Construction of the exact Fisher information matrix of Gaussian time series models by means of matrix differential rules

The Fisher information matrix is of fundamental importance for the analysis of parameter estimation of time series models. In this paper the exact information matrix of a multivariate Gaussian time series model expressed in state space form is derived. A computationally efficient procedure is used by applying matrix differential rules for the derivatives of a matrix function J=J(θ) with respect to its vector argument. An algorithm is given. It is sketched for the general state space structure without specifying a parametrization. It is then detailed for the vector autoregressive moving average (VARMA) model, with a given parametrization, where explicit recurrent relations are developed.