On orders of long AR models for ARMA estimation

Long intermediate AR models are used in Durbin's two algorithms for ARMA estimation. The order of that long AR model should be co in the asymptotical theory, but very high AR orders lead to inaccurate ARMA models in practice. A theoretical derivation is given for the requirements of AR orders for known ARMA processes. Two different applications of AR models each have their own best theoretical order. The first is the AR order that is optimal for prediction with a purely autoregressive model; this order also gives the best results in Durbin's first ARMA algorithm with reconstructed excitations. The second theoretical value for the AR order is higher and should be used if the AR parameters themselves are important, e.g. for estimating the MA parameters with Durbin's second ARMA method. A Sliding Window (SW) algorithm is presented that enables use of the theory for known processes to determine good long AR orders for data of unknown processes.