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

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.