On Efficient Exact Maximum Likelihood Estimation of High-Order ARMA Models

Abstract

The evaluation of the exact likelihood function of an autoregressive moving average (ARMA) process via Kalman filtering requires the specification of the unconditional covariance matrix of the state vector. Gardner et al. (1980) suggest a derivation which is widely used in empirical work. It involves finding the solution to a Lyapunov-type equation and amounts to solving a system of linear equations. Depending on the order of the ARMA model, the number of unknowns in this system can be rather large, since it corresponds to the number of unique elements in the unconditional state covariance matrix. An alternative algorithm for constructing the covariance matrix for univariate models which considers the specific structure of the state vector is given in Jones (1980). Using a state space representation that is different from the one in Gardner et al. and commonly used in the systems literature,¹ Mélard (1984) and Shea (1987) apply the Chandrasekhar version of the Kalman filter (see Morf et al., 1974) to compute the exact likelihood of ARMA models. This approach is computationally more efficient and requires only the first block column of the state covariance matrix rather than the full matrix. The Chandrasekhar algorithm is, however, not applicable in exact maximum likelihood procedures designed for missing-data situations as suggested by Ansley and Kohn (1983), for example.