IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v198y2023ics0167715223000603.html
   My bibliography  Save this article

Proximal causal inference without uniqueness assumptions

Author

Listed:
  • Zhang, Jeffrey
  • Li, Wei
  • Miao, Wang
  • Tchetgen Tchetgen, Eric

Abstract

We consider identification and inference about a counterfactual outcome mean when there is unmeasured confounding using tools from proximal causal inference (Miao et al., 2018, Tchetgen Tchetgen et al., 2020). Proximal causal inference requires existence of solutions to at least one of two integral equations. We motivate the existence of solutions to the integral equations from proximal causal inference by demonstrating that, assuming the existence of a solution to one of the integral equations, n-estimability of a linear functional (such as its mean) of that solution requires the existence of a solution to the other integral equation. Solutions to the integral equations may not be unique, which complicates estimation and inference. We construct a consistent estimator for the solution set for one of the integral equations and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator for a uniquely defined solution. A debiased estimator for the counterfactual mean is shown to be root-n consistent, regular, and asymptotically semiparametrically locally efficient under additional regularity conditions.

Suggested Citation

  • Zhang, Jeffrey & Li, Wei & Miao, Wang & Tchetgen Tchetgen, Eric, 2023. "Proximal causal inference without uniqueness assumptions," Statistics & Probability Letters, Elsevier, vol. 198(C).
    • Handle: RePEc:eee:stapro:v:198:y:2023:i:c:s0167715223000603
    DOI: 10.1016/j.spl.2023.109836
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715223000603
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2023.109836?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Xiaohong Chen & Demian Pouzo, 2012. "Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals," Econometrica, Econometric Society, vol. 80(1), pages 277-321, January.
    2. Ben Deaner, 2018. "Proxy Controls and Panel Data," Papers 1810.00283, arXiv.org, revised Nov 2023.
    3. Santos, Andres, 2011. "Instrumental variable methods for recovering continuous linear functionals," Journal of Econometrics, Elsevier, vol. 161(2), pages 129-146, April.
    4. Severini, Thomas A. & Tripathi, Gautam, 2012. "Efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors," Journal of Econometrics, Elsevier, vol. 170(2), pages 491-498.
    5. Xu Shi & Wang Miao & Jennifer C. Nelson & Eric J. Tchetgen Tchetgen, 2020. "Multiply robust causal inference with double‐negative control adjustment for categorical unmeasured confounding," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(2), pages 521-540, April.
    6. Wang Miao & Zhi Geng & Eric J Tchetgen Tchetgen, 2018. "Identifying causal effects with proxy variables of an unmeasured confounder," Biometrika, Biometrika Trust, vol. 105(4), pages 987-993.
    7. Wang Miao & Eric J. Tchetgen Tchetgen, 2016. "On varieties of doubly robust estimators under missingness not at random with a shadow variable," Biometrika, Biometrika Trust, vol. 103(2), pages 475-482.
    8. AmirEmad Ghassami & Andrew Ying & Ilya Shpitser & Eric Tchetgen Tchetgen, 2021. "Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference," Papers 2104.02929, arXiv.org, revised Mar 2022.
    9. Victor Chernozhukov & Whitney K. Newey & Rahul Singh, 2022. "Automatic Debiased Machine Learning of Causal and Structural Effects," Econometrica, Econometric Society, vol. 90(3), pages 967-1027, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Andrew Bennett & Nathan Kallus & Xiaojie Mao & Whitney Newey & Vasilis Syrgkanis & Masatoshi Uehara, 2022. "Inference on Strongly Identified Functionals of Weakly Identified Functions," Papers 2208.08291, arXiv.org, revised Jun 2023.
