Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-l69ms Total loading time: 0.371 Render date: 2022-08-15T02:07:03.609Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

The behaviour of the likelihood function for ARMA models

Published online by Cambridge University Press:  01 July 2016

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.

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.

Type
Research Article
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, 8394.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, 11871208.CrossRefGoogle Scholar
Deistler, M., Dunsmuir, W. and Hannan, E. J. (1978) Vector linear time series models: corrections and extensions. Adv. Appl. Prob. 10, 360372.CrossRefGoogle Scholar
Deistler, M. and Hannan, E. J. (1981) Some properties of the parametrization of ARMA systems with unknown order. J. Multivariate Anal. 11, 474484.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, 4360.Google Scholar
Dunsmuir, W. and Hannan, E. J. (1976) Vector linear time series models. Adv. Appl. Prob. 8, 339364.CrossRefGoogle Scholar
Franklin, S. P. (1965) Spaces in which sequences suffice. Fundamenta Math. 57, 107115.Google Scholar
Rozanov, Yu. V. (1967) Stationary Random Processes. Holden-Day, San Francisco.Google Scholar
1
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The behaviour of the likelihood function for ARMA models
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

The behaviour of the likelihood function for ARMA models
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

The behaviour of the likelihood function for ARMA models
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *