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Mirror-Image and Invariant Distributions in ARMA Models

Author

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  • Cryer, Jonathan D.
  • Nankervis, John C.
  • Savin, N.E.

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.

Suggested Citation

  • Cryer, Jonathan D. & Nankervis, John C. & Savin, N.E., 1989. "Mirror-Image and Invariant Distributions in ARMA Models," Econometric Theory, Cambridge University Press, vol. 5(1), pages 36-52, April.
    • Handle: RePEc:cup:etheor:v:5:y:1989:i:01:p:36-52_01
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    Cited by:

    1. Neil Kellard & Denise Osborn & Jerry Coakley & Nathan E. (Gene) Savin, 2015. "Papers with John," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(5), pages 663-671, September.
    2. Mukhtar Ali, 2002. "Distribution Of The Least Squares Estimator In A First-Order Autoregressive Model," Econometric Reviews, Taylor & Francis Journals, vol. 21(1), pages 89-119.
    3. Magnus, J.R. & Pesaran, B., 1990. "Forecasting, misspecification and unit roots : The case of Ar(1) versus ARMA(1,1)," Discussion Paper 1990-2, Tilburg University, Center for Economic Research.
    4. Haldrup, Niels, 1996. "Mirror image distributions and the Dickey-Fuller regression with a maintained trend," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 301-312.
    5. Mukhtar M. Ali, 1996. "Distribution of the Least Squares Estimator in a First-Order Autoregressive Model," Econometrics 9610004, University Library of Munich, Germany.
    6. Mukhtar M. Ali, 1996. "Exact Distribution of the Least Squares Estimator in a First- Order Autoregressive Model," Econometrics 9604001, University Library of Munich, Germany.

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