Optimal instrumental variable estimates of the AR parameters of an ARMA process

The modified Yule-Walker (MYW) equations for estimating the AR parameters of an ARMA process are presented as a special case of an instrumental variable (IV) method. The consistency and accuracy of the AR parameter estimates are studied. It is shown that estimation accuracy increases monotonically with the number of MYW equations for an optimal choice of the weighting matrix used in the least-squares solution of these equations. The asymptotic error covariance of the optimal IV method equals that of the prediction error method. The results of this paper verify experimental results reported in the literature regarding the performance of the MYW method, and provide the necessary accuracy analysis. Furthermore, they suggest several simple, asymptotically efficient, multistep algorithms for estimating the AR parameters, which are presented in a companion paper.