Some Exact Formulae for Autoregressive Moving Average Processes

This paper demonstrates that for a finite stationary autoregressive moving average process the inverse of the covariance matrix differs from the matrix of the covariances of the inverse process by a matrix of low rank. The formula for the exact inverse of the covariance matrix of the scalar or multivariate process is provided. We obtain approximations based on this formula and evaluate some of the approximate results in the existing literature. Applications to computational algorithms and to the distributions of two-step estimators are discussed. In addition the paper contains the formula for the determinant of the covariance matrix which is useful in exact maximum likelihood estimation; it also lists the expressions for the autocovariances of scalar autoregressive moving average processes.