1. Introduction

Count time series with excess zeros are commonly encountered in a variety of research fields. In principle, both zero-inflation and autocorrelation may be present in such series. Failing to adequately accommodate temporal dynamics and a high frequency of zeros can lead to

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

incorrect inferential conclusions. Developing a general modeling framework that accounts for these characteristics poses a daunting challenge.

In characterizing data comprised of counts with excess zeroes, two types of models are commonplace: a model that assumes a Poisson mixture distribution, and a model that assumes a binomial mixture distribution. A considerable literature exists for regression models based on the zero-inflated Poisson (ZIP) distribution to deal with count data that are independently distributed [1]. Many researchers have extended the classical ZIP model to analyze repeated measures data by incorporating independent random effects, as these can account for within-subject correlation and between-subject heterogeneity [2, 3]. To deal with count time series with excess zeros, some researchers have proposed parameter-driven ZIP models that accommodate the temporal dynamics by incorporating correlated random effects, which can be represented by a latent autoregressive process [4, 5]. However, for data arising from a binomial mixture distribution, a survey of the literature for analogous frameworks reflects an absence of work dealing with binomial time series with excess zeros. To handle such data, we propose two general classes of models: a class of observation-driven ZIB (ODZIB) models, and a class of parameter-driven ZIB (PDZIB) models. The inspiration for the two proposed modeling frameworks arises from the work of Hall [6], Yau et al. [4], and Yang et al. [5, 7].

Depending on how the temporal correlation is conceptualized, count time series models can be classified as either observation-driven or parameter-driven [8]. For the former, serial correlation is characterized by specifying that the conditional mean of the current response depends explicitly on its past values [9–14]. For the latter, such correlation is characterized through an unobservable underlying process [15–19]. In this chapter, we employ the partial likelihood framework to formulate the ODZIB model, as this largely simplifies parameter estimation with negligible loss of information. The ODZIB model can be viewed as an extension of the observation-driven binomial model [20]. Such a model is often fit using standard statistical software available for classical ZIB regression models. For the PDZIB model, we employ a state-space approach, as this framework allows for the investigation of the underlying latent processes that govern the temporal correlation and zero inflation. Due to the non-Gaussian distribution of the count response, and the nonlinear nature of modeling its conditional mean, traditional state-space methods using the Kalman filter and the Kalman smoother are not available for parameter estimation. We thereby adopt a Monte Carlo Expectation Maximization (MCEM) algorithm based on the particle filter [21] and the particle smoother [22].

The remainder of the chapter is organized as follows. In Section 2, we briefly introduce a class of observation-driven models for a zero-inflated count time series that arises from a binomial mixture. Section 3 proposes a class of parameter-driven models in the statespace framework, and presents the MCEM algorithm devised to fit such models. A comprehensive simulation study is provided in Section 4. In Section 5, we illustrate the proposed methodology through a practical application. Section 6 concludes with a brief discussion.
