8.4 Vector ARMA Models
Univariate ARMA models can also be generalized to handle vector time series. The resulting models are called VARMA models. The generalization, however, encounters some new issues that do not occur in developing VAR and VMA models. One of the issues is the identifiability problem. Unlike the univariate ARMA models, VARMA models may not be uniquely defined. For example, the VMA(1) model
is identical to the VAR(1) model
The equivalence of the two models can easily be seen by examining their component models. For the VMA(1) model, we have
For the VAR(1) model, the equations are
From the model for r2t, we have r2, t−1 = a2, t−1. Therefore, the models for r1t are identical. This type of identifiability problem is harmless because either model can be used in a real application.
Another type of identifiability problem is more troublesome. Consider the VARMA(1,1) model
This model is identical to the VARMA(1,1) model
for any nonzero ω and η. In this particular instance, ...