Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA(1) Models

Extending the results in Sargan (1976) and Tanaka (1984), we derive the asymptotic expansions, of the Edgeworth and Nagar type, of the MM and QML estimators of the 1^{st} order autocorrelation and the MA parameter for the MA(1) model. It turns out that the asymptotic properties of the estimators depend on whether the mean of the process is known or estimated. A comparison of the Nagar expansions, either in terms of bias or MSE, reveals that there is not uniform superiority of neither of the estimators, when the mean of the process is estimated. This is also confirmed by simulations. In the zero-mean case, and on theoretical grounds, the QMLEs are superior to the MM ones in both bias and MSE terms. The results presented here are important for deciding on the estimation method we choose, as well as for bias reduction and increasing the efficiency of the estimators.