  The state-space model and the Kalman filter provide us with unified and
computationaly efficient procedure for computing the log-likelihood of the
diverse type of time series models. This paper presents an algorithm for
computing the gradient and the Hessian matrix of the log-likelihood by
extending the Kalman filter without resorting to the numerical difference.
Different from the previous paper(Kitagawa 2020), it is assumed that the
observation noise variance R=1. It is known that for univariate time series, by
maximizing the log-likelihood of this restricted model, we can obtain the same
estimates as the ones for the original state-space model. By this modification,
the algorithm for computing the gradient and the Hessian becomes somewhat
complicated. However, the dimension of the parameter vector is reduce by one
and thus has a significant merit in estimating the parameter of the state-space
model especially for relatively low dimentional parameter vector. Three
examples of nonstationary time seirres models, i.e., trend model, statndard
seasonal adjustment model and the seasonal adjustment model with AR componet
are presented to exemplified the specification of structural matrices.
