6. Process or Product Monitoring and Control 6.4. Introduction to Time Series Analysis 6.4.5. Multivariate Time Series Models 

If each time series observation is a vector of numbers, you can model them using a multivariate form of the BoxJenkins model 
The multivariate form of the BoxJenkins univariate models is
sometimes called the ARMAV model, for AutoRegressive Moving Average
Vector or simply vector ARMA process.
The ARMAV model for a stationary multivariate time series, with a zero mean vector, represented by $$ x_t = (x_{1t}, \, x_{2t}, \, \ldots, \, x_{nt})^T, \,\,\,\,\,\,\,\, \infty < t < \infty $$ is of the form $$ \begin{eqnarray} x_t & = & \phi_1 x_{t1} + \phi_2 x_{t2} + \cdots + \phi_p x_{tp} + \\ & & a_t  \theta_1 a_{t1}  \theta_2 a_{t2}  \cdots  \theta_q a_{tq} \, , \end{eqnarray} $$ where

Estimation of parameters and covariance matrix difficult 
The estimation of the matrix parameters and
covariance
matrix is complicated and very difficult without computer
software. The estimation of the Moving Average matrices is
especially an ordeal. If we opt to ignore the MA component(s) we
are left with the ARV model given by:
$$ x_{t} = \phi_{1}x_{t1} + \phi_{2}x_{t2} + \ldots + \phi_{p}x_{tp} + a_{t} \, , $$
The parameter matrices may be estimated by multivariate least squares, but there are other methods such as maximium likelihood estimation. 
Interesting properties of parameter matrices 
There are a few interesting properties associated with the phi or AR
parameter matrices. Consider the following example for a bivariate
series with \(n=1\), \(p=2\), and \(q = 0\).
The ARMAV(2,0) model is:
$$ \left( \begin{array}{c} x_{t} \\ y_{t} \end{array} \right) = \left( \begin{array}{cc} \phi_{1.11} & \phi_{1.12} \\ \phi_{1.21} & \phi_{1.22} \end{array} \right) \left( \begin{array}{c} x_{t1} \\ y_{t1} \end{array} \right) + \left( \begin{array}{cc} \phi_{2.11} & \phi_{2.12} \\ \phi_{2.21} & \phi_{2.22} \end{array} \right) \left( \begin{array}{c} x_{t2} \\ y_{t2} \end{array} \right) + \left( \begin{array}{c} a_{1t} \\ a_{2t} \end{array} \right) \, .$$ Without loss of generality, assume that the \(X\) series is input and the \(Y\) series are output and that the mean vector is \((0,0)\). Therefore, tranform the observation by subtracting their respective averages. 
Diagonal terms of Phi matrix 
The diagonal terms of each Phi matrix are the scalar
estimates for each series, in this case:
\(\phi_{1.22}, \, \phi_{2.22}\) for the output series \(Y\). 
Transfer mechanism 
The lower offdiagonal elements represent the influence of the
input on the output.
This is called the "transfer" mechanism or transferfunction model as discussed by Box and Jenkins in Chapter 11. The \(\phi\) terms here correspond to their \(\delta\) terms. The upper offdiagonal terms represent the influence of the output on the input. 
Feedback 
This is called "feedback". The presence of feedback can also be seen
as a high value for a coefficient in the correlation matrix of the
residuals. A "true" transfer model exists when there is no feedback.
This can be seen by expressing the matrix form into scalar form: $$ \begin{eqnarray} x_t & = & \phi_{1.11}x_{t1} + \phi_{2.11}x_{t2} + \phi_{1.12}y_{t1} + \phi_{2.12}y_{t2} + a_{1t} \\ & & \\ y_t & = & \phi_{1.22}y_{t1} + \phi_{2.22}y_{t2} + \phi_{1.21}x_{t1} + \phi_{2.21}x_{t2} + a_{2t} \end{eqnarray} $$ 
Delay 
Finally, delay or "dead" time can be measured by studying the lower
offdiagonal elements again.
If, for example, \(\phi_{1.21}\) is nonsignificant, the delay is 1 time period. 