l − s(e*(ϵ))) = fηs(h − s(e*(ϵ))) for each ability type ϵ. Concluding that fηs is periodic on the whole interval [l−b, h−a] requires that the set {s(e*(ϵ))} covers [a, b]. That needs ϵ to have connected support and the solution to be interior with enough regularity; Assumption 2.3 only says support is identical across groups. With a two-point ability distribution you get equality at two pairs of points, not periodicity. So the only-if direction fails as stated. This is not fatal to the paper's main identification theorem, which does not use Proposition 2, but the claim needs to be weakened to a sufficient condition or given extra assumptions.

The bounds themselves are on firmer ground. The lower bound uses dominance, whose endpoint version is testable; the upper bound uses monotonicity of the lower-cutoff control function on (l, h), which is untestable. The paper acknowledges this in Remark 8, yet the upper bound is still load-bearing for the headline results, so applied readers should treat it as a substantive assumption, not a mild shape restriction. The empirical illustrations are informative: the SPP example shows tight bounds, and the ACCES example usefully shows the bounds ruling out large negative parametric extrapolations. The simulation and fuzzy RD extension are careful. The only other annoyance is that the R code is mentioned but not verifiable from the text; a commit hash would help.

For whom? Applied microeconometricians working with multi-cutoff RD and methodologists interested in partial identification will both get value. I would send this to a serious referee: the main theorem and inference procedure deserve a full review, and the referee should be asked to fix or qualify Proposition 2 and require code deposit.

Referee Report

2 major / 5 minor

Summary. This paper studies extrapolation of treatment effects away from the cutoff in multi-cutoff sharp regression discontinuity designs. It first builds a micro-founded decision model in which agents choose effort that affects a test score and a future outcome; the model is used to argue that the constant-bias assumption of Cattaneo et al. (2021) is plausible when the running variable is non-manipulable and cutoff assignment is as-if random (Proposition 1), but that it can fail when the running variable is partially manipulable (Proposition 2 and Examples 1-2). The paper then proposes a complementary partial-identification strategy: under continuity, monotonicity of the control outcome functions mu_{0,c}, and dominance mu_{0,l} <= mu_{0,h} on (l,h), the extrapolated effect tau_l(xbar) is pointwise sharply bounded by mu_{1,l}(xbar)-mu_{0,h}(xbar) and mu_{1,l}(xbar)-mu_{0,l}(l) (Theorem 1). The paper develops local-linear estimation with robust bias correction, pointwise confidence intervals, and a multiplier-bootstrap uniform confidence band, and illustrates the methods with the SPP and ACCES scholarship programs. The online appendix contains proofs, a fuzzy-RD extension, and simulations.

Significance. Conditional on the maintained assumptions, Theorem 1 is a clean and practically useful result: the bounds are simple, pointwise sharp, invariant to increasing monotone transformations of the outcome, and the paper honestly discloses that they are not uniformly sharp. The uniform inference procedure is nontrivial and is supported by a detailed asymptotic argument in the online appendix, simulations, and replication code, all of which are valuable. The identification assumptions are stated ex ante rather than fitted to the data used to evaluate the claims, and the numerical examples are clearly illustrative rather than calibrated. The main weakness is Proposition 2: its 'if and only if' is not proved under the stated assumptions, and because that proposition underpins the paper's negative message about partial manipulability, the manuscript needs a substantive revision before that part of the contribution is convincing. The Section 3 identification contribution does not depend on Proposition 2 and therefore survives the revision.

major comments (2)

