Accurate Estimation of Moving Average Models with Durbin's method

The best accuracy for estimated spectra is obtained with parsimonious time series models, which have the smallest number of parameters to guarantee unbiased models. Durbin 's method for the estimation of moving average (MA) parameters uses the estimated parameters of a long autoregressive (AR) model to calculate MA parameters. Probably all pejorative remarks on the quality of Durbin's method in the literature are based on suboptimal or wrong choices for the method of AR estimation or for the order of the intermediate AR model. That AR order should be considerably higher than the order of the best predicting AR model and it should grow with the sample size. Furthermore, the Burg estimates for the AR parameters give the best results because they have the smallest variance of all AR methods with a small bias. The triangular autocorrelation bias of the popular Yule-Walker method of AR estimation can cause large bias errors in finite samples, which makes it unsuited. Durbin's method applied to the proper number of AR parameters estimated with Burg's method outperforms all other known MA estimation methods, asymptotically as well as in finite samples. The accuracy is generally close to the Cramer-Rao bound.