Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-11T14:54:45.532Z Has data issue: false hasContentIssue false
  • English
  • Français

The behaviour of the likelihood function for ARMA models

Published online by Cambridge University Press:  01 July 2016

M. Deistler  and
B. M. Pötscher
M. Deistler*
Affiliation:
University of Technology, Vienna
B. M. Pötscher*
Affiliation:
University of Technology, Vienna
*
∗Postal address: Institute of Econometrics and Operations Research, University of Technology, A-1040 Vienna, Argentinierstr. 8, Austria.
∗Postal address: Institute of Econometrics and Operations Research, University of Technology, A-1040 Vienna, Argentinierstr. 8, Austria.
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper deals with some properties of the (Gaussian) likelihood function for multivariable ARMA models. Its behaviour at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed. We have not been able to show in general the existence of the maximum over the usual parameter spaces. However, the maximum always exists over a suitably enlarged parameter space (given that the data are non-degenerate), which includes parameters corresponding to processes with discrete spectral components.

Keywords

EXISTENCE OF A MAXIMUMMAXIMUM LIKELIHOOD ESTIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 4 , December 1984 , pp. 843 - 866
Copyright
Copyright © Applied Probability Trust 1984 

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.)

Footnotes

Support by ‘Fonds zur Förderung der wissenschaftlichen Forschung', project No. 4393, is gratefully acknowledged.

References

Anderson, T. W. and Mentz, R. P. (1980) On the structure of the likelihood function of autoregressive and moving average models. J. Time Series Analysis 1, 83–94.CrossRefGoogle Scholar
Bourbaki, N. (1966) General Topology. Hermann, Paris.Google Scholar
Deistler, M. (1983) The properties of the parametrization of ARMAX systems and their relevance for structural estimation and dynamic specification. Econometrica 51, 1187–1208.CrossRefGoogle Scholar
Deistler, M., Dunsmuir, W. and Hannan, E. J. (1978) Vector linear time series models: corrections and extensions. Adv. Appl. Prob. 10, 360–372.Google Scholar
Deistler, M. and Hannan, E. J. (1981) Some properties of the parametrization of ARMA systems with unknown order. J. Multivariate Anal. 11, 474–484.CrossRefGoogle Scholar
Deistler, M., Ploberger, W. and Pötscher, B. M. (1982) Identifiability and inference in ARMA systems. In Time Series Analysis: Theory and Practice 2, ed. Anderson, O. D., North-Holland, Amsterdam, 43–60.Google Scholar
Dunsmuir, W. and Hannan, E. J. (1976) Vector linear time series models. Adv. Appl. Prob. 8, 339–364.Google Scholar
Franklin, S. P. (1965) Spaces in which sequences suffice. Fundamenta Math. 57, 107–115.Google Scholar
Rozanov, Yu. V. (1967) Stationary Random Processes. Holden-Day, San Francisco.Google Scholar