Copula Markov Models for Count Series with Excess Zeros

Abstract

Count time series data are observed in several applied disciplines such as environmental science, biostatistics, economics, public health, and finance. In some cases, a specific count, say zero, may occur more often than usual. Additionally, serial dependence might be found among these counts if they are recorded over time. Overlooking the frequent occurrence of zeros and the serial dependence could lead to false inference. In this chapter, Markov zero-inflated count time series models based on a joint distribution of consecutive observations are proposed. The joint distribution function of the consecutive observations is constructed through copula functions. First- or second-order Markov chains are considered with the univariate margins of zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), or zero-inflated Conway–Maxwell–Poisson (ZICMP) distributions. Under the Markov models, bivariate copula functions such as the bivariate Gaussian, Frank, and Gumbel are chosen to construct a bivariate distribution of two consecutive observations. Moreover, the trivariate Gaussian and max-infinitely divisible copula functions are considered to build the joint distribution of three consecutive observations. Likelihood-based inference is performed and asymptotic properties are studied. The proposed class of models is applied to arson counts example, which suggests that the proposed models are superior to some of the models in the literature.