In this paper, the computation of the exact Fisher information
matrix of a large class of Gaussian time series models is considered. This
class, which is often called the single-input–single-output (SISO)
model, includes dynamic regression with autocorrelated errors and the transfer
function model, with autoregressive moving average errors. The method is based
on a combination of two computational procedures: recursions for the covariance
matrix of the derivatives of the state vector with respect to the parameters,
and the fast Kalman filter recursions used in the evaluation of the likelihood
function. It is much faster than existing procedures. An expression for the
asymptotic information matrix is also given.