The determination of optimum structures for the state space representation of multivariate stochastic processes

When identifying a model for a multivariate stationary stochastic process, an important problem is that of determining the structure of the state-variable model. Several "overlapping" parameterizations can usually be fitted to a given process, and the question arises as to which structure leads to the most accurate parameter estimates. The accuracy of parameter estimates is often measured by the determinant of the Fisher information matrix. We show that all admissible structures will give asymptotically the same value to this criterion. For finite data some structures may still be better than others, and two heuristic structure estimation methods are analyzed. Some simulation results are also presented.