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REVIEW 2 major objections 5 minor 121 references

Conformalized Lee Inference: Distribution-Free Individual Treatment Effect Intervals under Monotone Sample Selection

T0 review · 2 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Under monotone selection, calibrating conformal scores at the (1−απ) treated-selected quantile yields finite-sample coverage for always-selected treated outcomes and individual treatment effects.

desk verdict Clean conversion of Lee monotonicity into a sharp, finite-sample conformal cutoff for always-selected counterfactuals and ITEs; the math holds. read the letter →

arxiv 2607.02898 v1 pith:BJD7XCON submitted 2026-07-03 econ.EM

Jung Hyub Lee This is my paper · ORCID
classification econ.EM
keywords conformalpredictionLeeboundssampleselectionindividualtreatmenteffectscounterfactualpartialidentificationdistribution-freeinferencemonotone
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

When treatment changes who is observed, the treated-selected sample is a mixture of always-selected units and units induced into selection by treatment. Ordinary conformal prediction therefore under-covers the always-selected target. This paper shows that the monotone-selection logic of Lee bounds identifies a sharp likelihood-ratio bound: the always-selected law can put at most 1/π times as much mass on any event as the treated-selected law, where π is the observed control-to-treatment selection ratio. The conformalized Lee procedure trains any score on treated-selected data and simply raises the calibration quantile from 1−α to 1−απ. The resulting intervals cover the missing treated outcome of a future always-selected unit at the nominal level in finite samples, without requiring the prediction rule to be correct. For a selected control the untreated outcome is observed, so subtracting it converts the interval into a valid individual-treatment-effect interval for the always-selected population. Simulations confirm that ordinary conformal under-covers under selection shift while the Lee adjustment restores coverage.

What carries the argument

The Lee ambiguity set Q(P,π) = {Q ≪ P : 0 ≤ dQ/dP ≤ 1/π}, the sharp distributional identification region for the always-selected treated law. It converts the coverage problem into a robust conformal calibration that simply replaces the usual 1−α score quantile with the higher 1−απ treated-selected quantile.

What would settle it

In a randomized experiment with known one-sided selection, compare empirical coverage of ordinary conformal versus Lee-adjusted intervals on a held-out always-selected sample; if selection concentrates high residual scores among always-selected units and ordinary coverage falls well below 1−α while Lee coverage stays near target, the claim holds; systematic Lee under-coverage would falsify it.

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Extended reading notes

Core claim

Under random assignment and monotone selection, the split-conformal threshold formed from the (1−απ) quantile of treated-selected nonconformity scores delivers finite-sample marginal coverage of the always-selected treated potential outcome uniformly over every law in the sharp Lee ambiguity set {Q ≪ P : 0 ≤ dQ/dP ≤ 1/π}, and that population quantile is the smallest threshold that guarantees the uniform guarantee.

Load-bearing premise

Treatment can only raise, never lower, the chance of being observed (no units selected under control but not under treatment).

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops conformalized Lee inference for distribution-free prediction of treated potential outcomes and individual treatment effects under randomized assignment and monotone sample selection. Treated-selected observations are a mixture of always-selected and marginal-in units, so ordinary split conformal calibration under the treated-selected law P does not control coverage under the always-selected target law Q0. The paper shows that Q0 lies in the sharp Lee ambiguity set Q(P, π) = {Q ≪ P : 0 ≤ dQ/dP ≤ 1/π}, with π = p0/p1 identified from selection rates (Propositions 1–2). The proposed procedure trains any score on treated-selected data and calibrates at the (1 − απ) treated-selected score quantile; Theorem 1 gives finite-sample marginal coverage uniformly over Q(P, π), and Corollary 1 translates this into ITE intervals for selected controls by subtracting observed Y(0). Propositions 3–4 establish that the population cutoff F_P^{−1}(1 − απ) is minimax optimal over the reduced-information ambiguity set. Simulations show that naive conformal undercovers under tail and smooth selection, while Lee-adjusted methods restore coverage.

