Volterra bootstrap: Resampling higher-order statistics for strictly stationary univariate time series

TL;DR

This paper introduces a novel bootstrap method based on Volterra series for nonparametric hypothesis testing of stationary time series, effectively handling higher-order statistics where traditional methods fail.

Contribution

It proposes a higher-order bootstrap scheme using Volterra series and kernel regression, addressing limitations of existing bootstrap methods for higher-moment dependent statistics.

Findings

Effective for both linear and nonlinear processes

Scales linearly with input dimensionality

Valid for statistics depending on higher moments

Abstract

We are concerned with nonparametric hypothesis testing of time series functionals. It is known that the popular autoregressive sieve bootstrap is, in general, not valid for statistics whose (asymptotic) distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or a nonlinear one. Inspired by nonlinear system theory we circumvent this non-validity by introducing a higher-order bootstrap scheme based on the Volterra series representation of the process. In order to estimate coefficients of such a representation efficiently, we rely on the alternative formulation of Volterra operators in reproducing kernel Hilbert space. We perform polynomial kernel regression which scales linearly with the input dimensionality and is independent of the degree of nonlinearity. We illustrate the applicability of the suggested Volterra-representation-based bootstrap procedure in a simulation study where we consider strictly stationary linear and nonlinear processes.