Vector Cross-Correlation in Time Series and Applications

Summary This paper presents a generalization of the concept of vector correlation proposed by Escoufier (1973) to the context of time series. For two jointly stationary multivariate stochastic processes {X,) and {Y,} respectively, we define a coefficient of vector cross-correlation at lag k, denoted by Ax,(k), and we describe its main properties. A sample analogue Afi(k) is also introduced and its asymptotic distribution is derived for a wide class of stationary time series. For Y, X,, Axx(k) is a coefficient of vector autocorrelation and the A~x(k)'s can be used, in particular, to test the hypothesis of white noise. First, we describe a test for white noise against serial dependence at each lag k and secondly we define a global test against serial dependence at several lags (say k = 1, .... M). A procedure for checking the independence of two jointly stationary multivariate time series is also presented.