A plug-in bandwidth selection procedure for long run covariance estimation with stationary functional time series

TL;DR

This paper introduces a data-driven method for selecting the bandwidth parameter in estimating the long run covariance function for stationary functional time series, improving inference accuracy in applications like environmental science and economics.

Contribution

It proposes a new plug-in bandwidth selection procedure that minimizes the asymptotic mean squared normed error, with proven asymptotic consistency and practical guidance from simulations.

Findings

The method achieves asymptotic consistency in bandwidth selection.

Simulation results demonstrate improved estimation accuracy.

Practical recommendations for bandwidth choice are provided.

Abstract

In arenas of application including environmental science, economics, and medicine, it is increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long run covariance function or operator, which is analogous to the long run covariance matrix familiar to finite dimensional time series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies crucially on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite dimensional setting, we propose a bandwidth selection method that aims to minimize the estimator's asymptotic mean squared normed error (AMSNE) in $L^2[0,1]^2$. As the AMSNE depends on unknown population quantities including the long run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.