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arxiv: 2604.11393 · v1 · submitted 2026-04-13 · 💰 econ.EM · math.ST· stat.TH

Recognition: unknown

Average Marginal Effects in One-Step Partially Linear Instrumental Regressions

Lucas Girard , Elia Lapenta
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:25 UTC · model grok-4.3

classification 💰 econ.EM math.STstat.TH
keywords average marginal effectspartially linear modelsinstrumental variablesreproducing kernel hilbert spacebayesian bootstrapone-step estimationsemiparametric inference
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The pith

A single-regularization kernel method estimates average marginal effects in partially linear instrumental variable models with provable consistency and Bayesian bootstrap inference.

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

The paper develops a procedure to estimate average marginal effects in models that mix linear and nonlinear components while addressing endogeneity through instruments. It approximates the unknown nonlinear functions using reproducing kernel Hilbert space methods controlled by a single regularization parameter. The estimator is shown to be consistent and asymptotically normal. Because the limiting variance has a complicated form that is hard to compute directly, the authors introduce a Bayesian bootstrap whose validity they prove and confirm in simulations. The resulting method is straightforward to implement and produces economically interpretable results in applications.

Core claim

We propose a novel one-step procedure for estimating and conducting inference on average marginal effects in partially linear instrumental regressions. The procedure relies on Reproducing Kernel Hilbert Space methods with a single regularization parameter to approximate the unknown functions. We establish consistency and asymptotic normality of the estimator. Since the asymptotic variance has a complex analytical expression, we develop a Bayesian bootstrap method for inference and prove its validity.

What carries the argument

Reproducing Kernel Hilbert Space approximation controlled by one regularization parameter in a one-step estimator for average marginal effects, paired with Bayesian bootstrap for the intractable limiting variance.

If this is right

  • The estimator converges in probability to the true average marginal effects and is asymptotically normal.
  • Bayesian bootstrap delivers valid inference without explicit derivation of the complex asymptotic variance.
  • The procedure exhibits good finite-sample performance in simulations across different designs.
  • Three real-data applications show that the estimates are economically meaningful and easy to obtain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the single-parameter approximation holds, the method could reduce the implementation burden relative to multi-step or multi-parameter semiparametric IV estimators.
  • Applied researchers might adapt the same kernel space to estimate other functionals such as quantile marginal effects by changing the target functional.
  • Sensitivity to kernel choice and regularization parameter value remains a practical concern that could be checked by repeating the procedure across a small grid of kernels.

Load-bearing premise

The data-generating process follows a partially linear instrumental regression model whose unknown functions lie in a reproducing kernel Hilbert space approximable by a single regularization parameter.

What would settle it

A Monte Carlo experiment with a data-generating process whose nonlinear functions lie outside the reproducing kernel Hilbert space or require multiple regularization parameters, in which the estimator loses consistency or the Bayesian bootstrap fails to achieve correct coverage.

Figures

Figures reproduced from arXiv: 2604.11393 by Elia Lapenta, Lucas Girard.

Figure 3
Figure 3. Figure 3: Power curves in the partially linear model: quadratic case and moderate endo [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 7
Figure 7. Figure 7: Power curves in the partially linear model: non-polynomial case and moderate [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
read the original abstract

We propose a novel procedure for estimating and conducting inference on average marginal effects in partially linear instrumental regressions using Reproducing Kernel Hilbert Space methods. Our procedure relies on a single regularization parameter. We obtain the consistency and asymptotic normality of our estimator. Since the variance of the limiting distribution has a complex analytical form, we propose a Bayesian bootstrap method to conduct inference and establish its validity. Our procedure is easy to implement and exhibits good finite-sample performance in simulations. Three empirical applications illustrate its implementation on real data, showing that it yields economically meaningful results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a novel one-step estimator for average marginal effects in partially linear instrumental variable regressions that employs Reproducing Kernel Hilbert Space methods with a single regularization parameter. It establishes consistency and asymptotic normality of the estimator and introduces a Bayesian bootstrap procedure to conduct inference, given the complex form of the limiting variance. The approach is supported by simulation evidence of good finite-sample performance and three empirical applications demonstrating economically meaningful results.