Significance. If the results hold, the paper supplies a clean, implementable bridge between Lee’s monotone-selection logic and modern conformal prediction for counterfactual outcomes and ITEs. The contribution is not merely a sensitivity wrapper: the robustness class is identified by selection rates rather than chosen by the analyst, the finite-sample coverage argument is fully proved with standard exchangeable-rank devices, and the cutoff is shown to be sharp over the reduced-information set. The method is model-agnostic for the prediction rule, works with residual and CQR scores, and includes practical plug-in and lower-bound constructions for π. That combination is useful for applied work with one-sided selection (e.g., employment, survey response, survival) where average Lee bounds are already used but unit-level predictive intervals are desired. The appendix full-law identification result also points to a natural covariate-adaptive extension.

major comments (2)
  1. [§4, Appendix 7.3, Tables 3–6] The main procedure deliberately uses only reduced information (P, p0, p1) even though Appendix 7.3 shows that the full observed law identifies a strictly smaller class Q_full with known always-selected covariate marginal H_X and local bounds 1/π(x). The paper acknowledges this and notes that the reduced procedure remains valid but may be conservative. For a methods paper whose selling point is sharp use of Lee information, the manuscript should either (i) implement and compare a simple covariate-adaptive version (e.g., weighted or stratified calibration using estimated π(x) or reweighting toward H_X) in the simulations, or (ii) quantify how much length is left on the table under the current designs. Without that, the practical claim that the method “uses the exact amount of uncertainty implied by” Lee logic is overstated relative to the information actually available in the data.
  2. [Corollary 1, Abstract, §1] Coverage is only marginal over the always-selected (selected-control) population, not conditional on (X, Y(0)). Corollary 1 is correctly stated, but applied readers will often want intervals that are valid given covariates or given the observed untreated outcome. The paper should either provide a conditional or approximate-conditional extension (e.g., via localized scores or binning) or more prominently caveat that the ITE guarantee does not control coverage for a fixed individual. This is load-bearing for the “individual treatment effect intervals” framing in the title and abstract.
minor comments (5)
  1. [§1–2] Section numbering is duplicated (“1 Introduction / 2 Introduction”). Clean up the front matter.
  2. [Algorithm 1, Theorem 1] Notation for the estimated share switches among ˆπ, bπ, and π_used; standardize and define the plug-in vs lower-bound versions once in Algorithm 1.
  3. [Figures 1–4] Figures 1–4 are described in the text but the manuscript would benefit from explicit axis labels and a short note that solid/dashed curves are m=100/200 so readers can parse the panels without the caption alone.
  4. [§2.1, §4.2] Related-work discussion of Jin et al. (2023) and weighted conformal methods is good; a one-sentence comparison of computational cost (order-statistic vs PAC envelope) would help practitioners.
  5. [Table 5, §5] Table 5 reports Pr(k=m+1) for Hoeffding-Lee at small π; the main text should flag more clearly that infinite intervals are valid but uninformative and how often they arise in the designs.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: coverage and minimax claims follow from the Lee LR bound plus exchangeable ranks under P, not from fitted targets or self-citation.

full rationale

The paper’s central chain is self-contained and non-circular. Monotone selection plus random assignment yield the mixture P = πQ0 + (1−π)R and the sharp Lee ambiguity set Q(P,π) = {Q ≪ P : 0 ≤ dQ/dP ≤ 1/π} (Propositions 1–2). The transfer inequality Q(At) ≤ P(At)/π then forces calibration at the treated-selected (1−απ) score quantile so that every admissible Q has tail at most α; Theorem 1 proves finite-sample coverage via an auxiliary P-draw and lexicographic ranks under exchangeability of calibration scores, and Propositions 3–4 show that F_P^{-1}(1−απ) is the smallest uniformly valid population threshold over that set. π is identified from selection rates (D,S) alone—oracle, plug-in, or one-sided Clopper–Pearson/Hoeffding lower bounds—independent of outcomes and of the nonconformity scores. Simulations evaluate coverage on independent always-selected draws, not on quantities used to fit the cutoff. Lee [2009] is an external citation (different author); there is no load-bearing self-citation, no uniqueness theorem imported from the present author, and no fitted parameter renamed as a prediction. The derivation does not reduce to its inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The result rests on three maintained domain assumptions (random assignment, monotone selection, positivity), standard conformal/exchangeability math, and the derived Lee ambiguity set. Free parameters are only user-chosen error levels (α, δ_π) and simulation design knobs, not constants fitted to force the coverage claim. No new physical entity is postulated; Q(P,π) is an identified set from observables.

