An efficient algorithm for calculating the likelihood and likelihood gradient of ARMA models

Exact analytical expressions are obtained for the likelihood and likelihood gradient stationary autoregressive moving average (ARMA) models. Denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p+2q+1)N +o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p+6q+2)N+o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2 p+6q+2)N+o(N) multiply-add operations per iteration