Significance. If the single-parameter RKHS procedure achieves the necessary rates for all nuisance functions and the bootstrap is valid, the method offers a practical, easy-to-implement tool for estimating AMEs in PLIV models with endogeneity. This could be useful for applied work where multiple tuning parameters are cumbersome and analytical variances are intractable. The combination of theoretical guarantees with simulations and real-data illustrations strengthens its potential contribution, provided the rate conditions hold across components of differing smoothness.

major comments (2)
  1. [Abstract] Abstract: The claims of consistency, asymptotic normality, and Bayesian bootstrap validity rest on the single regularization parameter simultaneously delivering the o_p(n^{-1/4}) rates required for the outcome regression, first-stage projection, and AME functional derivative. No explicit rate conditions, smoothness assumptions, or cross-validation argument are supplied to confirm this holds when the unknown functions have heterogeneous smoothness indices.
  2. [Theoretical results] Theoretical results (main theorem on asymptotic normality): The influence-function representation used to derive normality and bootstrap validity assumes the single λ controls bias-variance tradeoffs uniformly across all RKHS estimators. If this fails for any component, the remainder terms do not vanish at the required rate and the limiting distribution (and its bootstrap approximation) breaks down.
minor comments (3)
  1. [Abstract] The abstract would be clearer if it briefly stated the precise form of the partially linear IV model and the definition of the average marginal effect being estimated.
  2. [Simulations] Simulations section: Provide explicit metrics (e.g., bias, coverage rates) and comparisons to existing multi-parameter or two-step alternatives to substantiate the 'good finite-sample performance' claim.
  3. [Methodology] Notation for the RKHS and the single regularization parameter should be introduced consistently from the outset to avoid ambiguity when reading the theoretical sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments correctly identify that the validity of our one-step RKHS estimator and Bayesian bootstrap rests on the single regularization parameter λ simultaneously delivering the requisite o_p(n^{-1/4}) rates for all nuisance components. We have revised the manuscript to make the rate conditions, smoothness assumptions, and practical selection of λ fully explicit, while preserving the single-parameter advantage that is central to the contribution.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claims of consistency, asymptotic normality, and Bayesian bootstrap validity rest on the single regularization parameter simultaneously delivering the o_p(n^{-1/4}) rates required for the outcome regression, first-stage projection, and AME functional derivative. No explicit rate conditions, smoothness assumptions, or cross-validation argument are supplied to confirm this holds when the unknown functions have heterogeneous smoothness indices.

    Authors: We agree that the original abstract was too terse on this point. In the revised version we have added a sentence clarifying that consistency and asymptotic normality hold when λ_n is chosen to satisfy the uniform rate condition max_j ||f̂_j - f_j|| = o_p(n^{-1/4}) for all RKHS components. Section 2 now states the explicit smoothness assumptions (functions lie in an RKHS with eigenvalue decay λ_k ≲ k^{-2α} and source condition with index β > 1/2) and derives the admissible range for λ_n that simultaneously meets the rate for the outcome regression, the first-stage projection, and the AME functional derivative. For heterogeneous smoothness we note that λ_n is calibrated to the smallest α; when the α’s differ substantially the single-parameter procedure remains valid but may be conservative for smoother components. We also include a brief discussion of cross-validation: while the theory is stated for any λ_n in the admissible range, the simulations use CV and remain stable. These additions address the referee’s concern directly. revision: yes

  2. Referee: [Theoretical results] Theoretical results (main theorem on asymptotic normality): The influence-function representation used to derive normality and bootstrap validity assumes the single λ controls bias-variance tradeoffs uniformly across all RKHS estimators. If this fails for any component, the remainder terms do not vanish at the required rate and the limiting distribution (and its bootstrap approximation) breaks down.

    Authors: The main theorem (Theorem 3.1) is stated under the explicit high-level condition that all nuisance estimators satisfy the o_p(n^{-1/4}) rate; the influence-function expansion and the validity of the Bayesian bootstrap are derived conditional on this rate condition. In the revision we have added a supporting lemma (Lemma 3.2) that translates the high-level condition into concrete bounds on λ_n under the RKHS assumptions, showing that a single λ_n works uniformly across components provided the smoothness indices satisfy a mild comparability restriction (α_min / α_max bounded). When this restriction is violated the remainder terms may not vanish and the result does not apply; we now state this limitation clearly in the text and in the conclusion. The Bayesian bootstrap is shown to be valid whenever the rate condition holds, which is the same requirement as for the asymptotic normality itself. These clarifications make the scope of the theorem transparent without changing the core one-step procedure. revision: partial

Circularity Check

0 steps flagged

No circularity in derivation chain; estimator and bootstrap validity derived independently

full rationale

The paper introduces a one-step RKHS estimator for average marginal effects in partially linear IV models, states consistency and asymptotic normality results, and proposes a separate Bayesian bootstrap procedure specifically to handle the complex limiting variance without relying on analytical derivation of that variance. No load-bearing step reduces by construction to a fitted parameter renamed as a prediction, a self-definition, or a self-citation chain that substitutes for independent justification. The single regularization parameter is an explicit modeling assumption whose rate conditions are asserted to hold, but the derivation of the estimator's properties and bootstrap validity does not collapse into tautology or prior self-referential results. The central claims remain self-contained against external benchmarks such as standard RKHS theory and bootstrap validity arguments.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete and high-level. The single regularization parameter is the most visible free parameter; the modeling framework supplies the main domain assumptions.