    2. AmirEmad Ghassami & Andrew Ying & Ilya Shpitser & Eric Tchetgen Tchetgen, 2021. "Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference," Papers 2104.02929, arXiv.org, revised Mar 2022.
    3. Ziyu Wang & Yucen Luo & Yueru Li & Jun Zhu & Bernhard Scholkopf, 2022. "Spectral Representation Learning for Conditional Moment Models," Papers 2210.16525, arXiv.org, revised Dec 2022.
    4. Centorrino, Samuele & Florens, Jean-Pierre, 2021. "Nonparametric Instrumental Variable Estimation of Binary Response Models with Continuous Endogenous Regressors," Econometrics and Statistics, Elsevier, vol. 17(C), pages 35-63.
    5. Juan Carlos Escanciano & Wei Li, 2013. "On the identification of structural linear functionals," CeMMAP working papers 48/13, Institute for Fiscal Studies.
    6. Andrew Bennett & Nathan Kallus & Xiaojie Mao & Whitney Newey & Vasilis Syrgkanis & Masatoshi Uehara, 2023. "Source Condition Double Robust Inference on Functionals of Inverse Problems," Papers 2307.13793, arXiv.org.
    7. Isaac Meza & Rahul Singh, 2021. "Nested Nonparametric Instrumental Variable Regression: Long Term, Mediated, and Time Varying Treatment Effects," Papers 2112.14249, arXiv.org, revised Mar 2024.
    8. Escanciano, Juan Carlos & Li, Wei, 2021. "Optimal Linear Instrumental Variables Approximations," Journal of Econometrics, Elsevier, vol. 221(1), pages 223-246.
    9. Chen, Qihui, 2021. "Robust and optimal estimation for partially linear instrumental variables models with partial identification," Journal of Econometrics, Elsevier, vol. 221(2), pages 368-380.
    10. Ben Deaner, 2021. "Many Proxy Controls," Papers 2110.03973, arXiv.org.
    11. Juan Carlos Escanciano & Wei Li, 2013. "On the identification of structural linear functionals," CeMMAP working papers CWP48/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Xiaohong Chen & Demian Pouzo, 2015. "Sieve Wald and QLR Inferences on Semi/Nonparametric Conditional Moment Models," Econometrica, Econometric Society, vol. 83(3), pages 1013-1079, May.
    13. Yilin Li & Wang Miao & Ilya Shpitser & Eric J. Tchetgen Tchetgen, 2023. "A self‐censoring model for multivariate nonignorable nonmonotone missing data," Biometrics, The International Biometric Society, vol. 79(4), pages 3203-3214, December.
    14. Hidehiko Ichimura & Whitney K. Newey, 2015. "The influence function of semiparametric estimators," CeMMAP working papers CWP44/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    15. Chen, Xiaohong & Pouzo, Demian & Powell, James L., 2019. "Penalized sieve GEL for weighted average derivatives of nonparametric quantile IV regressions," Journal of Econometrics, Elsevier, vol. 213(1), pages 30-53.
    16. Xiaohong Chen & Victor Chernozhukov & Sokbae Lee & Whitney K. Newey, 2014. "Local Identification of Nonparametric and Semiparametric Models," Econometrica, Econometric Society, vol. 82(2), pages 785-809, March.
    17. Andrew Bennett & Nathan Kallus & Xiaojie Mao & Whitney Newey & Vasilis Syrgkanis & Masatoshi Uehara, 2023. "Minimax Instrumental Variable Regression and $L_2$ Convergence Guarantees without Identification or Closedness," Papers 2302.05404, arXiv.org.
    18. Victor Chernozhukov & Whitney K. Newey & Andres Santos, 2023. "Constrained Conditional Moment Restriction Models," Econometrica, Econometric Society, vol. 91(2), pages 709-736, March.
    19. Hidehiko Ichimura & Whitney K. Newey, 2017. "The influence function of semiparametric estimators," CeMMAP working papers 06/17, Institute for Fiscal Studies.
    20. Rahul Singh, 2020. "Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments," Papers 2012.10315, arXiv.org, revised Mar 2023.

    More about this item

    Keywords

    Proximal causal inference; n-estimability;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:198:y:2023:i:c:s0167715223000603. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.