[Section 2.3.2; Appendix A.1] The proof of Proposition 2 does not establish the stated 'if and only if' under Assumption 2.3. In the only-if direction, the first-order condition gives f_eta^s(l - s(e*(epsilon_i))) = f_eta^s(h - s(e*(epsilon_i))) for each epsilon_i, and the proof then asserts that f_eta^s is periodic on [l-b, h-a], where a = inf_epsilon s(e*(epsilon)) and b = sup_epsilon s(e*(epsilon)). This inference requires the set {s(e*(epsilon)) : epsilon in supp(epsilon)} to be the entire interval [a,b], which needs supp(epsilon) to be connected and e* to be continuous on the support. Assumption 2.3(v) only states that the support of epsilon is identical across groups; it does not require connectedness, and continuity of e* is not guaranteed by the stated strict concavity and differentiability conditions alone (the implicit function theorem needs a nonzero second derivative at the optimum). With a disconnected support, for example two ability types, the equality pins down f_eta^s only at finitely many pairs of points, far short of periodicity on the whole interval. The converse direction is also incomplete: the proof claims that under periodicity the probability identity holds 'for any e_i', but the assumed periodic interval is defined through the optimal efforts. For a feasible effort with s(e_i) outside [a,b], the endpoints l - s(e_i) and h - s(e_i) lie outside the periodic interval, so the quantity Q is not shown to be constant in e_i. Thus the equivalence of the two decision problems is not proved without additional assumptions on the range of s over the whole choice set. Because Proposition 2 is the theoretical basis for the claim that partial manipulability generically breaks the constant-bias assumption, the authors should either add explicit support-connectedness, interiority, and range conditions and prove the required continuity, or restate Proposition 2 as a weaker necessary condition plus a separate sufficiency result.
[Section 3.1.1; Remark 8; Section 4.2.3] Assumption 3.1 requires mu_{0,l} to be weakly increasing on (l,h), but mu_{0,l} is not observed for x > l in a sharp RD design because the treated outcome is the one observed there; Remark 8 honestly admits this untestability. This assumption is load-bearing for the upper bound in Theorem 1, because the bound mu_{1,l}(xbar) - mu_{0,l}(l) is valid only if mu_{0,l}(xbar) >= mu_{0,l}(l). The empirical summaries in Sections 4.1.3 and 4.2.3, however, draw substantive policy conclusions from the full bounds (for example, that large negative effects are ruled out). To make the reported message match the identifying power actually available, the paper should either prominently qualify that the upper bound depends on the untestable monotonicity of mu_{0,l}, or report a lower-bound-only analysis that relies only on Assumption 3.2 and is therefore robust to violations of that monotonicity.

minor comments (5)

[Section 3.2.3, Step 4] The displayed confidence band uses bV_L^{-1/2}(x) and bV_U^{-1/2}(x), while Step 3 and the proof in Online Appendix S1.1.1 use bV_L^{1/2}(x) and bV_U^{1/2}(x); the notation should be made consistent.
[Appendix A.1] In the first-order condition in the proof of Proposition 2, the first term appears to have a missing parenthesis: it should be u'(s(e_i^*)) s'(e_i^*).
[Proposition 2] The statement of Proposition 2 uses inf_epsilon s(e*(epsilon)) and sup_epsilon s(e*(epsilon)) without first defining a and b and without specifying that the infimum and supremum are taken over the support of epsilon; this should be made explicit.
[Online Appendix S1.2] The demonstration that the bounds are not uniformly sharp is informal and based on a figure; a formal counterexample with explicit functions would make the claim easier to verify.
[Remark 4] The invariance claim should specify that the identification conditions are invariant to increasing monotone transformations of the outcome; for decreasing transformations the monotonicity and dominance directions reverse, and Corollary 1 applies instead.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the bounds and inference follow from explicitly stated assumptions; no fitted parameter is relabeled as a prediction and no load-bearing self-citation is present.

full rationale

The derivation chain is self-contained. Theorem 1 bounds the unobserved µ0,l(x̄) between µ0,l(l) (by monotonicity) and µ0,h(x̄) (by dominance), and the lower/upper bound estimators in Section 3.2 are direct nonparametric estimates of these identified quantities; nothing is fitted to the outcome being bounded and then relabeled as a prediction. Sharpness is demonstrated by explicit construction of control functions satisfying the assumptions, not by importing a uniqueness result. The empirical sections estimate the same quantities and do not derive their conclusions from a parameter fit. The only substantive fragility—the untestability of the monotonicity of µ0,l, acknowledged in Remark 8—is an identification/soundness concern, not circularity. The proof of Proposition 2 also contains a regularity gap (equality of fηs at paired points implies periodicity only if s(e*(ϵ)) fills an interval, which is not guaranteed by Assumption 2.3(v)), but this is a mathematical proof gap, not a reduction of a result to its own input. No load-bearing self-citation or renaming of a known result is present.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central identification theorem introduces two assumptions, monotonicity and dominance, that are not derived from data or from the economic model. The model's illustrative examples rely on hand-picked functional forms but do not feed back into the formal bounds. No new unobserved entities are introduced. Proposition 2's 'if and only if' is stronger than the proof establishes; see the red flag.

free parameters (2)

