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Maximum Likelihood Estimators In The Multivariate Autoregressive Moving‐Average Model From A Generalized Least Squares Viewpoint

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  • Gregory C. Reinsel
  • Sabyasachi Basu
  • Sook Fwe Yap

Abstract

. Explicit expressions are derived for the gradient vector and (approximate) Hessian matrix of the log likelihood function for the multivariate autoregressive moving‐average (ARMA) model. Based on these expressions an explicit description of the Gauss‐Newton iterative procedure to obtain maximum likelihood (ML) estimates of the parameters in the multivariate ARMA model is presented. The resulting computational procedure has the form of a generalized least squares (GLS) estimation involving lagged values of the observed vector series and of the residual series as independent variables. This direct form of the estimator is found to be appealing and useful in understanding and interpreting the ML estimation procedure from a regression point of view, and in comparing the ML procedure with other ‘linear’ estimation procedures that have recently been presented. Simulation results are also presented for a univariate and a multivariate ARMA model to illustrate the ML‐GLS estimation procedure and to compare it with other linear estimation procedures.

Suggested Citation

  • Gregory C. Reinsel & Sabyasachi Basu & Sook Fwe Yap, 1992. "Maximum Likelihood Estimators In The Multivariate Autoregressive Moving‐Average Model From A Generalized Least Squares Viewpoint," Journal of Time Series Analysis, Wiley Blackwell, vol. 13(2), pages 133-145, March.
    • Handle: RePEc:bla:jtsera:v:13:y:1992:i:2:p:133-145
    DOI: 10.1111/j.1467-9892.1992.tb00099.x
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    Cited by:

    1. Jean-Marie Dufour & Tarek Jouini, 2011. "Asymptotic Distributions for Some Quasi-Efficient Estimators in Echelon VARMA Models," CIRANO Working Papers 2011s-25, CIRANO.
    2. Dufour, Jean-Marie & Jouini, Tarek, 2014. "Asymptotic distributions for quasi-efficient estimators in echelon VARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 73(C), pages 69-86.
    3. Ursu, Eugen & Duchesne, Pierre, 2009. "On multiplicative seasonal modelling for vector time series," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 2045-2052, October.
    4. Alfredo García-Hiernaux & José Casals & Miguel Jerez, 2012. "Estimating the system order by subspace methods," Computational Statistics, Springer, vol. 27(3), pages 411-425, September.
    5. Boularouk, Y. & Djeddour, K., 2015. "New approximation for ARMA parameters estimate," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 118(C), pages 116-122.
    6. Jan G. Gooijer, 2021. "Asymmetric vector moving average models: estimation and testing," Computational Statistics, Springer, vol. 36(2), pages 1437-1460, June.
    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. Dong Wan Shin & Sahadeb Sarkar, 1995. "Estimation Of The Multivariate Autoregressive Moving Average Having Parameter Restrictions And An Application To Rotational Sampling," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(4), pages 431-444, July.
    9. D. S. Poskitt & M. O. Salau, 1995. "On The Relationship Between Generalized Least Squares And Gaussian Estimation Of Vector Arma Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(6), pages 617-645, November.

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