If each time series observation is a vector of numbers, you can model them using a multivariate form of the Box-Jenkins model
The multivariate form of the Box-Jenkins 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
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:
– xt is a vector of observations, x1t,x2t,…,xnt at time t,
– at is a vector of white noise, a1t,a2t,…,ant at time t,
is a n×n matrix of autoregressive parameters,
where Σa is the dispersion or covariance matrix.
A model with p autoregressive matrix parameters is an ARV(p) model or a vector AR model.
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:
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:
ϕ1.11,ϕ2.11 for the input series X,
ϕ1.22,ϕ2.22 for the output series Y.
The lower off-diagonal elements represent the influence of the input on the output.
This is called the “transfer” mechanism or transfer-function model as discussed by Box and Jenkins in Chapter 11. The ϕ terms here correspond to their δ terms.
The upper off-diagonal terms represent the influence of the output on the input.
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:
xt = ϕ1.11xt−1+ϕ2.11xt−2+ϕ1.12yt−1+ϕ2.12yt−2+a1t
yt = ϕ1.22yt−1+ϕ2.22yt−2+ϕ1.21xt−1+ϕ2.21xt−2+a2t
Finally, delay or “dead” time can be measured by studying the lower off-diagonal elements again.
If, for example, ϕ1.21 is non-significant, the delay is 1 time period