6.4.5. Multivariate Time Series Models

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 Box-Jenkins model

The multivariate form of the Box-Jenkins univariate models is sometimes called the ARMAV model, for AutoRegressive Moving Average Vector or plainly vector ARMA process.
The ARMAV model for a stationary multivariate time series, with a zero mean vector, represented by

The estimation of the matrix parameters and covariance matrix is complicated and impossible without computer software. Especially the estimation of the Moving Average matrices is an ordeal. If we opt to ignore the MA component(s) we are left with the ARV model given by:

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.

There are a few interesting properties associated with the phi or AR parameter matrices. Consider the following ARV(2) model:

Without los of generality assume that the X series are input and the Y series are output and that the mean vector = (0,0).
Therefore tranform the observation by subtracting their respective average.

The diagonal terms of each Phi matrix are the scalar estimates for each series, in this case:

f_1.11 , f_2.11 for the input series X,
f._2.11, f _2.22 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 f terms here correpsond to their d 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:

Finally, delay or "dead' time can be measured by studying the lower off-diagonal elements again.

If, for example, f _1.21 is non-significant, the delay is 1 time period.