Explicit vector expression of exact score for time series models in state space form

Koopman and Shephard (1992) [1] and Segal and Weinstein (1989) [4] propose a formula for calculating the exact score vector for a general form of linear Gaussian state space models. However, for applying their method, one needs to calculate the derivatives of functions with respect to vectors and matrices, which can be intractable in many practical cases. Koopman and Shephard (1992) [1] derive its explicit expression only for a particular case. In this note, we complement Koopman and Shephard (1992) [1] and Segal and Weinstein [4] by deriving an explicit vector expression of the exact score vector for the general form of linear Gaussian state space models.