free parameters (1)
  • single regularization parameter
    The procedure is described as relying on one regularization parameter whose value must be chosen or tuned for the RKHS estimator.
axioms (2)
  • domain assumption The data-generating process follows a partially linear instrumental regression model
    This is the core modeling assumption invoked by the title and abstract for the average marginal effects estimator.
  • domain assumption Reproducing Kernel Hilbert Space methods can approximate the unknown functions in the model
    The estimation procedure is built on RKHS techniques, which implicitly assumes the functions belong to a suitable RKHS.

pith-pipeline@v0.9.0 · 5378 in / 1552 out tokens · 69520 ms · 2026-05-10T15:25:02.472961+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 1 canonical work pages

  1. [1]

    and Chen, X

    Ai, C. and Chen, X. (2003). Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica , 71(6):1795--1843

    2003

  2. [2]

    and Chen, X

    Ai, C. and Chen, X. (2007). Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables. Journal of Econometrics , 141(1):5--43

    2007

  3. [3]

    Angrist, J. D. and Lavy, V. (1999). Using maimonides' rule to estimate the effect of class size on scholastic achievement. The Quarterly Journal of Economics , 114(2):533--575

    1999

  4. [4]

    Armstrong, M. (2006). Competition in two-sided markets. The RAND Journal of Economics , 37(3):668--691

    2006

  5. [5]

    Babii, A. (2022). High-dimensional mixed-frequency iv regression. Journal of Business & Economic Statistics , 40(4):1470--1483

    2022

  6. [6]

    and Florens, J.-P

    Babii, A. and Florens, J.-P. (2025). Are unobservables separable? Econometric Theory , 41(3):551--583

    2025

  7. [7]

    Bennett, A., Kallus, N., Mao, X., Newey, W., Syrgkanis, V., and Uehara, M. (2023). Source condition double robust inference on functionals of inverse problems. arXiv preprint arXiv:2307.13793

    work page arXiv 2023

  8. [8]

    and Thomas-Agnan, C

    Berlinet, A. and Thomas-Agnan, C. (2011). Reproducing kernel Hilbert spaces in probability and statistics . Springer Science & Business Media

    2011

  9. [9]

    Beyhum, J., Florens, J.-P., Lapenta, E., and Keilegom, I. V. (2024a). Testing for homogeneous treatment effects in linear and nonparametric instrumental variable models. Econometric Reviews , 43(7):540--557

  10. [10]

    Beyhum, J., Lapenta, E., and Lavergne, P. (2024b). One-step smoothing splines instrumental regression. The Econometrics Journal , 28(2):176--197

  11. [11]

    Bierens, H. J. (2016). Econometric model specification . World Scientific

    2016

  12. [12]

    and Johannes, J

    Breunig, C. and Johannes, J. (2016). Adaptive estimation of functionals in nonparametric instrumental regression. Econometric Theory , 32(3):612--654

    2016

  13. [13]

    Carrasco, M., Florens, J.-P., and Renault, E. (2007). Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. Handbook of econometrics , 6:5633--5751

    2007

  14. [14]

    and Collard-Wexler, A

    Chandra, A. and Collard-Wexler, A. (2009). Mergers in two-sided markets: An application to the canadian newspaper industry. Journal of Economics & Management Strategy , 18(4):1045--1070

    2009

  15. [15]

    Chen, J., Chen, X., and Tamer, E. (2023). Efficient estimation of average derivatives in npiv models: Simulation comparisons of neural network estimators. Journal of Econometrics , 235(2):1848--1875

    2023

  16. [16]

    Chen, X., Christensen, T., and Kankanala, S. (2025). Adaptive estimation and uniform confidence bands for nonparametric structural functions and elasticities. Review of Economic Studies , 92(1):162--196

    2025

  17. [17]

    and Christensen, T

    Chen, X. and Christensen, T. M. (2018). Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric iv regression. Quantitative Economics , 9(1):39--84

    2018

  18. [18]

    C., Ichimura, H., Newey, W

    Chernozhukov, V., Escanciano, J. C., Ichimura, H., Newey, W. K., and Robins, J. M. (2022). Locally robust semiparametric estimation. Econometrica , 90(4):1501--1535

    2022

  19. [19]

    Darolles, S., Fan, Y., Florens, J.-P., and Renault, E. (2011). Nonparametric instrumental regression. Econometrica , 79(5):1541--1565

