Accurate ARMA models with Durbin´s second method

Long intermediate AR models are used in Durbin´s (1959) algorithms for ARMA estimation. The order of that long AR model is infinite in the asymptotical theory, but very high AR orders are known to give inaccurate ARMA models in practice. A theoretical derivation is given for two different finite AR orders, as a function of the sample size. The first is the AR order optimal for prediction with a purely autoregressive model. The second theoretical AR order is higher and applies if the previously estimated AR parameters are used for estimating the MA parameters in Durbin´s (1960) second, iterative, ARMA method. A sliding window (SW) algorithm is presented that selects good long AR orders for data of unknown processes. With a proper choice of the AR order, the accuracy of Durbin´s second method approaches the Cramer-Rao bound for the integrated spectrum and the quality remains excellent if less observations are available