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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
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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

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