A frequency domain bootstrap for general multivariate stationary processes

Developing valid frequency domain bootstrap procedures for integrated periodogram statistics for multivariate time series is a challenging problem. This is mainly due to the fact that the distribution of such statistics depends on the fourth-order moment structure of the underlying multivariate process in nearly every scenario. Exceptions are some very special cases like nonparametric estimators of the spectral density matrix or Gaussian time series. In contrast to the univariate case, even additional structural assumptions - such as linearity of the multivariate process or a standardization of the statistic of interest - do not solve the problem. This paper proposes a new frequency domain bootstrap procedure for multivariate time series, the multivariate frequency domain hybrid bootstrap (MFHB), for integrated periodogram statistics as well as for functions thereof. Asymptotic validity of the MFHB procedure is established for these statistics and for a class of stationary multivariate processes satisfying rather weak dependence conditions ranging clearly beyond linear processes. The finite sample performance of the MFHB is investigated by means of simulations.