I am currently developing a Vector Autoregressive Model, and I have the model fully specified as follows:
where $X$ and $Z$ are $n \times 1$ column vectors, and $A$ is an $n\times n$ matrix.
In one dimension, I know how to use a best linear predictor by looking at the autocovariance matrix. However, it is not so clear how to do this in multiple dimensions, since there is not only an autocovariance matrix for each dimension, but also cross terms in the serial covariances. I have a hunch about how this could be done, but would really appreciate it if anyone had any tips or a reference to look at regarding best linear predictors.
On a closely related note, I did find this paper: http://faculty.washington.edu/ezivot/econ584/notes/varModels.pdf
which explains how to forecast using VAR models (see section 11.3). However, this forecast simply uses an iterative procedure in order to get the prediction. So another question is: is this approach equivalent to the best linear predictor method?