Adaptive non-parametric instrumental regression in the presence of dependence

TL;DR

This paper develops a fully data-driven non-parametric instrumental regression estimator that adapts to dependence in data, achieving minimax-optimal rates under weak dependence conditions.

Contribution

It introduces a new adaptive thresholded least squares estimator for dependent data, with non-asymptotic risk bounds and minimax optimality guarantees.

Findings

Estimator attains minimax-optimal convergence rates.

Risk bounds hold under weak dependence with fast decay of mixing coefficients.

Method is effective for both iid and dependent data scenarios.

Abstract

We consider the estimation of a structural function which models a non-parametric relationship between a response and an endogenous regressor given an instrument in presence of dependence in the data generating process. Assuming an independent and identically distributed (iid.) sample it has been shown in Johannes and Schwarz (2010) that a least squares estimator based on dimension reduction and thresholding can attain minimax-optimal rates of convergence up to a constant. As this estimation procedure requires an optimal choice of a dimension parameter with regard amongst others to certain characteristics of the unknown structural function we investigate its fully data-driven choice based on a combination of model selection and Lepski's method inspired by Goldenshluger and Lepski (2011). For the resulting fully data-driven thresholded least squares estimator a non-asymptotic oracle risk bound is derived by considering either an iid. sample or by dismissing the independence assumption. In both cases the derived risk bounds coincide up to a constant assuming sufficiently weak dependence characterised by a fast decay of the mixing coefficients. Employing the risk bounds the minimax optimality up to constant of the estimator is established over a variety of classes of structural functions.