On The Identification And Estimation Of Nonstationary And Cointegrated Armax Systems

This paper extends current theory on the identification and estimation of vector time series models to nonstationary processes. It examines the structure of dynamic simultaneous equations systems or ARMAX processes that start from a given set of initial conditions and evolve over a given, possibly infinite, future time horizon. The analysis proceeds by deriving the echelon canonical form for such processes. The results are obtained by amalgamating ideas from the theory of stochastic difference equations with adaptations of the Kronecker index theory of dynamic systems. An extension of these results to the analysis of unit-root, partially nonstationary (cointegrated) time series models is also presented, leading to straightforward identification conditions for the error correction, echelon canonical form. An innovations algorithm for the evaluation of the exact Gaussian likelihood is given. The asymptotic properties of the approximate Gaussian estimator and the exact maximum likelihood estimator based upon the algorithm are derived for the cointegrated case. Examples illustrating the theory are discussed, and some experimental evidence is also presented.I thank two referees for insightful comments and helpful suggestions on the content and presentation of this paper. I am particularly grateful for the correction of errors in earlier drafts and reference to the work of B. Hanzon. Financial support under ARC grant DP0343811 is gratefully acknowledged.