An algorithm for the exact Fisher information matrix of vector ARMAX time series processes

In this paper an algorithm is developed for the exact Fisher information matrix of a vector ARMAX Gaussian process, VARMAX. The algorithm developed in this paper is composed by recursion equations at a vector-matrix level and some of these recursions consist of derivatives. For that purpose appropriate differential rules are applied. The derivatives are derived from a state space model for a vector process.
The chosen representation is such that the recursions extracted from the proposed state space model are given in terms of expectations of derivatives of innovations and not the process and observation disturbances. This enables us to produce an implementable algorithm for the VARMAX process. The algorithm will be illustrated by an example.