Sieve bootstrap for functional time series

Abstract

A bootstrap procedure for functional time series is proposed which exploits a general vector autoregressive representation of the time series of Fourier coefficients appearing in the Karhunen–Loeve expansion of the functional process. A double sieve-type bootstrap method is developed, which avoids the estimation of process operators and generates functional pseudo-time series that appropriately mimics the dependence structure of the functional time series at hand. The method uses a finite set of functional principal components to capture the essential driving parts of the infinite dimensional process and a finite order vector autoregressive process to imitate the temporal dependence structure of the corresponding vector time series of Fourier coefficients. By allowing the number of functional principal components as well as the autoregressive order used to increase to infinity (at some appropriate rate) as the sample size increases, consistency of the functional sieve bootstrap can be established. We demonstrate this by proving a basic bootstrap central limit theorem for functional finite Fourier transforms and by establishing bootstrap validity in the context of a fully functional testing problem. A novel procedure to select the number of functional principal components is introduced while simulations illustrate the good finite sample performance of the new bootstrap method proposed.