  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.
