I've found all sorts of papers on dimension reduction for time series such as this (which doesn't seem to work anymore) and this.
All the papers seem to start with a discussion about how their method is superior to some other simpler method.
I have never had to think about dimension reduction in time series before so I am not familiar with any of the methods. So I got to thinking about what the simplest approach would be.
Objective: To arrive at a VARMA description of some multivariate time series data, whilst avoiding the need to estimate a huge number of variables with a small amount of data. Ultimately I want to end up with a VARMA equation in full dimension. My immediate problem is actually only a VMA(1) model though so if you know of something that only works for a VMA(1) model, then that is fine.
Is anything like the following method possible:
Use principal Components Analysis (PCA) to reduce the dimension of the original data
Fit a VARMA model on the most significant components of the transformed data
Somehow use the rotation matrix from the PCA to convert the estimated time series parameters from the rotated space back to the original space.
I have found some info on structural specifications like scalar component models (SCMs) but this doesn't really reduce the dimension as much as PCA would, and quite frankly, SCM is really difficult to understand!