Problems with the estimation of moving average processes

As a consequence of the well-known underidentification of the moving average model unless the parameter space is restricted, Maximum Likelihood and other estimators possess properties which can pose problems for estimation when a root of the process is close to the unit circle. The behaviour of the estimators is studied both through the analytic properties of their criterion functions and by Monte Carlo simulation. Conclusions about the choice of estimator are drawn, in particular regarding the treatment of the pre-sample residuals.