Computation of the Gradient and the Hessian of the Log-likelihood of the State-space Model by the Kalman Filter

TL;DR

This paper introduces an algorithm that efficiently computes the gradient and Hessian of the log-likelihood for state-space models using the Kalman filter, avoiding numerical differentiation.

Contribution

It extends the Kalman filter to directly calculate derivatives of the log-likelihood, improving accuracy and computational efficiency over previous methods.

Findings

Algorithm successfully computes gradients and Hessians for ARMA and seasonal adjustment models.

Provides examples demonstrating the specification of structural and initial matrices.

Enhances maximum likelihood estimation procedures for state-space models.

Abstract

The maximum likelihood estimates of an ARMA model can be obtained by the Kalman filter based on the state-space representation of the model. This paper presents an algorithm for computing gradient of the log-likelihood by an extending the Kalman filter without resorting to the numerical difference. Three examples of seasonal adjustment model and ARMA model are presented to exemplified the specification of structural matrices and initial matrices. An extension of the algorithm to compute the Hessian matrix is also shown.