Estimation Of Autoregressive Moving‐Average Models Via High‐Order Autoregressive Approximations

. In this paper the problem of estimating autoregressive moving‐average (ARMA) models is dealt with by first estimating a high‐order autoregressive (AR) approximation and then using the AR estimate to form the ARMA estimate. We show how to obtain an efficient ARMA estimate by allowing the order of the AR estimate to tend to infinity as the number of observations tends to infinity. This approach is closely related to the work of Durbin. By transforming the approach into the frequency domain, we can view it as an L2‐norm model approximation of the relative error of the spectral factors. It can also be seen as replacing the periodogram estimate in the Whittle approach by a high‐order AR spectral density estimate. Since L2‐norm approximation is a difficult task, we replace it by a modification of a recent model approximation technique called balanced model reduction. By an example, we show that this technique gives almost efficient ARMA estimates without the use of numerical optimization routines.