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Multivariate Time Series Model with Hierarchical Structure for Over-dispersed Discrete Outcomes

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  • Nobuhiko Terui
  • Masataka Ban
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    Abstract

    In this paper, we propose a multivariate time series model for over-dispersed discrete data to explore the market structure based on sales count dynamics. We first discuss the microstructure to show that over-dispersion is inherent in the modeling of market structure based on sales count data. The model is built on the likelihood function induced by decomposing sales count response variables according to products' competitiveness and conditioning on their sum of variables, and it augments them to higher levels by using Poisson-Multinomial relationship in a hierarchical way, represented as a tree structure for the market definition. State space priors are applied to the structured likelihood to develop dynamic generalized linear models for discrete outcomes. For over-dispersion problem, Gamma compound Poisson variables for product sales counts and Dirichlet compound multinomial variables for their shares are connected in a hierarchical fashion. Instead of the density function of compound distributions, we propose a data augmentation approach for more efficient posterior computations in terms of the generated augmented variables particularly for generating forecasts and predictive density. We present the empirical application using weekly product sales time series in a store to compare the proposed models accommodating over-dispersion with alternative no over-dispersed models by several model selection criteria, including in-sample fit, out-of-sample forecasting errors, and information criterion. The empirical results show that the proposed modeling works well for the over-dispersed models based on compound Poisson variables and they provide improved results than models with no consideration of over-dispersion.

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    File URL: http://hdl.handle.net/10097/56551
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    File URL: http://ir.library.tohoku.ac.jp/re/bitstream/10097/56551/1/tmarg113.pdf
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    Bibliographic Info

    Paper provided by Graduate School of Economics and Management, Tohoku University in its series TMARG Discussion Papers with number 113.

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    Length: 32 pages
    Date of creation: Jan 2013
    Date of revision: Aug 2013
    Handle: RePEc:toh:tmarga:113

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    1. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika van der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639.
    2. Park, C Whan & Lessig, V Parker, 1977. " Students and Housewives: Differences in Susceptibility to Reference Group Influence," Journal of Consumer Research, University of Chicago Press, vol. 4(2), pages 102-10, Se.
    3. Harvey, Andrew C & Fernandes, C, 1989. "Time Series Models for Count or Qualitative Observations: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(4), pages 422, October.
    4. Nobuhiko Terui & Masataka Ban & Toshihiko Maki, 2010. "Finding market structure by sales count dynamics—Multivariate structural time series models with hierarchical structure for count data—," Annals of the Institute of Statistical Mathematics, Springer, vol. 62(1), pages 91-107, February.
    5. Bearden, William O & Etzel, Michael J, 1982. " Reference Group Influence on Product and Brand Purchase Decisions," Journal of Consumer Research, University of Chicago Press, vol. 9(2), pages 183-94, September.
    6. Harvey, Andrew C & Fernandes, C, 1989. "Time Series Models for Count or Qualitative Observations," Journal of Business & Economic Statistics, American Statistical Association, vol. 7(4), pages 407-17, October.
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