free parameters (3)
  • target miscoverage α
    User-chosen nominal level for prediction sets; not estimated from data but required to define the cutoff 1−απ.
  • confidence budget δ_π for lower bounds on π
    Hand-chosen (simulations use 0.05) split between arms for Clopper–Pearson/Hoeffding lower bounds; trades coverage guarantee 1−α−δ_π against interval width.
  • train/calibration split of treated-selected sample
    Random split size m is a design choice affecting finite-sample length and stability; not identified by theory.
assumptions (5)
  • domain assumption Random assignment: (Y(1),Y(0),S(1),S(0),X) ⊥ D (Assumption 1).
    Identifies potential-outcome and selection laws from treatment arms; stated in Section 3.
  • domain assumption Monotone selection: S(1) ≥ S(0) a.s. (Assumption 2).
    Creates principal strata A/M/N and identifies π = p0/p1 as always-selected share among treated-selected units.
  • domain assumption Positivity: selection rates p0,p1 bounded away from 0 and 1 (Assumption 3).
    Keeps π well-defined and selected populations nondegenerate.
  • standard math Split conformal exchangeability of calibration scores under the treated-selected law P, with lexicographic tie-breaking.
    Used in Theorem 1 proof via auxiliary draw and uniform rank; standard conformal device.
  • standard math Standard Borel space for Z = (X,Y(1)) so Radon–Nikodym derivatives and regular conditionals exist.
    Invoked for Q(P,π) and full-law Q_full constructions.
invented entities (1)
  • Lee ambiguity set Q(P,π) independent evidence
    purpose: Collects all always-selected target laws consistent with treated-selected law P and share π under monotone selection.
    Derived as necessary and sharp from reduced observables (Props. 1–2), not postulated as a free sensitivity class; independent handle is the observed selection rates and treated-selected distribution.

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Pith. "Pith review of Conformalized Lee Inference: Distribution-Free Individual Treatment Effect Intervals under Monotone Sample Selection." pith.science (2026). https://pith.science/paper/BJD7XCON

@misc{pith2026260702898,
  author       = {Pith},
  title        = {Pith review of: Conformalized Lee Inference: Distribution-Free Individual Treatment Effect Intervals under Monotone Sample Selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJD7XCON}},
  note         = {Machine review of arXiv:2607.02898}
}
read the original abstract

Empirical studies often observe outcomes only for selected units, and treatment may change who is observed. This paper studies prediction in randomized studies with one-sided selection. Standard prediction intervals can fail because treated selected observations are not the same group as selected controls. The paper asks how to predict missing treated outcomes and individual treatment effects for always-observed units. The proposed conformalized Lee procedure uses treated selected observations to train and check any prediction rule, then adjusts the cutoff using the observed treatment-control selection gap. For selected controls, the missing treated-outcome interval is shifted by the observed untreated outcome to produce an individual treatment-effect interval. The method provides reliable coverage without requiring the prediction rule to be correctly specified. The key result shows that the proposed adjustment uses the exact amount of uncertainty implied by the monotone selection logic of Lee [2009]. In simulations, ordinary conformal prediction demonstrates a lower coverage rate under selection-induced distribution shift, while the Lee-adjusted methods achieve the desired coverage rate. The results show that the proposed selection correction method can support reliable counterfactual prediction, while retaining practical implementation with modern prediction tools.

Figures

Figures reproduced from arXiv: 2607.02898 by the authors.

Figure 1
Figure 1. Coverage curves under benign selection. Columns correspond to naive conformal, plug [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Coverage curves under conditional-tail selection. Naive conformal under-covers sharply [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Coverage curves under unconditional-tail selection. This design induces both residual [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Coverage curves under smooth heterogeneous selection. Naive conformal under-covers [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]

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Reference graph

Works this paper leans on

121 extracted references · 6 linked inside Pith

  1. [1]

    Journal of the American Statistical Association , volume =

    Andrew Gelman and Aki Vehtari , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2021.1938081 , URL =

    doi:10.1080/01621459.2021.1938081 2021

  2. [2]

    Broman and Kara H

    Karl W. Broman and Kara H. Woo , title =. The American Statistician , volume =. 2018 , publisher =. doi:10.1080/00031305.2017.1375989 , URL =

    arXiv doi:10.1080/00031305.2017.1375989 2018

  3. [3]

    Bishop, Yvonne M. M. and Fienberg, Stephen E. and Holland, Paul W. , title =. 1975 , pages =

    1975

  4. [4]

    Journal of Econometrics , year=

    Estimation of treatment effects under endogenous heteroskedasticity , author=. Journal of Econometrics , year=

  5. [5]

