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

In this paper an algorithm is developed for the exact Fisher information matrix of a Gaussian vector ARMAX or VARMAX process. The algorithm proposed in this paper is composed by Chandrasekhar recursion equations at a vector–matrix level, and some of these recursions consist of derivatives based on appropriate differential rules applied to a state space model for a vector process. The chosen representation is such that the recursions extracted from the state space model are given in terms of expectations of derivatives of innovations, and not the process and observation disturbances. The algorithm will be illustrated by an example. On that example, a comparison is made with results from E4, a toolbox for Matlab, and with the asymptotic information matrix.