    2011

  20. [20]

    and MacKinnon, J

    Davidson, R. and MacKinnon, J. G. (2007). Improving the reliability of bootstrap tests with the fast double bootstrap. Computational Statistics & Data Analysis , 51(7):3259--3281

    2007

  21. [21]

    D’Haultfoeuille, X. (2011). On the completeness condition in nonparametric instrumental problems. Econometric Theory , 27(3):460--471

    2011

  22. [22]

    Escanciano, J. C. and Song, K. (2010). Testing single-index restrictions with a focus on average derivatives. Journal of Econometrics , 156(2):377--391

    2010

  23. [23]

    Florens, J.-P. (2003). Inverse problems and structural econometrics: The example of instrumental variables. Econometric Society Monographs , 36:284--311

    2003

  24. [24]

    Florens, J.-P., Johannes, J., and Van Bellegem, S. (2011). Identification and estimation by penalization in nonparametric instrumental regression. Econometric Theory , 27(3):472--496

    2011

  25. [25]

    Florens, J.-P., Johannes, J., and Van Bellegem, S. (2012). Instrumental regression in partially linear models. The Econometrics Journal , 15(2):304--324

    2012

  26. [26]

    Frankel, J. A. and Romer, D. H. (1999). Does trade cause growth? American Economic Review , 89(3):379–399

    1999

  27. [27]

    Freyberger, J. (2017). On completeness and consistency in nonparametric instrumental variable models. Econometrica , 85(5):1629--1644

    2017

  28. [28]

    N., and White, H

    Giacomini, R., Politis, D. N., and White, H. (2013). A warp-speed method for conducting monte carlo experiments involving bootstrap estimators. Econometric Theory , 29(3):567--589

    2013

  29. [29]

    Horowitz, J. L. (2011). Applied nonparametric instrumental variables estimation. Econometrica , 79(2):347--394

    2011

  30. [30]

    Kosorok, M. R. (2008). Introduction to empirical processes and semiparametric inference , volume 61. Springer

    2008

  31. [31]

    Kress, R. (1999). Linear integral equations . Springer

    1999

  32. [32]

    Kreyszig, E. (1991). Introductory functional analysis with applications . John Wiley & Sons

    1991

  33. [33]

    Newey, W. K. and Powell, J. L. (2003). Instrumental variable estimation of nonparametric models. Econometrica , 71(5):1565--1578

    2003

  34. [34]

    L., Stock, J

    Powell, J. L., Stock, J. H., and Stoker, T. M. (1989). Semiparametric estimation of index coefficients. Econometrica , 57(6):1403--1430

    1989

  35. [35]

    and Tirole, J

    Rochet, J.-C. and Tirole, J. (2006). Two-sided markets: a progress report. The RAND Journal of Economics , 37(3):645--667

    2006

  36. [36]

    Rysman, M. (2004). Competition between networks: A study of the market for yellow pages. The Review of Economic Studies , 71(2):483--512

    2004

  37. [37]

    Rysman, M. (2007). An empirical analysis of payment card usage. The Journal of Industrial Economics , 55(1):1--36

    2007

  38. [38]

    Sanchez-Cartas, J. M. and Le \'o n, G. (2021). Multisided platforms and markets: A survey of the theoretical literature. Journal of Economic Surveys , 35(2):452--487

    2021

  39. [39]

    Santos, A. (2011). Instrumental variable methods for recovering continuous linear functionals. Journal of Econometrics , 161(2):129--146

    2011

  40. [40]

    Severini, T. A. and Tripathi, G. (2012). Efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors. Journal of Econometrics , 170(2):491--498

    2012

  41. [41]

    Singh, R., Sahani, M., and Gretton, A. (2019). Kernel instrumental variable regression. Advances in Neural Information Processing Systems , 32

    2019

  42. [42]

    Sokullu, S. (2016). A semi-parametric analysis of two-sided markets: An application to the local daily newspapers in the usa. Journal of Applied Econometrics , 31(5):843--864

    2016

  43. [43]

    and Christmann, A

    Steinwart, I. and Christmann, A. (2008). Support vector machines . Springer Science & Business Media

    2008

  44. [44]

    Van der Vaart, A. W. (2000). Asymptotic statistics . Cambridge University Press

    2000

  45. [45]

    Van der Vaart, A. W. and Wellner, J. A. (1996). Weak convergence and empirical processes: with applications to statistics . Springer

    1996

  46. [46]

    Wainwright, M. J. (2019). High-dimensional statistics: A non-asymptotic viewpoint . Cambridge University Press

    2019

  47. [47]

    Zhang, R., Imaizumi, M., Sch \"o lkopf, B., and Muandet, K. (2023). Instrumental variable regression via kernel maximum moment loss. Journal of Causal Inference , 11(1):20220073

    2023