Bootstrap for testing the equality of selfsimilarity exponents across multivariate time series

Because of the ever-increasing collections of multivariate data, multivariate selfsimilarity has become a widely used model for scale-free dynamics, with successful applications in numerous different fields. Multivariate selfsimilarity exponent estimation has therefore received considerable attention, with notably an original procedure recently proposed and based on the eigenvalues of the covariance random matrices of the wavelet coefficients at fixed scales. Expanding on preliminary work aiming to test for the equality of the selfsimilarity exponents in bivariate time series, we propose and study here a truly multivariate procedure that permits, from a single observation of multivariate time series, to test for the equality of several, possibly many, selfsimilarity exponents. It is based on an original bootstrap procedure, applied in a multivariate time-scale domain and designed to effectively capture the scale-dependent joint covariance structure of multivariate wavelet coefficients as well as the associated wavelet eigenvalue structures. Extensive simulations conducted on synthetic data, modeled by operator fractional Brownian motions, the reference multivariate selfsimi-larity model, permit to show that the proposed multivariate timescale domain bootstrap based test yields the targeted significance level under the null hypothesis (all selfsimilarity exponents are equal) and to assess the power of the test for several alternative hypotheses. This analysis leads us to conclude that the proposed test for the equality of multivariate selfsimilarity exponents is effective and ready for use on (a single time series of) real data.