Likelihood-Based Analysis of Linear State-Space Models Using the Cholesky Decomposition

Abstract

Data associated with the linear state-space model can be assembled as a matrix whose Cholesky decomposition leads directly to a likelihood evaluation. It is possible to build several matrices for which this is true. Although the chosen matrix or assemblage can be very large, rows and columns can usually be rearranged so that sparse matrix factorization is feasible and provides an alternative to the Kalman filter. Moreover, technology for calculating derivatives of the log-likelihood using backward differentiation is available, and hence it is possible to maximize the likelihood using the Newton–Raphson approach. Emphasis is given to the estimation of dispersion parameters by both maximum likelihood and restricted maximum likelihood, and an illustration is provided for an ARMA(1,1) model.