Tests of serial dependence for multivariate time series with arbitrary distributions

Abstract

In this paper, one studies tests of serial independence using a fixed number p of consecutive observations from a stationary time series, first in the univariate case, and then in the multivariate case, where even high-dimensional vectors can be used. The common distribution function is not assumed to be continuous, and the test statistics are based on the multilinear copula process. A bootstrap procedure based on multipliers is also proposed and shown to be valid. Tests based on Spearman’s rho and Kendall’s tau statistics are also considered, extending the results known for the case of continuous distributions. Contiguity results are obtained for some specific models and sufficient conditions for consistency of test statistics are stated, as well as a data-driven procedure to select p. Also, numerical experiments are performed to assess the finite sample level and power of the proposed tests. A case study using a time series of Arctic sea ice extent images is used to illustrate the usefulness of the proposed methodologies. The R package MixedIndTests (Nasri et al., 2022) includes all the methodologies proposed in this article.