Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:58:11.412Z Has data issue: false hasContentIssue false
  • English
  • Français

Mirror-Image and Invariant Distributions in ARMA Models

Published online by Cambridge University Press:  18 October 2010

Jonathan D. Cryer ,
John C. Nankervis  and
N.E. Savin
Jonathan D. Cryer
Affiliation:
University of Iowa
John C. Nankervis
Affiliation:
City of London Polytechnic
N.E. Savin
Affiliation:
University of Iowa
Get access Rights & Permissions [Opens in a new window]

Abstract

The finite sample distributions of estimators and test statistics in ARMA time series models are generally unknown. For typical sample sizes, the approximations provided by asymptotic distributions are often unsatisfactory. Hence simulation or numerical integration methods are used to investigate the distributions. In practice only a limited part of the parameter space is examined using these methods. Thus any results which allow us to infer properties from one portion of the parameter space to another or to establish symmetry are most welcome.

For the ARMA model estimated with no intercept term, we show that the least-squares and maximum likelihood estimators have mirror-image invariant or symmetric distributions. The F, t, likelihood ratio, Wald, and Lagrange multiplier statistics are also shown to have distributions with certain mirror-image invariant or symmetry properties. The analysis is extended to misspecified models as well as to ARMA spectral densities.

These properties would have been helpful in either simplifying or extending much earlier work in this area.

Type
Articles
Information
Econometric Theory , Volume 5 , Issue 1 , April 1989 , pp. 36 - 52
Copyright
Copyright © Cambridge University Press 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Ahtola, J. & Tiao, G.C.. Distributions of least squares estimators of autoregressive parameters for a process with complex roots on the unit circle. Journal of Time Series Analysis 8 (1987): 1–14.CrossRefGoogle Scholar
2.Aitchison, J. & Silvey, S.D.. Maximum likelihood estimation of parameters subject to restraints. Annals of Mathematical Statistics 29 (1958): 813–828.CrossRefGoogle Scholar
3.Andrews, D.W.K.A note on the unbiasedness of feasible GLS, quasi-maximum likelihood, robust, adaptive and spectral estimators of the linear model. Econometrica 54 (1986): 687–698.CrossRefGoogle Scholar
4.Ansley, C.F. & Newbold, P.. Finite sample properties of estimators for ARMA models. Journal of Econometrics 13 (1980): 159–183.CrossRefGoogle Scholar
5.Chesher, A. & Peters, S.. Symmetry, regression design and sampling distributions. Discussion Paper No. 88/202, University of Bristol, 14 March 1988.Google Scholar
6.Cryer, J.D. & Ledolter, J.. Small-sample properties of the maximum likelihood estimator in the first-order moving average model. Biometrika 68 (1981): 691–694.Google Scholar
7.Dent, W. & Min, A.. A Monte Carlo study of autoregressive integrated moving average processes. Journal of Econometrics 7 (1978): 23–55.CrossRefGoogle Scholar
8.Dickey, D.A. Estimation and hypothesis testing for nonstationary time series. Unpublished Ph.D. dissertation, Iowa State University, Ames, Iowa, 1976.Google Scholar
9.Dickey, D.A., Hasza, D.P., & Fuller, W.A.. Testing for unit roots in seasonal time series. Journal of the American Statistical Association 79 (1984): 355–367.CrossRefGoogle Scholar
10.Engle, R.F. Wald, likelihood ratio, and Lagrange multiplier tests in econometrics. In Griliches, Z. and Intriligator, M.D. (eds.), Handbook of Econometrics, Vol. II. New York: Elsevier Science Publishers, 1984.Google Scholar
11.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
12.Fuller, Wayne A.Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
13.Osborn, D.R.On the criterion functions for the estimation of moving average processes. Journal of the American Statistical Association 77 (1982): 388–392.CrossRefGoogle Scholar
14.Phillips, P.C.B.Approximations to some finite sample distributions associated with a first-order stochastic difference equation. Econometrica 45 (1977): 463–486.CrossRefGoogle Scholar
15.Pollock, D.S.G. & Pitta, E.. The misspecification of dynamic regression models. Working paper, Department of Economics, Queen Mary College, University of London, 1987.Google Scholar
16.Sawa, T.The exact moments of the least squares estimator for the autoregressive model. Journal of Econometrics 8 (1978): 159–172.CrossRefGoogle Scholar