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A bias bound approach to nonparametric inference

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  • Susanne M. Schennach
Abstract
The traditional approach to obtain valid confidence intervals for nonparametric quantities is to select a smoothing parameter such that the bias of the estimator is negligible relative to its standard deviation. While this approach is apparently simple, it has two drawbacks: First, the question of optimal bandwidth selection is no longer well-defined, as it is not clear what ratio of bias to standard deviation should be considered negligible. Second, since the bandwidth choice necessarily deviates from the optimal (mean squares-minimizing) bandwidth, such a confidence interval is very inefficient. To address these issues, we construct valid confidence intervals that account for the presence of a nonnegligible bias and thus make it possible to perform inference with optimal mean squared error minimizing bandwidths. The key difficulty in achieving this involves finding a strict, yet feasible, bound on the bias of a nonparametric estimator. It is well-known that it is not possible to consistently estimate the point-wise bias of an optimal nonparametric estimator (for otherwise, one could subtract it and obtain a faster convergence rate violating Stone's bounds on optimal convergence rate). Nevertheless, we find that, under minimal primitive assumptions, it is possible to consistently estimate an upper bound on the magnitude of the bias, which is sufficient to deliver a valid confidence interval whose length decreases at the optimal rate and which does not contradict Stone’s results.

Suggested Citation

  • Susanne M. Schennach, 2015. "A bias bound approach to nonparametric inference," CeMMAP working papers 71/15, Institute for Fiscal Studies.
    • Handle: RePEc:azt:cemmap:71/15
    DOI: 10.1920/wp.cem.2015.7115
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    1. 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.
    2. 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.
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    4. 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.
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    8. 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.
    9. Timothy B. Armstrong, 2014. "Adaptive Testing on a Regression Function at a Point," Cowles Foundation Discussion Papers 1957R, Cowles Foundation for Research in Economics, Yale University, revised Feb 2015.
    10. Pagan,Adrian & Ullah,Aman, 1999. "Nonparametric Econometrics," Cambridge Books, Cambridge University Press, number 9780521355643, October.
    11. Sebastian Calonico & Matias D. Cattaneo & Rocío Titiunik, 2015. "Optimal Data-Driven Regression Discontinuity Plots," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1753-1769, December.
    12. Schennach, Susanne M., 2004. "Nonparametric Regression In The Presence Of Measurement Error," Econometric Theory, Cambridge University Press, vol. 20(6), pages 1046-1093, December.
    13. Politis, Dimitris N. & Romano, Joseph P., 1999. "Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 1-25, January.
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    Cited by:

    1. Timothy B. Armstrong & Michal Kolesár, 2018. "Optimal Inference in a Class of Regression Models," Econometrica, Econometric Society, vol. 86(2), pages 655-683, 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. Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Robust inference in deconvolution," Quantitative Economics, Econometric Society, vol. 12(1), pages 109-142, January.
    4. Sebastian Calonico & Matias D. Cattaneo & Max H. Farrell, 2018. "On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 767-779, April.
    5. Harold D. Chiang & Kengo Kato & Yuya Sasaki & Takuya Ura, 2021. "Linear programming approach to nonparametric inference under shape restrictions: with an application to regression kink designs," Papers 2102.06586, arXiv.org.
    6. Timothy B. Armstrong, 2018. "Adaptation Bounds for Confidence Bands under Self-Similarity," Cowles Foundation Discussion Papers 2146, Cowles Foundation for Research in Economics, Yale University.
    7. Byunghoon Kang, 2018. "Inference in Nonparametric Series Estimation with Specification Searches for the Number of Series Terms," Working Papers 240829404, Lancaster University Management School, Economics Department.
    8. Byunghoon Kang, 2017. "Inference in Nonparametric Series Estimation with Data-Dependent Undersmoothing," Working Papers 170712442, Lancaster University Management School, Economics Department.
    9. Yuya Sasaki & Takuya Ura, 2021. "Slow Movers in Panel Data," Papers 2110.12041, arXiv.org.
    10. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    11. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.
    12. Sebastian Calonico & Matias D. Cattaneo & Max H. Farrell, 2018. "Coverage Error Optimal Confidence Intervals for Local Polynomial Regression," Papers 1808.01398, arXiv.org, revised Jul 2021.
    13. Byunghoon Kang, 2019. "Inference in Nonparametric Series Estimation with Specification Searches for the Number of Series Terms," Papers 1909.12162, arXiv.org, revised Feb 2020.

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