On the expectation of estimators for general ARMA processes

Abstract

A simple technique is presented for obtaining explicit expressions for the approximate expectation of estimators for general stationary autoregressive moving average (ARMA) processes. The technique is based on a Taylor series expansion of the log-likelihood function in terms of the expected values of the sample autocovariances or in terms of the expected value of the periodogram ordinates, depending upon whether the estimation makes use of time-domain methods. For illustrative purpose the expectation of estimators arising from two particular approximations of the likelihood function is obtained , although the proposed technique can be easily extended to other types of log-likelihood functions as well. The results may be used to provide information about the bias of estimators in non-seasonal and seasonal ARMA models.