Multivariate Time Series Model with Hierarchical Structure for Over-Dispersed Discrete Outcomes
Nobuhiko Terui1, Masataka Ban2, Toshihiko Maki1
1Graduate School of Ecpnomics and Management, Tohoku University, Sendai, Miyagi, Japan; 2Faculty of Business Administration, Mejiro University, Tokyo, Shinjuku, Japan

In this study, we propose a multivariate time series model for over-dispersed discrete data. We extend the model for market structure based on sales count dynamics by Terui, Ban, and Maki (2010) to accommodate the over-dispersion problem. We first discuss a microstructure of sales count data to show that over-dispersion is inherent in the modeling of market structure based on sales count dynamics. The model is build on the likelihoods generated by decomposing sales count response variables according to products' competitiveness and conditioning on their sum of variables, and it builds them up to higher levels. State space priors are applied to the likelihood function to develop dynamic generalized linear models for discrete outcomes.

The Gamma compound Poisson variable for product sales counts and Dirichlet compound multinomial variables for their shares are connected in a hierarchical fashion, represented as a tree structure for a market definition. Instead of the direct use of the density function of compound distributions, we introduce two levels of a hierarchical model to deal with mixtures. We first define the original model by Terui et al. (2010) when the mixing parameters are given at the upper level, and then the lower level model describes the dynamics of mixing parameters when the original parameters are provided as observed data. The proposed method has the advantage of linking the parameters with managerially meaningful covariate and trend terms, as the original parameters are not integrated out but maintained in the model.

As an application using point of sales (POS) time series in a store, we compare the proposed model with the one without over-dispersion by using several model selection criteria, including in-sample fit, out-of-sample forecasting errors, and information criteria to show that the proposed model improves the results.

Keywords: Compound Poisson; Discrete Outcomes; Dynamic Generalized Linear Model; Over-dispersion

Biography: Nobuhiko Terui is a professor at Graduate School of Economics and Management, Tohoku University (Japan). He has published in journals such as Marketing Science, Quantitative Marketing and Economics, Journal of Interactive Marketing, Geographical Analysis, Econometric Theory, Journal of Time Series Analysis, and International Journal of Forecasting. His current research interests are in the modeling of nonlinear response of heterogeneous consumers, dynamic ecoometric and marketing models and related decision problems.