A Fast Algorithm for the Repeated Evaluation of the Likelihood of a General Linear Process for Long Series

The likelihood function of a stationary and invertible autoregressive moving average (ARMA) process for a given series is a function of the determinant of the covariance matrix of the series and a quadratic form involving the inverse of this matrix. In this article the quadratic form is transformed into a new quadratic form, whose matrix is evaluated readily and has constant elements along its diagonals; that is, it is a Symmetrie Toeplitz matrix. It allows the computation of the quadratic form to be performed conveniently, by multiplying these constant diagonals by the sums of the products of the elements placed at constant lags in the vector appearing in the new quadratic form. This approach leads to an algorithm with the following main properties: 1. Once the autocovariances of the series have been computed, the number of arithmetic operations required to obtain a good approximation to the likelihood function is independent of the size of the observed series. 2. The algorithm can cope with any general ...