Functional linear instrumental regression under second order stationarity

TL;DR

This paper develops a nonparametric estimator for the slope in functional linear instrumental regression under second order stationarity, achieving minimax optimal convergence rates for Sobolev space parameters.

Contribution

It introduces a new estimator leveraging stationarity assumptions and analyzes its optimal convergence rates in various smoothing scenarios.

Findings

Estimator achieves minimax optimal convergence rates.

Handles finitely and infinitely smoothing cross-covariance operators.

Provides theoretical guarantees for Sobolev space parameters.

Abstract

We consider the problem of estimating the slope parameter in functional linear instrumental regression, where in the presence of an instrument W, i.e., an exogenous random function, a scalar response Y is modeled in dependence of an endogenous random function X. Assuming second order stationarity jointly for X and W a nonparametric estimator of the functional slope parameter and its derivatives is proposed based on an n-sample of (Y,X,W). In this paper the minimax optimal rate of convergence of the estimator is derived assuming that the slope parameter belongs to the well-known Sobolev space of periodic functions. We discuss the cases that the cross-covariance operator associated to the random functions X and W is finitely, infinitely or in some general form smoothing.