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.