Example 1 and 2 structural constants = τ̃=1, β=1, γ=0; u(s)=s, s=5√e, y=10√e, K=15(2-ε)e
Hand-picked in Section 2.3.2 to obtain a closed-form optimal effort function; illustrates failure of constant bias but is not estimated and does not appear in Theorem 1.
Example 2 ability distributions = ε∼Uniform(0,1) for C=2; ε∼Uniform(2/3,5/3) for C=3
Chosen to show the failure survives group differences in ability; illustrative only, not part of the formal theorems.

assumptions (7)

standard math Assumption 2.1: µd,c(x) is continuous at x=c for each cutoff c
Standard RD identification condition inherited from Hahn et al. (2001); used to identify boundary values such as µ0,l(l).
domain assumption Assumption 2.2 and the Cattaneo et al. (2021) constant bias formula
The paper analyzes this prior identification condition rather than assuming it for its own bounds; it is the object of the model-based critique.
domain assumption Assumptions 2.3(i)-(v): concavity, differentiability, density existence, convex cost, common support of ability
Regularity conditions for the microeconomic decision model in Section 2.2; needed for Proposition 2 and the numerical examples.
domain assumption Assumption 2.4: common subjective belief τ̃_i=τ̃
Strong homogeneity assumption; the paper notes it is strong and then shows constant bias can fail even under it.
domain assumption Assumption 3.1: µ0,c(x) is weakly increasing on (l,h) for c∈{l,h}
Untestable shape restriction on the lower-cutoff group's control mean; load-bearing for the upper bound and for pointwise sharpness.
domain assumption Assumption 3.2: µ0,l(x) ≤ µ0,h(x) on (l,h)
Dominance restriction across cutoff groups; load-bearing for the lower bound; only its endpoint version at x=l is testable under continuity.
standard math Assumptions S1.1-S1.8: i.i.d. sampling, kernel and density conditions, smoothness, bandwidth rates 1/9<η<1/2, bounded residual support
High-level conditions used only for the uniform confidence band in the online appendix; standard in local polynomial inference.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On Extrapolation of Treatment Effects in Multiple-Cutoff Regression Discontinuity Designs." pith.science (2026). https://pith.science/paper/BI7OYLQW

@misc{pith2026241204265,
  author       = {Pith},
  title        = {Pith review of: On Extrapolation of Treatment Effects in Multiple-Cutoff Regression Discontinuity Designs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BI7OYLQW}},
  note         = {Machine review of arXiv:2412.04265}
}

read the original abstract

We investigate how to learn treatment effects away from the cutoff in multiple-cutoff regression discontinuity designs. Using a microeconomic model, we demonstrate that the parallel-trend type assumption proposed in the literature is justified when cutoff positions are assigned as if randomly and the running variable is non-manipulable (e.g., parental income). However, when the running variable is partially manipulable (e.g., test scores), extrapolations based on that assumption can be biased. As a complementary strategy, we propose a novel partial identification approach based on empirically motivated assumptions. We also develop a uniform inference procedure and provide two empirical illustrations.

Figures

Figures reproduced from arXiv: 2412.04265 by the authors.

**Figure 2.** Examples of the Error Distribution fη s Note: We set infϵ s(e ∗ (ϵ)) = 2, supϵ s(e ∗ (ϵ)) = 8, l = 4, and h = 6, which are defined in Proposition 2. The black solid line indicates the density of the standard Gaussian distribution. In this case, e ∗ i depends on Ci . The red dashed line represents the density of the uniform distribution on [−4, 4], while the blue dot-dashed line is a periodic sine function. In these … view at source ↗

**Figure 3.** E [Yi(0)|Xi = x] in Numerical Examples: Partially Manipulable Running Variable cutoffs is determined after the entrance exam has been administered, then all groups can be seen as facing the same decision problem at the time of their effort choice. In such cases, the distributional equivalence of ϵi implies the constant bias assumption. ⌟ 3 Alternative Identification Results In the previous section, we demonstrated t… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Bounds of Theorem 1 Corollary 2. Take an arbitrary closed interval X ⊂ (l, h). Then, under the same assumptions in Theorem 1, the bounds ∇ l(x) and ∇l(x) are both attainable as a function over the interval X , in the sense that ∇ l(x) and ∇l(x) are consistent with the …

**Figure 5.** Figure 5: SPP (N = 20551) potential deviations from constancy by comparing them to the extrapolated RD effects under the constant bias assumption. 4.1.3 Results and Implications We consider the closed interval I = [−55.0, −42.5] ⊂ (−57.21, −40.75) in our analysis. Figure 5a pres…