    Journal of the Royal Statistical Society

    Doubly robust estimation of the local average treatment effect curve , author=. Journal of the Royal Statistical Society. Series B, Statistical methodology , volume=. 2015 , publisher=

    2015

  6. [6]

    2018 , publisher=

    Double/debiased machine learning for treatment and structural parameters , author=. 2018 , publisher=

    2018

  7. [7]

    Review of Economic Studies , volume=

    Asymptotic efficiency of semiparametric two-step GMM , author=. Review of Economic Studies , volume=. 2014 , publisher=

    2014

  8. [8]

    Journal of Business & Economic Statistics , volume=

    Identification of the direction of a causal effect by instrumental variables , author=. Journal of Business & Economic Statistics , volume=. 2016 , publisher=

    2016

Show all 121 references
  1. [9]

    Journal of the Royal Statistical Society , volume=

    Direct and Indirect Treatment Effects–causal Chains and Mediation Analysis with Instrumental Variables , author=. Journal of the Royal Statistical Society , volume=

  2. [10]

    Econometrica , volume=

    Testing for causal effects in a generalized regression model with endogenous regressors , author=. Econometrica , volume=. 2010 , publisher=

    2010

  3. [11]

    Sociological methodology , volume=

    Asymptotic confidence intervals for indirect effects in structural equation models , author=. Sociological methodology , volume=. 1982 , publisher=

    1982

  4. [12]

    Journal of Educational and Behavioral Statistics , volume=

    Identification of causal parameters in randomized studies with mediating variables , author=. Journal of Educational and Behavioral Statistics , volume=. 2008 , publisher=

    2008

  5. [13]

    Journal of Econometrics , volume=

    Non-parametric analysis of a generalized regression model: the maximum rank correlation estimator , author=. Journal of Econometrics , volume=. 1987 , publisher=

    1987

  6. [14]

    , author=

    The moderator--mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. , author=. Journal of personality and social psychology , volume=. 1986 , publisher=

    1986

  7. [15]

    , author=

    A comparison of methods to test mediation and other intervening variable effects. , author=. Psychological methods , volume=. 2002 , publisher=

    2002

  8. [16]

    Journal of Economic Theory , volume=

    Heterogeneous quantal response equilibrium and cognitive hierarchies , author=. Journal of Economic Theory , volume=. 2009 , publisher=

    2009

  9. [17]

    Handbook of econometrics , volume=

    Large sample estimation and hypothesis testing , author=. Handbook of econometrics , volume=. 1994 , publisher=

    1994

  10. [18]

    The Annals of Statistics , volume=

    Maximal inequalities for degenerate U -processes with applications to optimization estimators , author=. The Annals of Statistics , volume=. 1994 , publisher=

    1994

  11. [19]

    Econometric Theory , volume=

    Local rank estimation of transformation models with functional coefficients , author=. Econometric Theory , volume=. 2010 , publisher=

    2010

  12. [20]

    The Econometrics Journal , volume=

    Rank estimation of partially linear index models , author=. The Econometrics Journal , volume=. 2011 , publisher=

    2011

  13. [21]

    The Annals of Statistics , volume=

    U-processes: rates of convergence , author=. The Annals of Statistics , volume=. 1987 , publisher=

    1987

  14. [22]

    Econometrica , volume=

    Simulation and the Asymptotics of Optimization Estimators , author=. Econometrica , volume=. 1989 , publisher=

    1989

  15. [23]

    Journal of Econometrics , volume=

    Two-stage rank estimation of quantile index models , author=. Journal of Econometrics , volume=. 2001 , publisher=

    2001

  16. [24]

    Econometric theory , volume=

    U-processes in the analysis of a generalized semiparametric regression estimator , author=. Econometric theory , volume=. 1994 , publisher=

    1994

  17. [25]

    Econometrica , volume=

    The Limiting Distribution of the Maximum Rank Correlation Estimator , author=. Econometrica , volume=. 1993 , publisher=

    1993

  18. [26]

    Econometrica , volume=

    Semiparametric Estimation of Index Coefficients , author=. Econometrica , volume=. 1989 , publisher=

    1989

  19. [27]

    Quantitative Economics , volume=

    Inference on semiparametric multinomial response models , author=. Quantitative Economics , volume=. 2021 , publisher=

    2021

  20. [28]

    Journal of Econometrics , volume=

    Rank estimators for monotonic index models , author=. Journal of Econometrics , volume=. 1998 , publisher=

    1998

  21. [29]

