Tests of serial independence for continuous multivariate time series based on a Mobius decomposition of the independence empirical copula process

Genest and R,millard have recently studied tests of randomness based on a decomposition of the serial independence empirical copula process into a finite number of asymptotically independent sub-processes. A generalization of this decomposition that can be used to test serial independence in the continuous multivariate time series framework is investigated. The weak limits of the Cram,r-von Mises statistics derived from the various processes under consideration are determined. As these statistics are not distribution-free, the consistency of the bootstrap methodology is investigated. Extensive simulations are used to study the finite-sample behavior of the tests for continuous time series of dimension one to three, and comparisons with the portmanteau test are provided, as well as, in the one-dimensional case, with the ranked-based version of the Brock, Dechert, and Scheinkman test. Finally, the studied tests are applied to a real trivariate financial time series.