**Figure 6.** Figure 6: ACCES (N = 1365) 4.2.3 Results and Implications Figure 6b displays the estimation results over the interval I = [−840, −590]. The bounds suggest that the ACCES program likely has a nonnegative effect overall, and the effects are roughly inbetween [0.00, 0.15] just abov…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 49 canonical work pages

[1]

and Cattaneo, M

Abadie, A. and Cattaneo, M. D. (2018). Econometric Methods for Program Evaluation . Annual Review of Economics , 10:465--503

work page 2018
[2]

Angrist, J. D. and Rokkanen, M. (2015). Wanna Get Away? Regression Discontinuity Estimation of Exam School Effects Away From the Cutoff . Journal of the American Statistical Association , 110(512):1331--1344

work page 2015
[3]

Arai, Y., Hsu, Y., Kitagawa, T., Mourifié, I., and Wan, Y. (2022). Testing identifying assumptions in fuzzy regression discontinuity designs . Quantitative Economics , 13(1)

work page 2022
[4]

and Kumar, R

Babii, A. and Kumar, R. (2023). Isotonic Regression Discontinuity Designs . Journal of Econometrics , 234(2):371--393

work page 2023
[5]

Bertanha, M. (2020). Regression Discontinuity Design with Many Thresholds . Journal of Econometrics , 218(1):216--241

work page 2020
[6]

and Imbens, G

Bertanha, M. and Imbens, G. W. (2020). External Validity in Fuzzy Regression Discontinuity Designs . Journal of Business & Economic Statistics , 38(3):593--612

work page 2020
[7]

W., Jackson, C

Beuermann, D. W., Jackson, C. K., Navarro-Sola, L., and Pardo, F. (2022). What is a Good School, and Can Parents Tell? Evidence on the Multidimensionality of School Output . The Review of Economic Studies , 90(1):65--101

work page 2022
[8]

Björklund, A., Lindahl, M., and Plug, E. (2006). The Origins of Intergenerational Associations: Lessons from Swedish Adoption Data . The Quarterly Journal of Economics , 121(3):999--1028

work page 2006

Show all 55 references

[9]

Bugni, F. A. and Canay, I. A. (2021). Testing Continuity of a Density via G-order Statistics in the Regression Discontinuity Design . Journal of Econometrics , 221(1):138--159

2021
[10]

D., and Farrell, M

Calonico, S., Cattaneo, M. D., and Farrell, M. H. (2018). On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference . Journal of the American Statistical Association , 113(522):767--779

2018
[11]

D., and Farrell, M

Calonico, S., Cattaneo, M. D., and Farrell, M. H. (2019). nprobust: Nonparametric Kernel-Based Estimation and Robust Bias-Corrected Inference . Journal of Statistical Software , 91(8):1–33

2019
[12]

D., and Farrell, M

Calonico, S., Cattaneo, M. D., and Farrell, M. H. (2020). Optimal Bandwidth Choice for Robust Bias-Corrected Inference in Regression Discontinuity Designs . The Econometrics Journal , 23(2):192--210

2020
[13]

D., and Farrell, M

Calonico, S., Cattaneo, M. D., and Farrell, M. H. (2022). Coverage Error Optimal Confidence Intervals for Local Polynomial Regression . Bernoulli , 28(4):2998--3022

2022
[14]

D., and Titiunik, R

Calonico, S., Cattaneo, M. D., and Titiunik, R. (2014). Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs . Econometrica , 82(6):2295--2326

2014
[15]

Canay, I. A. and Kamat, V. (2017). Approximate Permutation Tests and Induced Order Statistics in the Regression Discontinuity Design . The Review of Economic Studies , 85(3):1577--1608

2017
[16]

D., Jansson, M., and Ma, X

Cattaneo, M. D., Jansson, M., and Ma, X. (2020). Simple Local Polynomial Density Estimators . Journal of the American Statistical Association , 115(531):1449--1455

2020
[17]

D., Keele, L., Titiunik, R., and Vazquez-Bare, G

Cattaneo, M. D., Keele, L., Titiunik, R., and Vazquez-Bare, G. (2016). Interpreting Regression Discontinuity Designs with Multiple Cutoffs . The Journal of Politics , 78(4):1229--1248

2016
[18]

D., Keele, L., Titiunik, R., and Vazquez-Bare, G

Cattaneo, M. D., Keele, L., Titiunik, R., and Vazquez-Bare, G. (2021). Extrapolating Treatment Effects in Multi-Cutoff Regression Discontinuity Designs . Journal of the American Statistical Association , 116(536):1941--1952

2021
[19]

