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Optimal Inference in a Class of Regression Models

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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 Kolesár, 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
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    File URL: http://cowles.yale.edu/sites/default/files/d20/d2043.pdf
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    1. 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.
    2. 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.
    3. repec:eee:econom:v:200:y:2017:i:1:p:17-35 is not listed on IDEAS
    4. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2014. "Inference on Treatment Effects after Selection among High-Dimensional Controlsâ€," Review of Economic Studies, Oxford University Press, vol. 81(2), pages 608-650.
    5. 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.
    6. 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.
    7. McCloskey, Adam, 2017. "Bonferroni-based size-correction for nonstandard testing problems," Journal of Econometrics, Elsevier, vol. 200(1), pages 17-35.
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    Cited by:

    1. Athey, Susan & Wager, Stefan, 2017. "Efficient Policy Learning," Research Papers 3506, Stanford University, Graduate School of Business.

    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

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