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

Extending the results in Sargan (1976 Sargan , J. D. ( 1976 ). Econometric estimators and the Edgeworth approximation . Econometrica 44 : 421 – 448 .Crossref], Web of Science ®] , Google Scholar]) and Tanaka (1984 Tanaka , K. ( 1984 ). An asymptotic expansion associated with the maximum likelihood estimators in ARMA models . J. Roy. Statist. Soc. B 46 : 58 – 67 . Google Scholar]), we derive the asymptotic expansions of the distribution, the bias and the mean squared error of the MM and QML estimators of the first-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 moment 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. We also discuss how the approximations are affected by moderate deviations from the unit root case. The results presented here are important for bias correction and increasing the efficiency of the estimators.