    Handbook of Labor, Human Resources and Population Economics , pages=

    Mediation analysis , author=. Handbook of Labor, Human Resources and Population Economics , pages=. 2020 , publisher=

    2020

  22. [30]

    Journal of Economic Surveys , volume=

    Causal mediation analysis in economics: Objectives, assumptions, models , author=. Journal of Economic Surveys , volume=. 2022 , publisher=

    2022

  23. [31]

    Comptes Rendus Mathematique , volume=

    On the distribution of the product of correlated normal random variables , author=. Comptes Rendus Mathematique , volume=. 2016 , publisher=

    2016

  24. [32]

    Economics letters , volume=

    Computation of the maximum rank correlation estimator , author=. Economics letters , volume=. 1999 , publisher=

    1999

  25. [33]

    Journal of the Royal Statistical Society: Series B (Methodological) , volume=

    On the behaviour of Wald statistics for the disjunction of two regular hypotheses , author=. Journal of the Royal Statistical Society: Series B (Methodological) , volume=. 1993 , publisher=

    1993

  26. [34]

    Journal of the American Statistical Association , volume=

    Large-scale hypothesis testing for causal mediation effects with applications in genome-wide epigenetic studies , author=. Journal of the American Statistical Association , volume=. 2022 , publisher=

    2022

  27. [35]

    arXiv preprint arXiv:2012.11342 , year=

    A nearly similar powerful test for mediation , author=. arXiv preprint arXiv:2012.11342 , year=

    2012 arXiv

  28. [36]

    Unpublished manuscript , year=

    Identification and estimation of causal mediation effects with treatment noncompliance , author=. Unpublished manuscript , year=

  29. [37]

    The Stata Journal , volume=

    Causal mediation analysis in instrumental-variables regressions , author=. The Stata Journal , volume=. 2020 , publisher=

    2020

  30. [38]

    The Economic Journal , volume=

    The effect of trade on workers and voters , author=. The Economic Journal , volume=. 2022 , publisher=

    2022

  31. [39]

    Bernoulli , volume=

    Wald tests of singular hypotheses , author=. Bernoulli , volume=

  32. [40]

    arXiv preprint arXiv:1312.0569 , year=

    Wald tests when restrictions are locally singular , author=. arXiv preprint arXiv:1312.0569 , year=

    arXiv

  33. [41]

    UvA-Econometrics Discussion Papers , year=

    Inference about the indirect effect: a likelihood approach , author=. UvA-Econometrics Discussion Papers , year=

  34. [42]

    IZA Discussion Papers , year=

    The marginal income effect of education on happiness: Estimating the direct and indirect effects of compulsory schooling on well-being in Australia , author=. IZA Discussion Papers , year=

  35. [43]

    , author=

    Epidemiologic Approaches to Understanding Mechanisms of Cardiovascular Diseases: Genes, Environment, and DNA Methylation. , author=. 2015 , journal =

    2015

  36. [44]

    Multivariate behavioral research , volume=

    Addressing moderated mediation hypotheses: Theory, methods, and prescriptions , author=. Multivariate behavioral research , volume=. 2007 , publisher=

    2007

  37. [45]

    American Economic Review , volume=

    Estimating average and local average treatment effects of education when compulsory schooling laws really matter , author=. American Economic Review , volume=. 2006 , publisher=

    2006

  38. [46]

    Health economics , volume=

    The causal effect of education on health: What is the role of health behaviors? , author=. Health economics , volume=. 2016 , publisher=

    2016

  39. [47]

    , author=

    A general approach to causal mediation analysis. , author=. Psychological methods , volume=. 2010 , publisher=

    2010

  40. [48]

    Statistical Science , volume =

    Identification, inference and sensitivity analysis for causal mediation effects , author=. Statistical Science , volume =

  41. [49]

    Journal of business & economic statistics , volume=

    A stochastic frontier model with endogenous treatment status and mediator , author=. Journal of business & economic statistics , volume=. 2020 , publisher=

    2020

  42. [50]

    International journal of epidemiology , volume=

    Network Mendelian randomization: using genetic variants as instrumental variables to investigate mediation in causal pathways , author=. International journal of epidemiology , volume=. 2015 , publisher=

    2015

  43. [51]

    Multivariate Behavioral Research , volume=

    Assessing mediational models: Testing and interval estimation for indirect effects , author=. Multivariate Behavioral Research , volume=. 2010 , publisher=

    2010

  44. [52]