Cattaneo, M. D. and Titiunik, R. (2022). Regression Discontinuity Designs . Annual Review of Economics , 14:821--851

2022
[20]

Cerulli, G., Dong, Y., Lewbel, A., and Poulsen, A. (2017). Testing Stability of Regression Discontinuity Models . In Regression Discontinuity Designs , volume 38 of Advances in Econometrics , pages 317--339. Emerald Publishing Limited

2017
[21]

Chernozhukov, V., Chetverikov, D., and Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors . The Annals of Statistics , 41(6):2786 -- 2819

2013
[22]

Chernozhukov, V., Chetverikov, D., and Kato, K. (2014a). Anti-concentration and honest, adaptive confidence bands . The Annals of Statistics , 42(5):1787 -- 1818

2014
[23]

Chernozhukov, V., Chetverikov, D., and Kato, K. (2014b). Gaussian approximation of suprema of empirical processes . The Annals of Statistics , 42(4):1564 -- 1597

2014
[24]

Chetverikov, D. (2019). TESTING REGRESSION MONOTONICITY IN ECONOMETRIC MODELS . Econometric Theory , 35(4):729–776

2019
[25]

and Kwon, S

Deaner, B. and Kwon, S. (2025). Extrapolation in Regression Discontinuity Design Using Comonotonicity . arXiv:2507.00289

2025 arXiv
[26]

and Lewbel, A

Dong, Y. and Lewbel, A. (2015). Identifying the Effect of Changing the Policy Threshold in Regression Discontinuity Models . The Review of Economics and Statistics , 97(5):1081--1092

2015
[27]

and Gijbels, I

Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . Chapman & Hall/CRC

1996
[28]

P., and Zhang, Y

Fan, Q., Hsu, Y.-C., Lieli, R. P., and Zhang, Y. (2022). Estimation of Conditional Average Treatment Effects With High-Dimensional Data . Journal of Business & Economic Statistics , 40(1):313--327

2022
[29]

and Levine, D

Fudenberg, D. and Levine, D. K. (2022). Learning in Games and the Interpretation of Natural Experiments . American Economic Journal: Microeconomics , 14(3):353--77

2022
[30]

Fusejima, K., Ishihara, T., and Sawada, M. (2025). A Unified Diagnostic Test for Regression Discontinuity Designs . Journal of Econometrics , 251:106074

2025
[31]

Gerard, F., Rokkanen, M., and Rothe, C. (2020). Bounds on Treatment Effects in Regression Discontinuity Designs with a Manipulated Running Variable . Quantitative Economics , 11(3):839--870

2020
[32]

Hahn, J., Todd, P., and Van der Klaauw, W. (2001). Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design . Econometrica , 69(1):201--209

2001
[33]

Imai, S., Qin, L., and Yanagi, T. (2025). Doubly Robust Uniform Confidence Bands for Group-Time Conditional Average Treatment Effects in Difference-in-Differences . arXiv:2305.02185

2025 arXiv
[34]

and Kalyanaraman, K

Imbens, G. and Kalyanaraman, K. (2011). Optimal Bandwidth Choice for the Regression Discontinuity Estimator . The Review of Economic Studies , 79(3):933--959

2011
[35]

Imbens, G. W. and Manski, C. F. (2004). Confidence Intervals for Partially Identified Parameters . Econometrica , 72(6):1845--1857

2004
[36]

Imbens, G. W. and Rubin, D. B. (2015). Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction . Cambridge University Press

2015
[37]

and Piterbarg, V

Konakov, V. and Piterbarg, V. (1984). On the convergence rate of maximal deviation distribution for kernel regression estimates . Journal of Multivariate Analysis , 15(3):279--294

1984
[38]

V., Gundersen, C., and Jolliffe, D

Kreider, B., Pepper, J. V., Gundersen, C., and Jolliffe, D. (2012). Identifying the Effects of SNAP (Food Stamps) on Child Health Outcomes When Participation Is Endogenous and Misreported . Journal of the American Statistical Association , 107(499):958--975

2012
[39]

Lee, D. S. (2008). Randomized experiments from non-random selection in U.S. House elections . Journal of Econometrics , 142(2):675--697. The regression discontinuity design: Theory and applications

2008
[40]

Lee, S., Okui, R., and Whang, Y.-J. (2017). Doubly Robust Uniform Confidence Band for the Conditional Average Treatment Effect Function . Journal of Applied Econometrics , 32(7):1207--1225

2017
[41]

