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Adaptive Estimation Of Functionals In Nonparametric Instrumental Regression

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  • Breunig, Christoph
  • Johannes, Jan

Abstract

We consider the problem of estimating the value ℓ(ϕ) of a linear functional, where the structural function ϕ models a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parameter m depending on certain characteristics of the structural function ϕ and the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice of m which combines model selection and Lepski’s method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural function ϕ.

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  • 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.
    • Handle: RePEc:cup:etheor:v:32:y:2016:i:03:p:612-654_00
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    Cited by:

    1. Breunig, Christoph & Mammen, Enno & Simoni, Anna, 2020. "Ill-posed estimation in high-dimensional models with instrumental variables," Journal of Econometrics, Elsevier, vol. 219(1), pages 171-200.
    2. Christoph Breunig & Xiaohong Chen, 2021. "Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models," Papers 2101.12282, arXiv.org, revised Feb 2022.
    3. Breunig, Christoph & Mammen, Enno & Simoni, Anna, 2018. "Nonparametric estimation in case of endogenous selection," Journal of Econometrics, Elsevier, vol. 202(2), pages 268-285.
    4. Christoph Breunig & Xiaohong Chen, 2020. "Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models," Papers 2006.09587, arXiv.org, revised Feb 2024.
    5. Escanciano, Juan Carlos & Li, Wei, 2021. "Optimal Linear Instrumental Variables Approximations," Journal of Econometrics, Elsevier, vol. 221(1), pages 223-246.
    6. Rodney V. Fonseca & Aluísio Pinheiro, 2020. "Wavelet estimation of the dimensionality of curve time series," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1175-1204, October.

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