(Generalised) autoregressive models and their trajectories

Abstract

The set of classical AR-models is not ‘compact’, so that a sequence of AR-models may fail to converge to anything reasonable. A compactification leads to AR-models with arbitrary singularity at infinity. These limit AR-models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition. It is shown that they produce trajectories in a very natural way. Then the classical equivalence theorem is extended to them, stating that two AR-models having the same behaviour are equivalent.

Keywords:

• AR-model
• behaviour
• transfer function
• initial condition
• frequency response

Acknowledgement

The author would like to thank the referees for their critical remarks.