Analysis of autoregressive-moving average models: Estimation and prediction

The exact likelihood function from a set of observations generated by univariate stationary normal autoregressive–moving average models is derived. Convenient methods are developed to evaluate the exact likelihood function, specifically the inverse and determinant of the T × T covariance matrix of the observations, ΣT and the quadratic form, Z′ΣT-1Z, where Z is a T × 1 vector of observations. Computations of the least squares and the maximum likelihood estimates of the parameters are discussed and illustrated. Several approximations to these estimators are suggested and their feasibility and quality are examined through a simulation. They seem promising alternatives from the limited cases that have been tried. The mean and variance of the predictive distribution of any future Z conditional on the observed Z∘s are conveniently expressed for exact computation.