In this section we introduce two examples of bivariate time series. A bivariate time series is a…

In this section we introduce two examples of  bivariate time series. A  bivariate time series is a series of two-dimensional vectors (Xt1,Xt2)_ observed at times (usually = 123, . . .). The two component series {Xt1} and {Xt2} could be studied independently as univariate time series, each characterized, from a second-order point of view, by its own mean and auto covariance function. Such an approach, however, fails to take into account possible dependence between the two component series, and such cross-dependence may be of great importance, for example in predicting future values of the two component series.

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We therefore consider the series of random vectors Xt(Xt1,Xt2)_ and define the mean vector

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