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Estimation of vector ARMAX models

Author

Listed:
  • Hannan, E. J.
  • Dunsmuir, W. T. M.
  • Deistler, M.

Abstract

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.

Suggested Citation

  • Hannan, E. J. & Dunsmuir, W. T. M. & Deistler, M., 1980. "Estimation of vector ARMAX models," Journal of Multivariate Analysis, Elsevier, vol. 10(3), pages 275-295, September.
    • Handle: RePEc:eee:jmvana:v:10:y:1980:i:3:p:275-295
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    Cited by:

    1. Cavicchioli, Maddalena, 2017. "Asymptotic Fisher information matrix of Markov switching VARMA models," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 124-135.
    2. Mélard, Guy, 2022. "An indirect proof for the asymptotic properties of VARMA model estimators," Econometrics and Statistics, Elsevier, vol. 21(C), pages 96-111.
    3. Peter Robinson & J. Vidal Sanz Vidal Sanz, 2003. "Modified whittle estimation of multilateral spatial models," CeMMAP working papers 18/03, Institute for Fiscal Studies.
    4. Peter Robinson & J. Vidal Sanz Vidal Sanz, 2003. "Modified whittle estimation of multilateral spatial models," CeMMAP working papers CWP18/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Findley, David F. & Potscher, Benedikt M. & Wei, Ching-Zong, 2004. "Modeling of time series arrays by multistep prediction or likelihood methods," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 151-187.
    6. Robinson, P.M. & Vidal Sanz, J., 2006. "Modified Whittle estimation of multilateral models on a lattice," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1090-1120, May.
    7. Marie-Christine Duker & David S. Matteson & Ruey S. Tsay & Ines Wilms, 2024. "Vector AutoRegressive Moving Average Models: A Review," Papers 2406.19702, arXiv.org.
    8. Yong Bao, 2018. "The asymptotic covariance matrix of the QMLE in ARMA models," Econometric Reviews, Taylor & Francis Journals, vol. 37(4), pages 309-324, April.
    9. M. Deistler & B. Pötscher & J. Schrader, 1984. "The uniqueness of the transfer function of linear systems from input-output observations," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 31(1), pages 157-181, December.
    10. Abdelhamid Ouakasse & Guy Mélard, 2017. "A New Recursive Estimation Method for Single Input Single Output Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 38(3), pages 417-457, May.
    11. Holger Fink & Andreas Fuest & Henry Port, 2018. "The Impact of Sovereign Yield Curve Differentials on Value-at-Risk Forecasts for Foreign Exchange Rates," Risks, MDPI, vol. 6(3), pages 1-19, August.
    12. Boubacar Mainassara, Y. & Francq, C., 2011. "Estimating structural VARMA models with uncorrelated but non-independent error terms," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 496-505, March.
    13. Zheng, Tingguo & Chen, Rong, 2017. "Dirichlet ARMA models for compositional time series," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 31-46.
    14. Andrés Martínez & Alfonso Salafranca & Ana E. Sipols & Clara Simon Blas & Daniel Hengel, 2024. "Distributed lags using elastic-net regularization for market response models: focus on predictive and explanatory capacity," Journal of Marketing Analytics, Palgrave Macmillan, vol. 12(2), pages 417-435, June.
    15. Virtue U. Ekhosuehi & David E. Omoregie, 2021. "Inspecting debt servicing mechanism in Nigeria using ARMAX model of the Koyck-kind," Operations Research and Decisions, Wroclaw University of Science Technology, Faculty of Management, vol. 31, pages 5-20.
    16. Virtue U. Ekhosuehi & David E. Omoregie, 2021. "Inspecting debt servicing mechanism in Nigeria using ARMAX model of the Koyck-kind," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 31(1), pages 5-20.
    17. Guy Melard, 2020. "An Indirect Proof for the Asymptotic Properties of VARMA Model Estimators," Working Papers ECARES 2020-10, ULB -- Universite Libre de Bruxelles.
    18. B. Pötscher, 1985. "The behaviour of the Lagrangian multiplier test in testing the orders of an ARMA-model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 32(1), pages 129-150, December.
    19. Pierre Duchesne, 2005. "On the asymptotic distribution of residual autocovariances in VARX models with applications," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(2), pages 449-473, December.

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