Parameter estimation of ARMA models using a computationally efficient maximum likelihood technique

A method is presented for estimating the parameters of a fixed order autoregressive moving average model, based on maximization of an appropriate likelihood function. The resulting static optimization is accomplished with a modified Newton numerical algorithm. Under suitable initial conditions for the model, the gradient and hessian matrices of each iteration can be compted analytically. Because of the highly nonlinear character of the likelihood function, starting values of the algorithm are important. Extensions of the method to more complicated models are described. Some numerical examples illustrate the properties of the method.