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The exact Gaussian likelihood estimation of time-dependent VARMA models

Author

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  • Alj, Abdelkamel
  • Jónasson, Kristján
  • Mélard, Guy

Abstract

An algorithm for the evaluation of the exact Gaussian likelihood of an r-dimensional vector autoregressive-moving average (VARMA) process of order (p, q), with time-dependent coefficients, including a time dependent innovation covariance matrix, is proposed. The elements of the matrices of coefficients and those of the innovation covariance matrix are deterministic functions of time and assumed to depend on a finite number of parameters. These parameters are estimated by maximizing the Gaussian likelihood function. The advantages of that approach is that the Gaussian likelihood function can be computed exactly and efficiently. The algorithm is based on the Cholesky decomposition method for block-band matrices. It is shown that the number of operations as a function of p, q and n, the size of the series, is barely doubled with respect to a VARMA model with constant coefficients. A detailed description of the algorithm followed by a data example is provided.

Suggested Citation

  • Alj, Abdelkamel & Jónasson, Kristján & Mélard, Guy, 2016. "The exact Gaussian likelihood estimation of time-dependent VARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 633-644.
    • Handle: RePEc:eee:csdana:v:100:y:2016:i:c:p:633-644
    DOI: 10.1016/j.csda.2014.07.006
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    References listed on IDEAS

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    1. Melard, Guy & Roy, Roch & Saidi, Abdessamad, 2006. "Exact maximum likelihood estimation of structured or unit root multivariate time series models," Computational Statistics & Data Analysis, Elsevier, vol. 50(11), pages 2958-2986, July.
    2. Rajae Azrak & Guy Melard, 1998. "The exact quasi-likelihood of time dependent ARMA models," ULB Institutional Repository 2013/13740, ULB -- Universite Libre de Bruxelles.
    3. Lanne, Markku & Saikkonen, Pentti, 2013. "Noncausal Vector Autoregression," Econometric Theory, Cambridge University Press, vol. 29(3), pages 447-481, June.
    4. Marc Hallin, 1986. "Nonstationary q-dependent processes and time-varying moving average models: invertibility properties and the forecasting problem," ULB Institutional Repository 2013/2005, ULB -- Universite Libre de Bruxelles.
    5. Rajae Azrak & Guy Mélard, 2006. "Asymptotic Properties of Quasi-Maximum Likelihood Estimators for ARMA Models with Time-Dependent Coefficients," Statistical Inference for Stochastic Processes, Springer, vol. 9(3), pages 279-330, October.
    6. Hallin, Marc, 1978. "Mixed autoregressive-moving average multivariate processes with time-dependent coefficients," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 567-572, December.
    7. B. L. Shea, 1989. "The Exact Likelihood of a Vector Autoregressive Moving Average Model," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 38(1), pages 161-184, March.
    8. Hindrayanto, Irma & Koopman, Siem Jan & Ooms, Marius, 2010. "Exact maximum likelihood estimation for non-stationary periodic time series models," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2641-2654, November.
    9. Rajae Azrak & Guy Melard, 2006. "Asymptotic properties of quasi-maximum likelihood estimators for ARMA models with time-dependent coefficients," ULB Institutional Repository 2013/13758, ULB -- Universite Libre de Bruxelles.
    10. Triantafyllopoulos, K. & Nason, G.P., 2007. "A Bayesian analysis of moving average processes with time-varying parameters," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 1025-1046, October.
    11. Chen, Yen-Hung & Hsu, Nan-Jung, 2014. "A frequency domain test for detecting nonstationary time series," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 179-189.
    12. José Alberto Mauricio, 2002. "An algorithm for the exact likelihood of a stationary vector autoregressive‐moving average model," Journal of Time Series Analysis, Wiley Blackwell, vol. 23(4), pages 473-486, July.
    13. Mauricio, Jose Alberto, 2006. "Exact maximum likelihood estimation of partially nonstationary vector ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 50(12), pages 3644-3662, August.
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    Cited by:

    1. Rajae Azrak & Guy Melard, 2017. "Autoregressive Models with Time-dependent Coefficients. A comparison between Several Approaches," Working Papers ECARES ECARES 2017-48, ULB -- Universite Libre de Bruxelles.
    2. Marie-Christine Duker & David S. Matteson & Ruey S. Tsay & Ines Wilms, 2024. "Vector AutoRegressive Moving Average Models: A Review," Papers 2406.19702, arXiv.org.
    3. Abdelkamel Alj & Rajae Azrak & Christophe Ley & Guy Mélard, 2017. "Asymptotic Properties of QML Estimators for VARMA Models with Time-dependent Coefficients," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 617-635, September.

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