# 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 *(X _{t}*

_{1}

*,X*

_{t}_{2}

*)*_ observed at times

*t*(usually

*t*= 1

*,*2

*,*3

*, . . .*). The two component series {

*X*

_{t}_{1}} and {

*X*

_{t}_{2}} 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.

We therefore consider the series of random vectors X* _{t}*=

*(X*

_{t}_{1}

*,X*

_{t}_{2}

*)*_ and define the mean vector