A State Space Canonical Form For Unit Root Processes

In this paper we develop a canonical state space representation of autoregressive moving average (ARMA) processes with unit roots with integer integration orders at arbitrary unit root frequencies. The developed representation utilizes a state process with a particularly simple dynamic structure, which in turn renders this representation highly suitable for unit root, cointegration, and polynomial cointegration analysis. We also propose a new definition of polynomial cointegration that overcomes limitations of existing definitions and extends the definition of multicointegration for I(2) processes of Granger and Lee (1989a, Journal of Applied Econometrics 4,145-159). A major purpose of the canonical representation for statistical analysis is the development of parameterizations of the sets of all state space systems of a given system order with specified unit root frequencies and integration orders. This is, e.g., useful for pseudo maximum likelihood estimation. In this respect an advantage of the state space representation, compared to ARMA representations, is that it easily allows one to put in place restrictions on the (co)integration properties. The results of the paper are exemplified for the cases of largest interest in applied work.