IDEAS home Printed from https://ideas.repec.org/p/cwl/cwldpp/2043.html
   My bibliography  Save this paper

Optimal Inference in a Class of Regression Models

Author

Listed:

Abstract

We consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression. Our main assumption is that the regression function is known to lie in a convex function class, which covers most smoothness and/or shape assumptions used in econometrics. We derive finite-sample optimal CIs and sharp efficiency bounds under normal errors with known variance. We show that these results translate to uniform (over the function class) asymptotic results when the error distribution is not known. When the function class is centrosymmetric, these efficiency bounds imply that minimax CIs are close to efficient at smooth regression functions. This implies, in particular, that it is impossible to form CIs that are tighter using data-dependent tuning parameters, and maintain coverage over the whole function class. We specialize our results to inference in a linear regression, and inference on the regression discontinuity parameter, and illustrate them in simulations and an empirical application.

Suggested Citation

  • Timothy B. Armstrong & Michal Koles�r, 2016. "Optimal Inference in a Class of Regression Models," Cowles Foundation Discussion Papers 2043, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:2043
    Note: Includes supplementary material
    as

    Download full text from publisher

    File URL: https://cowles.yale.edu/sites/default/files/d20/d2043.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Victor Chernozhukov & Sokbae Lee & Adam M. Rosen, 2013. "Intersection Bounds: Estimation and Inference," Econometrica, Econometric Society, vol. 81(2), pages 667-737, March.
    2. Drees, Holger, 1999. "On fixed-length confidence intervals for a bounded normal mean," Statistics & Probability Letters, Elsevier, vol. 44(4), pages 399-404, October.
    3. Imbens, Guido W. & Lemieux, Thomas, 2008. "Regression discontinuity designs: A guide to practice," Journal of Econometrics, Elsevier, vol. 142(2), pages 615-635, February.
    4. Sebastian Calonico & Matias D. Cattaneo & Rocio Titiunik, 2014. "Robust Nonparametric Confidence Intervals for Regression‐Discontinuity Designs," Econometrica, Econometric Society, vol. 82, pages 2295-2326, November.
    5. T. Tony Cai & Mark Low & Zongming Ma, 2014. "Adaptive Confidence Bands for Nonparametric Regression Functions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1054-1070, September.
    6. Ichimura, Hidehiko & Todd, Petra E., 2007. "Implementing Nonparametric and Semiparametric Estimators," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 74, Elsevier.
    7. Lee, David S. & Card, David, 2008. "Regression discontinuity inference with specification error," Journal of Econometrics, Elsevier, vol. 142(2), pages 655-674, February.
    8. McCloskey, Adam, 2017. "Bonferroni-based size-correction for nonstandard testing problems," Journal of Econometrics, Elsevier, vol. 200(1), pages 17-35.
    9. Leeb, Hannes & Pötscher, Benedikt M., 2005. "Model Selection And Inference: Facts And Fiction," Econometric Theory, Cambridge University Press, vol. 21(1), pages 21-59, February.
    10. Guido Imbens & Karthik Kalyanaraman, 2012. "Optimal Bandwidth Choice for the Regression Discontinuity Estimator," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 79(3), pages 933-959.
    11. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
    12. Andrews, Donald W K, 2001. "Testing When a Parameter Is on the Boundary of the Maintained Hypothesis," Econometrica, Econometric Society, vol. 69(3), pages 683-734, May.
    13. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2014. "Inference on Treatment Effects after Selection among High-Dimensional Controlsâ€," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 81(2), pages 608-650.
    14. Susanne M Schennach, 2020. "A Bias Bound Approach to Non-parametric Inference," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 87(5), pages 2439-2472.
    15. Timothy B. Armstrong, 2014. "Adaptive Testing on a Regression Function at a Point," Cowles Foundation Discussion Papers 1957, Cowles Foundation for Research in Economics, Yale University, revised Oct 2014.
    16. Peter Hall & Joel L. Horowitz, 2013. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers CWP29/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    17. Lee, David S., 2008. "Randomized experiments from non-random selection in U.S. House elections," Journal of Econometrics, Elsevier, vol. 142(2), pages 675-697, February.
    18. Alberto Abadie & Guido W. Imbens, 2006. "Large Sample Properties of Matching Estimators for Average Treatment Effects," Econometrica, Econometric Society, vol. 74(1), pages 235-267, January.
