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Model specification and selection for multivariate time series

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  • Bhansali, Rajendra J.

Abstract

Three major difficulties are identified with an established echelon form approach (see Hannan (1987)) to specifying a Vector Autoregressive Moving Average, VARMA, model for an observed time series. A family of state space representations, valid for each integer, h, is introduced, and collectively referred to as multistep state space representations. This family includes as its special case, with h=0, a state space representation introduced earlier by Akaike (1974), and, with h=1, that introduced by Cooper and Wood (1982). Appropriate generalizations of the notions of minimality, McMillan degree, left matrix fraction description and Kronecker indices, as applicable individually to each member of this family, are presented. The reverse echelon form and state space representation corresponding to the Kronecker indices for each h are derived, and the former illustrated with three examples of standard VARMA processes. The question of how the presence of zero constraints on the coefficients of a reverse echelon form may be detected solely from an inspection of the Kronecker indices is examined. A canonical correlation procedure proposed originally by Akaike (1976) for h=0 is considered for estimating the Kronecker indices with each h. The efficacy of the estimation procedure is investigated by a simulation study. A procedure is suggested for implementing the new approach introduced in this paper with an observed time series, and three different applications of this approach are outlined. This approach is also related to some of its alternatives, including the Kronecker invariants of Poskitt (1992) and the scalar component approach of Tiao and Tsay (1989).

Suggested Citation

  • Bhansali, Rajendra J., 2020. "Model specification and selection for multivariate time series," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
  • Handle: RePEc:eee:jmvana:v:175:y:2020:i:c:s0047259x18303762
    DOI: 10.1016/j.jmva.2019.104539
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    References listed on IDEAS

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    1. Ruey S. Tsay, 1989. "Identifying Multivariate Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(4), pages 357-372, July.
    2. Christopher A. Sims & Tao Zha, 1999. "Error Bands for Impulse Responses," Econometrica, Econometric Society, vol. 67(5), pages 1113-1156, September.
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    5. Lutkepohl, Helmut & Poskitt, D S, 1996. "Specification of Echelon-Form VARMA Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(1), pages 69-79, January.
    6. D. M. Cooper & E. F. Wood, 1982. "Identifying Multivariate Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 3(3), pages 153-164, May.
    7. Tsay, Ruey S, 1989. "Parsimonious Parameterization of Vector Autoregressive Moving Average Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(3), pages 327-341, July.
    8. K. L. Vaninskii & A. M. Yaglom, 1990. "Stationary Processes With A Finite Number Of Non‐Zero Canonical Correlations Between Future And Past," Journal of Time Series Analysis, Wiley Blackwell, vol. 11(4), pages 361-375, July.
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    Cited by:

    1. Marie-Christine Duker & David S. Matteson & Ruey S. Tsay & Ines Wilms, 2024. "Vector AutoRegressive Moving Average Models: A Review," Papers 2406.19702, arXiv.org.

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