    Epidemiology , volume=

    Identifiability and exchangeability for direct and indirect effects , author=. Epidemiology , volume=. 1992 , publisher=

    1992

  45. [53]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=

    Identifying direct and indirect effects in a non-counterfactual framework , author=. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume=. 2007 , publisher=

    2007

  46. [54]

    The international journal of biostatistics , volume=

    Extended instrumental variables estimation for overall effects , author=. The international journal of biostatistics , volume=. 2008 , publisher=

    2008

  47. [55]

    Econometric reviews , volume=

    Econometric mediation analyses: Identifying the sources of treatment effects from experimentally estimated production technologies with unmeasured and mismeasured inputs , author=. Econometric reviews , volume=. 2015 , publisher=

    2015

  48. [56]

    American Economic Review , volume=

    Understanding the mechanisms through which an influential early childhood program boosted adult outcomes , author=. American Economic Review , volume=. 2013 , publisher=

    2013

  49. [57]

    Probabilistic and Causal Inference: The Works of Judea Pearl , pages=

    Direct and Indirect Effects , author=. Probabilistic and Causal Inference: The Works of Judea Pearl , pages=. 2022 , publisher=

    2022

  50. [58]

    Evaluation review , volume=

    Process analysis: Estimating mediation in treatment evaluations , author=. Evaluation review , volume=. 1981 , publisher=

    1981

  51. [59]

    Journal of Human Resources , volume=

    The impact of family composition on educational achievement , author=. Journal of Human Resources , volume=. 2019 , publisher=

    2019

  52. [60]

    Econometrics , volume=

    Multiple discrete endogenous variables in weakly-separable triangular models , author=. Econometrics , volume=. 2016 , publisher=

    2016

  53. [61]

    Journal of Econometrics , volume=

    Semiparametric bounds on treatment effects , author=. Journal of Econometrics , volume=. 2010 , publisher=

    2010

  54. [62]

    Journal of Econometrics , volume=

    Instrumental variables and the sign of the average treatment effect , author=. Journal of Econometrics , volume=. 2019 , publisher=

    2019

  55. [63]

    American Economic Review , volume=

    Treatment effect bounds under monotonicity assumptions: an application to Swan-Ganz catheterization , author=. American Economic Review , volume=. 2008 , publisher=

    2008

  56. [64]

    Econometrica , volume=

    Partial identification in triangular systems of equations with binary dependent variables , author=. Econometrica , volume=. 2011 , publisher=

    2011

  57. [65]

    2005 , publisher=

    Threshold crossing models and bounds on treatment effects: a nonparametric analysis , author=. 2005 , publisher=

    2005

  58. [66]

    Journal of econometrics , volume=

    Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator , author=. Journal of econometrics , volume=. 1985 , publisher=

    1985

  59. [67]

    Econometrica , volume=

    An Efficient Semiparametric Estimator for Binary Response Models , author=. Econometrica , volume=. 1993 , publisher=

    1993

  60. [68]

    Journal of Econometrics , volume=

    Rank estimation of a generalized fixed-effects regression model , author=. Journal of Econometrics , volume=. 2000 , publisher=

    2000

  61. [69]

    Journal of econometrics , volume=

    Semiparametric least squares (SLS) and weighted SLS estimation of single-index models , author=. Journal of econometrics , volume=. 1993 , publisher=

    1993

  62. [70]

    Econometrica , volume=

    Identification in nonseparable models , author=. Econometrica , volume=. 2003 , publisher=

    2003

  63. [71]

    Econometrica , volume=

    Nonparametric identification under discrete variation , author=. Econometrica , volume=. 2005 , publisher=

    2005

  64. [72]

    Econometrica , volume=

    Identification and estimation of triangular simultaneous equations models without additivity , author=. Econometrica , volume=. 2009 , publisher=

    2009

  65. [73]

    arXiv preprint arXiv:2110.04388 , year=

    Estimating high dimensional monotone index models by iterative convex optimization , author=. arXiv preprint arXiv:2110.04388 , year=

    arXiv

  66. [74]

    Statistical Science , volume=

    The emperor’s new tests , author=. Statistical Science , volume=. 1999 , publisher=

    1999

  67. [75]

    Review of Economics and Statistics , pages=

    A nearly similar powerful test for mediation , author=. Review of Economics and Statistics , pages=. 2024 , publisher=

    2024

  68. [76]

    Journal of Econometrics , volume =

    Estimating high dimensional monotone index models by iterative convex optimization , author=. Journal of Econometrics , volume =. 2024 , publisher=

