Mirror-Image and Invariant Distributions in Arma Models

JONATHAND. CRYER
University of lo wa
JOHNC. NANKERVIS
City of London Polytechnic
N.E. SAVIN
University of lo wa
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 i 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.