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

Kyriakopoulou, Dimitra and Demos, Antonis (2010): Edgeworth and Moment Approximations: The Case of MM and QML Estimators for the MA (1) Models. Published in:

Abstract

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Edgeworth expansion, moving average process, method of moments, Quasi Maximum Likelihood, autocorrelation, asymptotic properties
• Subjects: C - Mathematical and Quantitative Methods > C0 - General; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling; Y - Miscellaneous Categories > Y1 - Data: Tables and Charts

References

Ali, M.M. 1984. Distributions of the sample autocorrelations when observations are from a stationary autoregressive-moving-average process. Journal of Business and Economic Statistics 2, 271-278.

Andrews, D.W.K. 1999. Estimation when a parameter is on a boundary. Econometrica 67, 1341-1383.

Andrews, D.W.K. and O. Lieberman 2005. Valid edgeworth expansions for the Whittle maximum likelihood estimator for stationary long-memory Gaussian time series. Econometric Theory 21, 710-734.

Arvanitis, S. and A. Demos 2006. Bias Properties of Three Indirect Inference Estimators, mimeo Athens University of Economics and Business.

Barndorff-Nielsen, O.E. and D.R. Cox 1989. Asymptotic Techniques for Use in Statistics. Chapman and Hall.

Bhattacharya, R.N. and J.K. Ghosh 1978. On the validity of the formal Edgeworth Expansion. The Annals of Statistics 6, 434-451.

Bao, Y. and A. Ullah 2007. The second-order bias and mean squared error of estimators in time-series models. Journal of Econometrics 140, 650-669.

Calzolari, G., G. Fiorentini and E. Sentana 2004. Constrained Indirect Estimation. Review of Economic Studies 71, 945-973.

Chambers, J.M. 1967. On methods of asymptotic approximation for multivariate distributions. Biometrika 54, 367-383.

Corradi, V. and E.M. Iglesias (2008). Bootstrap refinements for QML estimators of the GARCH(1,1) parameters. Journal of Econometrics 144, 400-510.

Edgeworth and Moment Expansions Davidson, J.E.H. (1981. Problems with the Estimation of Moving Average Processes. Journal of Econometrics 16, 295-310.

Davis, R. and S. Resnick 1986. Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics 14, 533-558.

Durbin, J. 1959. E¢ cient estimation of parameters in Moving-Average models. Biometrika 46, 306-316.

Durbin, J. 1980. Approximations for densities of su¢ cient estimators. Biometrika 67, 311-333.

Gotze, F. and C. Hipp 1983. Asymptotic Expansions fro Sums of Weakly De- pendent Random Variables. Theory of Probability and its Applications 64, 211-239.

Gotze, F. and C. Hipp 1994. Asymptotic distribution of statistics in time series. The Annals of Statistics 22, 2062-2088.

Gourieroux, C., A. Monfort and E. Renault 1993. Indirect Inference. Journal of Applied Econometrics 8, S85-S118.

Hall, P. 1992. The bootstrap and edgeworth expansion. Springer.

Hall, P. and J.L. Horowitz 1996. Bootstrap Critical Values for Tests Based on Generelized-Method-of Moments Estimators. Econometrica 64, 891-916.

Iglesias, E.M. and O.B. Linton 2007. Higher order asymptotic theory when a parameter is on a boundary with an application to GARCH models. Econometric Theory 23, 1136-1161.

Iglesias, E.M. and G.D.A. Phillips 2008. Finite sample theory of QMLE in ARCH models with dynamics in the mean equation. Journal of Time Series Analysis 29, 719-737.

Kakizawa, Y. 1999. Valid Edgeworth expansions of some estimators and bootstrap conÖdence intervals in Örst-order autoregression. Journal of Time Series Analysis 20, 343-359.

Kan, R. and X. Wang 2010. On the distribution of the sample autocorrelation coefficients. Journal of Econometrics 154, 101-121.

Lieberman, O., J. Rousseau and D.M. Zucker 2003. Valid asymptotic expansions Edgeworth and Moment Expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The Annals of Statistics 31, 586-612.

Linnik, Y.V. and N.M. Mitrofanova 1965. Some asymptotic expansions for the distribution of the maximum likelihood estimate. Sankhya A 27, 73-82.

Linton, O. 1997. An asymptotic expansion in the GARCH(1,1) model. Econo- metric Theory 13, 558-581.

Mitrofanova, N.M. 1967. An asymptotic expansion for the maximum likelihood estimate of a vector parameter. Theory of Probability and its Applications 12, 364-372.

Nagar, A.L. 1959. The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 575-595.

Ogasawara, H. 2006. Asymptotic expansion of the sample correlation coefficient under nonnormality. Computational Statistics and Data Analysis 50, 891-910.

Phillips, P.C.B. 1977. Approximations to some Önite sample distributions associated with a Örst-order stochastic di§erence equation. Econometrica 45, 463-485.

Rothenberg, T.J. 1986. Approximating the distributions of econometric estimators and test statistics, In The Handbook of Econometrics. vol. II, Amsterdam: North-Holland.

Sargan, J.D. 1974. Validity of Nagarís expansion. Econometrica 42, 169-176.

Sargan, J.D. 1976. Econometric estimators and the Edgeworth approximation. Econometrica 44, 421-448.

Sargan, J.D. 1977. Erratum "Econometric estimators and the Edgeworth ap- proximation". Econometrica 45, 272.

Sargan, J.D. 1988. Econometric estimators and the Edgeworth approximation. Contributions to Econometrics vol. 2, 98-132. E. Maasoumi editor. Cambridge University Press.

Sargan, J.D. and S.E. Satchell 1986. A theorem of validity for Edgeworth ex- pansions. Econometrica 54, 189-213.

Tanaka, K. 1983. Asymptotic expansions associated with the AR(1) model with Edgeworth and Moment Expansions unknown mean. Econometrica 51, 1221-1232.

Tanaka, K. 1984. An asymptotic expansion associated with the maximum likelihood estimators in ARMA models. Journal of Royal Statistical Society B 46, 58-67.

Taniguchi, M. 1987. Validity of Edgeworth expansions of minimum contrast estimators for Gaussian ARMA processes. Journal of Multivariate Analysis 21, 1-28.

Taniguchi, M. and Y. Kakizawa 2000. Asymptotic theory of statistical inference for time series. Springer Series in Statistics.