    2024

  69. [77]

    Journal of Applied Econometrics , volume=

    The dynamics of health in the British Household Panel Survey , author=. Journal of Applied Econometrics , volume=. 2004 , publisher=

    2004

  70. [78]

    Journal of Epidemiology & Community Health , volume=

    Employment transitions and mental health: an analysis from the British household panel survey , author=. Journal of Epidemiology & Community Health , volume=. 2005 , publisher=

    2005

  71. [79]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume=

    Health-related non-response in the British Household Panel Survey and European Community Household Panel: using inverse-probability-weighted estimators in non-linear models , author=. Journal of the Royal Statistical Society Series A: Statistics in Society , volume=. 2006 , publisher=

    2006

  72. [80]

    The Annals of Statistics , volume=

    Wald tests when restrictions are locally singular , author=. The Annals of Statistics , volume=. 2025 , publisher=

    2025

  73. [81]

    arXiv preprint arXiv:2404.11739 , year=

    Testing mechanisms , author=. arXiv preprint arXiv:2404.11739 , year=

  74. [82]

    Journal of Applied Econometrics , year=

    Testing sign congruence between two parameters , author=. Journal of Applied Econometrics , year=

  75. [83]

    Journal of Business & Economic Statistics , volume=

    The finite sample performance of estimators for mediation analysis under sequential conditional independence , author=. Journal of Business & Economic Statistics , volume=. 2016 , publisher=

    2016

  76. [84]

    Journal of Business & Economic Statistics , volume=

    Estimations and tests for generalized mediation models with high-dimensional potential mediators , author=. Journal of Business & Economic Statistics , volume=. 2024 , publisher=

    2024

  77. [85]

    Journal of Business & Economic Statistics , volume=

    Direct and indirect effects based on difference-in-differences with an application to political preferences following the Vietnam draft lottery , author=. Journal of Business & Economic Statistics , volume=. 2019 , publisher=

    2019

  78. [86]

    Journal of Business & Economic Statistics , volume=

    Direct and indirect effects based on changes-in-changes , author=. Journal of Business & Economic Statistics , volume=. 2022 , publisher=

    2022

  79. [87]

    Journal of the American Statistical Association , volume=

    Uniformly more powerful tests for hypotheses concerning linear inequalities and normal means , author=. Journal of the American Statistical Association , volume=. 1989 , publisher=

    1989

  80. [88]

    Journal of Business & Economic Statistics , volume=

    Varying coefficient mediation model and application to analysis of behavioral economics data , author=. Journal of Business & Economic Statistics , volume=. 2022 , publisher=

    2022

  81. [89]

    Journal of Business & Economic Statistics , volume=

    Partial identification of local average treatment effects with an invalid instrument , author=. Journal of Business & Economic Statistics , volume=. 2013 , publisher=

    2013

  82. [90]

    Journal of Business & Economic Statistics , volume=

    A framework for separating individual-level treatment effects from spillover effects , author=. Journal of Business & Economic Statistics , volume=. 2021 , publisher=

    2021

  83. [91]

    Journal of Applied Econometrics , volume=

    Direct and indirect effects of continuous treatments based on generalized propensity score weighting , author=. Journal of Applied Econometrics , volume=. 2020 , publisher=

    2020

  84. [92]

    The Econometrics Journal , volume=

    Causal mediation analysis with double machine learning , author=. The Econometrics Journal , volume=. 2022 , publisher=

    2022

  85. [93]

    Journal of Business & Economic Statistics , volume=

    Exploring encouragement, treatment, and spillover effects using principal stratification, with application to a field experiment on teens’ museum attendance , author=. Journal of Business & Economic Statistics , volume=. 2021 , publisher=

    2021

  86. [94]

    The Review of Economic Studies , volume=

    Training, Wages, and Sample Selection: Estimating Sharp Bounds on Treatment Effects , author=. The Review of Economic Studies , volume=. 2009 , publisher=

    2009

  87. [95]

    Wiley Series in Probability and Mathematical Statistics , year=

    Robust statistics , author=. Wiley Series in Probability and Mathematical Statistics , year=

  88. [96]

    Econometrica , volume=

    Identification and Robustness with Contaminated and Corrupted Data , author=. Econometrica , volume=. 1995 , publisher=

    1995

  89. [97]

    arXiv preprint arXiv:2005.06095 , year=

    Exchangeability, conformal prediction, and rank tests , author=. arXiv preprint arXiv:2005.06095 , year=