Londoño-Vélez, J., Rodríguez, C., and Sánchez, F. (2020). Upstream and Downstream Impacts of College Merit-Based Financial Aid for Low-Income Students: Ser Pilo Paga in Colombia . American Economic Journal: Economic Policy , 12(2):193–227

2020
[42]

Mammen, E. (1993). Bootstrap and Wild Bootstrap for High Dimensional Linear Models . The Annals of Statistics , 21(1):255--285

1993
[43]

Manski, C. F. (1997). Monotone Treatment Response . Econometrica , 65(6):1311--1334

1997
[44]

Manski, C. F. and Pepper, J. V. (2018). How Do Right-to-Carry Laws Affect Crime Rates? Coping with Ambiguity Using Bounded-Variation Assumptions . The Review of Economics and Statistics , 100(2):232--244

2018
[45]

Marx, P., Tamer, E., and Tang, X. (2024). Parallel Trends and Dynamic Choices . Journal of Political Economy Microeconomics , 2(1):129--171

2024
[46]

McCrary, J. (2008). Manipulation of the Running Variable in the Regression Discontinuity Design: A Density Test . Journal of Econometrics , 142(2):698--714

2008
[47]

Mehta, N. (2019). An Economic Approach to Generalizing Findings from Regression-Discontinuity Designs . Journal of Human Resources , 54(4):953--985

2019
[48]

Melguizo, T., Sanchez, F., and Velasco, T. (2016). Credit for Low-Income Students and Access to and Academic Performance in Higher Education in Colombia: A Regression Discontinuity Approach . World Development , 80:61--77

2016
[49]

Oreopoulos, P. (2006). Estimating Average and Local Average Treatment Effects of Education when Compulsory Schooling Laws Really Matter . American Economic Review , 96(1):152–175

2006
[50]

and Roth, J

Rambachan, A. and Roth, J. (2023). A More Credible Approach to Parallel Trends . The Review of Economic Studies , 90(5):2555--2591

2023
[51]

and Sant'Anna, P

Roth, J. and Sant'Anna, P. H. C. (2023). When Is Parallel Trends Sensitive to Functional Form? Econometrica , 91(2):737--747

2023
[52]

Stoye, J. (2009). More on Confidence Intervals for Partially Identified Parameters . Econometrica , 77(4):1299--1315

2009
[53]

Sun, Y. (2023). Extrapolating Away from the Cutoff in Regression Discontinuity Designs . arXiv:2311.18136

2023 arXiv
[54]

Todd, P. E. and Wolpin, K. I. (2023). The Best of Both Worlds: Combining Randomized Controlled Trials with Structural Modeling . Journal of Economic Literature , 61(1):41–85

2023
[55]

van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics . Springer Series in Statistics. Springer Science+Business Media, New York

1996

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

and Cattaneo, M

Abadie, A. and Cattaneo, M. D. (2018). Econometric Methods for Program Evaluation . Annual Review of Economics , 10:465--503

work page 2018

[2] [2]

Angrist, J. D. and Rokkanen, M. (2015). Wanna Get Away? Regression Discontinuity Estimation of Exam School Effects Away From the Cutoff . Journal of the American Statistical Association , 110(512):1331--1344

work page 2015

[3] [3]

Arai, Y., Hsu, Y., Kitagawa, T., Mourifié, I., and Wan, Y. (2022). Testing identifying assumptions in fuzzy regression discontinuity designs . Quantitative Economics , 13(1)

work page 2022

[4] [4]

and Kumar, R

Babii, A. and Kumar, R. (2023). Isotonic Regression Discontinuity Designs . Journal of Econometrics , 234(2):371--393

work page 2023

[5] [5]

Bertanha, M. (2020). Regression Discontinuity Design with Many Thresholds . Journal of Econometrics , 218(1):216--241

work page 2020

[6] [6]

and Imbens, G

Bertanha, M. and Imbens, G. W. (2020). External Validity in Fuzzy Regression Discontinuity Designs . Journal of Business & Economic Statistics , 38(3):593--612

work page 2020

[7] [7]

W., Jackson, C

Beuermann, D. W., Jackson, C. K., Navarro-Sola, L., and Pardo, F. (2022). What is a Good School, and Can Parents Tell? Evidence on the Multidimensionality of School Output . The Review of Economic Studies , 90(1):65--101

work page 2022

[8] [8]

Björklund, A., Lindahl, M., and Plug, E. (2006). The Origins of Intergenerational Associations: Lessons from Swedish Adoption Data . The Quarterly Journal of Economics , 121(3):999--1028

work page 2006