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ADAPTIVE ESTIMATION OF FUNCTIONALS IN NONPARAMETRIC INSTRUMENTAL REGRESSION

Published online by Cambridge University Press:  30 March 2015

Christoph Breunig  and
Jan Johannes
Christoph Breunig*
Affiliation:
Humboldt-Universität zu Berlin
Jan Johannes
Affiliation:
CREST-Ensai and Université catholique de Louvain
*
*Address correspondence to Christoph Breunig, Humboldt-Universität zu Berlin, Department of Economics, Spandauer Str. 1, 10178 Berlin, Germany, e-mail: christoph.breunig@hu-berlin.de.
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Abstract

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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.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 3 , June 2016 , pp. 612 - 654
Copyright
Copyright © Cambridge University Press 2015 

References

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