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Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models

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  • Christoph Breunig
  • Xiaohong Chen

Abstract

This paper considers adaptive, minimax estimation of a quadratic functional in a nonparametric instrumental variables (NPIV) model, which is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional can attain a convergence rate that coincides with the lower bound previously derived in Chen and Christensen [2018]. The minimax rate is achieved by the optimal choice of the sieve dimension (a key tuning parameter) that depends on the smoothness of the NPIV function and the degree of ill-posedness, both are unknown in practice. We next propose a Lepski-type data-driven choice of the key sieve dimension adaptive to the unknown NPIV model features. The adaptive estimator of the quadratic functional is shown to attain the minimax optimal rate in the severely ill-posed case and in the regular mildly ill-posed case, but up to a multiplicative $\sqrt{\log n}$ factor in the irregular mildly ill-posed case.

Suggested Citation

  • Christoph Breunig & Xiaohong Chen, 2021. "Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models," Papers 2101.12282, arXiv.org, revised Feb 2022.
  • Handle: RePEc:arx:papers:2101.12282
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    References listed on IDEAS

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    1. Breunig, Christoph & Johannes, Jan, 2016. "Adaptive Estimation Of Functionals In Nonparametric Instrumental Regression," Econometric Theory, Cambridge University Press, vol. 32(3), pages 612-654, June.
    2. Xiaohong Chen & Timothy M. Christensen, 2018. "Optimal sup‐norm rates and uniform inference on nonlinear functionals of nonparametric IV regression," Quantitative Economics, Econometric Society, vol. 9(1), pages 39-84, March.
    3. Xiaohong Chen & Timothy Christensen, 2013. "Optimal Sup-norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation," Cowles Foundation Discussion Papers 1923R, Cowles Foundation for Research in Economics, Yale University, revised Apr 2015.
    4. Fabian Dunker, 2015. "Adaptive estimation for some nonparametric instrumental variable models," Papers 1511.03977, arXiv.org, revised Aug 2021.
    5. Richard Blundell & Xiaohong Chen & Dennis Kristensen, 2007. "Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves," Econometrica, Econometric Society, vol. 75(6), pages 1613-1669, November.
    6. Christoph Breunig & Xiaohong Chen, 2020. "Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models," Papers 2006.09587, arXiv.org, revised Nov 2024.
    7. Carrasco, Marine & Florens, Jean-Pierre & Renault, Eric, 2007. "Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 77, Elsevier.
    8. Whitney K. Newey & James L. Powell, 2003. "Instrumental Variable Estimation of Nonparametric Models," Econometrica, Econometric Society, vol. 71(5), pages 1565-1578, September.
    9. Horowitz, Joel L., 2014. "Adaptive nonparametric instrumental variables estimation: Empirical choice of the regularization parameter," Journal of Econometrics, Elsevier, vol. 180(2), pages 158-173.
    10. Breunig, Christoph & Johannes, Jan, 2016. "Adaptive estimation of functionals in nonparametric instrumental regression," LIDAM Reprints ISBA 2016041, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Joel L. Horowitz, 2011. "Applied Nonparametric Instrumental Variables Estimation," Econometrica, Econometric Society, vol. 79(2), pages 347-394, March.
    12. Christoph Breunig & Xiaohong Chen, 2020. "Adaptive, Rate-Optimal Testing in Instrumental Variables Models," Cowles Foundation Discussion Papers 2238, Cowles Foundation for Research in Economics, Yale University.
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