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2015): “Adaptive estimation of functionals in nonparametric instrumental regression,” Econometric Theory, forthcoming
"... We consider the problem of estimating the value `(ϕ) of a linear functional, where the structural function ϕ models a nonparametric relationship in presence of instrumental variables. We propose a plugin estimator which is based on a dimension reduction technique and additional thresholding. It is ..."
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We consider the problem of estimating the value `(ϕ) of a linear functional, where the structural function ϕ models a nonparametric relationship in presence of instrumental variables. We propose a plugin 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.
ADAPTIVE NONPARAMETRIC INSTRUMENTAL VARIABLES ESTIMATION: EMPIRICAL CHOICE OF THE REGULARIZATION PARAMETER
, 2010
"... In nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularized (that is, modified) to make consistent estimation possible. The amount of modification is controlled by a regularization parameter. The optimal va ..."
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In nonparametric instrumental variables estimation, the mapping that identifies the function of interest, g say, is discontinuous and must be regularized (that is, modified) to make consistent estimation possible. The amount of modification is controlled by a regularization parameter. The optimal value of this parameter depends on unknown population characteristics and cannot be calculated in applications. Theoretically justified methods for choosing the regularization parameter empirically in applications are not yet available. This paper presents, apparently for the first time, such a method for use in series estimation, where the regularization parameter is the number of terms in a series approximation to g. The method does not require knowledge of the smoothness of g or of other unknown functions. It adapts to their unknown smoothness. The estimator of g based on the empirically selected regularization parameter converges in probability at a rate that is at least as fast as the asymptotically optimal rate 1/2 multiplied by (log n), where n is the sample size. The asymptotic integrated meansquare error (AIMSE) of the estimator is within a specified factor of the optimal AIMSE.
Asymptotic Normal Inference in Linear Inverse Problems Marine Carrascoy
, 2013
"... This chapter has been prepared for the Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics, edited by Aman Ullah, Je ¤ Racine, and Liangjun Su and to be published by Oxford University Press of New York. We thank the editors and one referee for helpful comments. ..."
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This chapter has been prepared for the Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics, edited by Aman Ullah, Je ¤ Racine, and Liangjun Su and to be published by Oxford University Press of New York. We thank the editors and one referee for helpful comments.
unknown title
, 2012
"... Goodnessoffit tests based on series estimators in nonparametric instrumental regression ..."
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Goodnessoffit tests based on series estimators in nonparametric instrumental regression