    19. Cun-Hui Zhang & Stephanie S. Zhang, 2014. "Confidence intervals for low dimensional parameters in high dimensional linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(1), pages 217-242, January.
    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. Guido W. Imbens & Jeffrey M. Wooldridge, 2009. "Recent Developments in the Econometrics of Program Evaluation," Journal of Economic Literature, American Economic Association, vol. 47(1), pages 5-86, March.
    2. Timothy B Armstrong & Michal Kolesár, 2018. "A Simple Adjustment for Bandwidth Snooping," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 85(2), pages 732-765.
    3. Huber, Martin, 2019. "An introduction to flexible methods for policy evaluation," FSES Working Papers 504, Faculty of Economics and Social Sciences, University of Freiburg/Fribourg Switzerland.
    4. Eduardo Fé & Bruce Hollingsworth, 2016. "Short- and long-run estimates of the local effects of retirement on health," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 179(4), pages 1051-1067, October.
    5. Christelis, Dimitris & Georgarakos, Dimitris & Sanz-de-Galdeano, Anna, 2020. "The impact of health insurance on stockholding: A regression discontinuity approach," Journal of Health Economics, Elsevier, vol. 69(C).
    6. Otávio Bartalotti, 2013. "Theory and Practice of Inference in Regression Discontinuity: A Fixed-Bandwidth Asymptotics Approach," Working Papers 1302, Tulane University, Department of Economics, revised Nov 2013.
    7. Joaquín Artés & Ignacio Jurado, 2018. "Government fragmentation and fiscal deficits: a regression discontinuity approach," Public Choice, Springer, vol. 175(3), pages 367-391, June.
    8. Mauricio Villamizar‐Villegas & Freddy A. Pinzon‐Puerto & Maria Alejandra Ruiz‐Sanchez, 2022. "A comprehensive history of regression discontinuity designs: An empirical survey of the last 60 years," Journal of Economic Surveys, Wiley Blackwell, vol. 36(4), pages 1130-1178, September.
    9. Yoici Arai & Taisuke Otsu & Myung Hwan Seo, 2019. "Causal inference on regression discontinuity designs by high-dimensional methods," STICERD - Econometrics Paper Series 601, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    10. Timothy B. Armstrong & Michal Kolesár, 2021. "Finite‐Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness," Econometrica, Econometric Society, vol. 89(3), pages 1141-1177, May.
    11. Takahide Yanagi, 2014. "The Effect of Measurement Error in the Sharp Regression Discontinuity Design," KIER Working Papers 910, Kyoto University, Institute of Economic Research.
    12. Montoya, Ana Maria & Noton, Carlos & Solis, Alex, 2018. "The Returns to College Choice: Loans, Scholarships and Labor Outcomes," Working Paper Series 2018:12, Uppsala University, Department of Economics.
    13. Blaise Melly & Rafael Lalive, 2020. "Estimation, Inference, and Interpretation in the Regression Discontinuity Design," Diskussionsschriften dp2016, Universitaet Bern, Departement Volkswirtschaft.
    14. Yoichi Arai & Yu‐Chin Hsu & Toru Kitagawa & Ismael Mourifié & Yuanyuan Wan, 2022. "Testing identifying assumptions in fuzzy regression discontinuity designs," Quantitative Economics, Econometric Society, vol. 13(1), pages 1-28, January.
    15. Myung Hwan Seo & Yoichi Arai & Taisuke Otsu, 2021. "Regression Discontinuity Design with Potentially Many Covariates," Working Paper Series no142, Institute of Economic Research, Seoul National University.
    16. Xu, Ke-Li, 2020. "Inference of local regression in the presence of nuisance parameters," Journal of Econometrics, Elsevier, vol. 218(2), pages 532-560.
    17. Susanne M Schennach, 2020. "A Bias Bound Approach to Non-parametric Inference," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 87(5), pages 2439-2472.
    18. Yang He & Otávio Bartalotti, 2020. "Wild bootstrap for fuzzy regression discontinuity designs: obtaining robust bias-corrected confidence intervals," The Econometrics Journal, Royal Economic Society, vol. 23(2), pages 211-231.
    19. Bartalotti Otávio, 2019. "Regression Discontinuity and Heteroskedasticity Robust Standard Errors: Evidence from a Fixed-Bandwidth Approximation," Journal of Econometric Methods, De Gruyter, vol. 8(1), pages 1-26, January.
    20. Bartalotti, Otávio C. & Calhoun, Gray & He, Yang, 2016. "Bootstrap Confidence Intervals for Sharp Regression Discontinuity Designs with the Uniform Kernel," Staff General Research Papers Archive 3394, Iowa State University, Department of Economics.

    More about this item

    Keywords

    Nonparametric inference; efficiency bounds;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:cwl:cwldpp:2043. 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: Brittany Ladd (email available below). General contact details of provider: https://edirc.repec.org/data/cowleus.html .

    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.