Maximum likelihood estimation for MA(1) processes with a root on or near the unit circle

This paper considers maximum likelihood estimation for the moving average parameter theta in an MA(1) model when theta is equal to or close to 1. A derivation of the limit distribution of the estimate <(theta)over cap(LM)>, defined as the largest of the local maximizers of the likelihood, is given here for the first time. The theory presented covers, in a unified way, cases where the true parameter is strictly inside the unit circle as well as the noninvertible case where it is on the unit circle. The asymptotic distribution of the maximum likelihood estimator <(theta)over cap(MLE)> is also described and shown to differ, but only slightly, from that of <(theta)over cap(LM)>. Of practical significance is the fact that the asymptotic distribution for either estimate is surprisingly accurate even for small sample sizes and for values of the moving average parameter considerably far from the unit circle.