Equivalence of representations for a class of nonstationary processes

The author considers models for nonstationary stochastic behavior of the type R(σ)w=0, where R(s) is a full low rank polynomial matrix in s and s^-1, σ denotes shift, and w belongs to a class of discrete-time stochastic processes called integrated processes. By definition, an integrated process is a process that can be reduced to stationarity by application of a filter that has all its zeros on the unit circle. For instance, the random walk belongs to this class. It is shown that two models of this type are equivalent, in the sense that the set of solutions is the same, if and only if the representing matrices are related by left multiplication by a matrix that is unimodular over the ring of ration functions having no poles on the unit circle