An Introduction to Discrete-Valued Time Series

Chapter 4INGARCH Models for Count Time Series

The models discussed in Chapter 3 used types of thinning operations to transfer the ARMA model to the count data case. Another popular approach for modeling such stationary processes of counts are the INGARCH models, the definition of which is related to linear regression models (also see Section 5.1). Despite their controversial name, these models are particularly attractive for overdispersed counts with an ARMA-like autocorrelation structure. Results concerning the basic model with a conditional Poisson distribution are presented, but generalizations with, for example, a binomial or negative binomial conditional distribution are also considered.

4.1 Poisson Autoregression

Due to the multiplication problem discussed in Section 2.1 , the ARMA models of Definition B.3.5 are not applicable to the count data case. The models presented in Chapters 2 and 3 circumvented this problem by replacing the multiplications with a type of thinning operation, that ensures that these modified model recursions always produce integer values. The INGARCH models to be presented in this chapter use another solution to the multiplication problem: a linear regression of the conditional means . To construct an AR(1)-like model, for instance, the AR(1) recursion is transferred to the level of conditional means as . Then the count at time is generated by ...