An Introduction to Discrete-Valued Time Series

Chapter 3Further Thinning-based Models for Count Time Series

After having introduced important tasks and approaches for analyzing count time series, we shall now return to the question of how to model the underlying process. The main characteristic of the INAR(1) model in Section 2.2 is the use of the binomial thinning operator as a substitute for the multiplication, to be able to transfer the AR(1) recursion to the count data case. In Section 3.1, we shall see that this approach can also be used to define higher-order ARMA-like models. Furthermore, different types of thinning operation have been developed for such models; see Section 3.2. Finally, various thinning-based models to deal with count time series with a finite range (Section 3.3) and multivariate count time series (Section 3.4) are also available in the literature.

3.1 Higher-order INARMA Models

The INAR(1) model, as introduced in Section 2.2, was developed as an integer-valued counterpart to the conventional AR(1) model, mainly by replacing the multiplication in the AR(1) recursion with the binomial thinning operator. This idea is not limited to the first-order autoregressive case, but can also be used to mimic higher-order ARMA models; see Appendix B. The resulting models are then referred to as INARMA models.

Example 3.1.1 (INMA(1) model)

An integer-valued counterpart to the MA(1) model was proposed by Al-Osh ...