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Vector cross-correlation in time series and applications. (English) Zbl 0826.62065

Summary: This paper presents a generalization of the concept of vector correlation proposed by Y. Escoufier [Biometrics 29, 751-760 (1973)] to the context of time series. For two jointly stationary multivariate stochastic processes \(\{{\mathbf X}_t\}\) and \(\{{\mathbf Y}_t\}\) respectively, we define a coefficient of vector cross-correlation at lag \(k\), denoted by \(\lambda_{xy} (k)\), and we describe its main properties. A sample analogue \(\widehat {\lambda}_{xy} (k)\) is also introduced and its asymptotic distribution is derived for a wide class of stationary time series. For \({\mathbf Y}_t \equiv {\mathbf X}_t\), \(\lambda_{xx} (k)\) is a coefficient of vector autocorrelation and the \(\widehat {\lambda}_{xx} (k)\)’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 \(k=1, \dots, M)\). A procedure for checking the independence of two jointly stationary multivariate time series is also presented.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M07 Non-Markovian processes: hypothesis testing
62E20 Asymptotic distribution theory in statistics
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