A Recursion Formula for Calculating the Exact Gradient Vector of the Loglikelihood for Linear Gaussian State Space Model

Abstract

Linear Gaussian state space models have been widely used in a variety of fields. In estimation and testing of these state space models, the gradient vector of the log-likelihood plays important roles. One usually calculates an approximate value of the gradient vector by numerically differentiating the log-likelihood. In this paper, we propose a recursive formula for computing the exact value of the gradient vector for a general form of linear Gaussian state space models. We also explicitly consider how to handle the initial condition in the context of computing the exact gradient vector when state variables are stationary, which is the issue that has not been considered in the the existing literature. We give two examples to illustrate how to apply the proposed formula.(A former title is A Recursion Formula for Calculating the Exact Gradient Vector of the Loglikelihood for Linear Gaussian State Space Models)