Parameter estimation in multivariate stochastic difference equations

We will review the principal methods of estimation of parameters in multivariate autoregressive moving average equations which have additional observable input terms in them and present some new methods of estimation as well. We begin with the conditions for the estimability of the parameters. In addition to the usual method of system representation, the canonical form I, we will present two new representations of the system equation, the so-called canonical forms II and III which are convenient for parameter estimation. We will mention, in some detail, the various methods of estimation like the various least-squares methods, the maximum likelihood methods, etc., and discuss them regarding their relative accuracy of the estimate and the corresponding computational complexity. We will introduce a new class of estimates, the so-called limited information estimates which utilizes the canonical forms II and III. The accuracy of these estimates is close to that of maximum likelihood, but their computation time is only a fraction of the computation time for the usual maximum likelihood estimates. We will present a few numerical examples to illustrate the various methods.