A Canonical Form for Unit Root Processes in the State Space Framework

In this paper we develop a canonical state space representation for rational stochastic processes containing unit roots with integer integration orders at arbitrary points on the unit circle. It is shown that the state space framework, which is – in a certain sense made precise in the paper – equivalent to the ARMA framework, is very suitable for the analysis of unit roots and cointegration issues. The advantages become especially prominent for systems with higher integration orders at the various roots on the unit circle. A unique state space representation is constructed that clearly reveals the integration and cointegration properties. The canonical form given in the paper can be used to construct a parameterization of the class of all rational processes with a given state space unit root structure, which is defined in the paper.