ADAPTIVE ESTIMATION OF FUNCTIONALS IN NONPARAMETRIC INSTRUMENTAL REGRESSION

Abstract

We consider the problem of estimating the valueℓ(ϕ) of a linear functional, where the structural functionϕmodels 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 parametermdepending on certain characteristics of the structural functionϕand the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice ofmwhich 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 functionϕ.