    2005 arXiv

  90. [98]

    Proceedings of the National Academy of Sciences , volume=

    Sensitivity analysis of individual treatment effects: A robust conformal inference approach , author=. Proceedings of the National Academy of Sciences , volume=. 2023 , publisher=

    2023

  91. [99]

    Journal of the American Statistical Association , volume=

    Conformal sensitivity analysis for individual treatment effects , author=. Journal of the American Statistical Association , volume=. 2024 , publisher=

    2024

  92. [100]

    Journal of Econometrics , volume=

    Generalized lee bounds , author=. Journal of Econometrics , volume=. 2025 , publisher=

    2025

  93. [101]

    Journal of Econometrics , volume=

    Heterogeneous treatment effect bounds under sample selection with an application to the effects of social media on political polarization , author=. Journal of Econometrics , volume=. 2024 , publisher=

    2024

  94. [102]

    arXiv preprint arXiv:2606.09223 , year=

    Sharp Bounds and Inference in Sample Selection Models with Treatment Endogeneity , author=. arXiv preprint arXiv:2606.09223 , year=

    arXiv

  95. [103]

    Biometrics , volume=

    Principal stratification in causal inference , author=. Biometrics , volume=. 2002 , publisher=

    2002

  96. [104]

    Journal of Educational and Behavioral Statistics , volume=

    Estimation of causal effects via principal stratification when some outcomes are truncated by “death” , author=. Journal of Educational and Behavioral Statistics , volume=. 2003 , publisher=

    2003

  97. [105]

    2005 , publisher=

    Algorithmic learning in a random world , author=. 2005 , publisher=

    2005

  98. [106]

    Journal of the American Statistical Association , volume=

    Distribution-free predictive inference for regression , author=. Journal of the American Statistical Association , volume=. 2018 , publisher=

    2018

  99. [107]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Conformal inference of counterfactuals and individual treatment effects , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2021 , publisher=

    2021

  100. [108]

    Journal of the American Statistical Association , volume=

    An exact and robust conformal inference method for counterfactual and synthetic controls , author=. Journal of the American Statistical Association , volume=. 2021 , publisher=

    2021

  101. [109]

    Econometrica , volume=

    Selection without exclusion , author=. Econometrica , volume=. 2020 , publisher=

    2020

  102. [110]

    Journal of Econometrics , volume=

    Sample selection models without exclusion restrictions: Parameter heterogeneity and partial identification , author=. Journal of Econometrics , volume=. 2024 , publisher=

    2024

  103. [111]

    arXiv preprint arXiv:2411.04312 , year=

    Lee bounds with a continuous treatment in sample selection , author=. arXiv preprint arXiv:2411.04312 , year=

  104. [112]

    arXiv preprint arXiv:2601.09453 , year=

    Lee Bounds for Random Objects , author=. arXiv preprint arXiv:2601.09453 , year=

  105. [113]

    Journal of the American Statistical Association , volume=

    Doubly-valid/doubly-sharp sensitivity analysis for causal inference with unmeasured confounding , author=. Journal of the American Statistical Association , volume=. 2025 , publisher=

    2025

  106. [114]

    Biometrika , volume=

    Sensitivity analysis for certain permutation inferences in matched observational studies , author=. Biometrika , volume=. 1987 , publisher=

    1987

  107. [115]

    Annals of statistics , volume=

    Bounds on the conditional and average treatment effect with unobserved confounding factors , author=. Annals of statistics , volume=

  108. [116]

    Journal of the American Statistical Association , volume=

    Sharp sensitivity analysis for inverse propensity weighting via quantile balancing , author=. Journal of the American Statistical Association , volume=. 2023 , publisher=

    2023

  109. [117]

    , author=

    Quantile regression forests. , author=. Journal of machine learning research , volume=

  110. [118]

    Advances in neural information processing systems , volume=

    Conformalized quantile regression , author=. Advances in neural information processing systems , volume=

  111. [119]

    Advances in neural information processing systems , volume=

    Conformal prediction under covariate shift , author=. Advances in neural information processing systems , volume=

  112. [120]

    The Annals of Statistics , volume=

    Conformal prediction beyond exchangeability , author=. The Annals of Statistics , volume=. 2023 , publisher=

    2023

  113. [121]

    2003 , publisher=

    Partial identification of probability distributions , author=. 2003 , publisher=

    2003

Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.