On the closed form of the covariance matrix and its inverse of the causal ARMA process

Derivation of the theoretical autocovariance function of a causal autoregressive moving-average process of order (p, q), ARMA(p, q), when q >= 1 is considered. A recursive relationship is established between the covariance matrices of an ARMA(p, q) process and its associated ARMA(p, q - 1) process. The obtained recursion is shown to produce the inverse of the covariance matrix and its determinant. Moreover, the introduced method can be easily implemented in any programming environment. Copyright 2004 Blackwell Publishing Ltd.