An implimentation of the Differential Filter for Computing Gradient and Hessian of the Log-likelihood of Nonstationary Time Series Models

Abstract: 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.