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6 An Introduction to Discrete-Valued Time Series
so on), they are at the core of several other models, such as those designed
for non-stationary processes or processes with a long memory. In particular,
the related generalized autoregressive conditional heteroskedasticity (GARCH)
model, with its potential for application to financial time series, has become
very popular in recent decades; see Appendix B.4.1 for further details. A com-
prehensive survey of models within the “ARMA alphabet soup” is provided by
Holan et al. (2010). A brief summary and references to introductory textbooks
in this field can be found in Appendix B.
In view of their important role in the modeling of real-valued time series, it
is quite natural to adapt such ARMA approaches to the case of discrete-valued
time series. is has been done both for the case of count data and for the
categorical case, and such ARMA-like models serve as the starting point of our
discussion in both Parts I and II. In fact, Part I starts with an integer-valued
counterpart to the specific case of an AR(1) model, the so-called INAR(1)
model, because this simple yet useful model allows us to introduce some
general principles for fitting models to a count time series and for checking the
model adequacy. Together with the discussion of forecasting count processes,
also provided in Chapter 2, we are thus able to transfer the Box–Jenkins
method to the count data case. In the context of introducing the INAR(1)
model, the typical features of count data are also discussed, and it will become
clear why integer-valued counterparts to the ARMA model are required; in
other words, why we cannot just use the conventional ARMA recursion (1.1)
for the modeling of time series of counts.
ARMA-like models using so-called “thinning operations”, commonly
referred to as IN ARMA models, are presented in Chapter 3. e INAR(1)
model also belongs to this class, while Chapter 4 deals with a modification of
the ARMA approach related to regression models; the latter are often termed
INGARCH models, although this is a somewhat misleading name. More gen-
eral regression models for count time series, and also hidden-Markov models,
are discussed in Chapter 5. As this book is intended to be an introductory
textbook on discrete-valued time series, its main focus is on simple models,
which nonetheless are quite powerful in real applications. However, references
to more elaborate models are also included for further reading.
In Part II of this book, we follow a similar path and first lay the foundations
for analyzing categorical time series by introducing appropriate tools, for
example for their visualization or the assessment of serial dependence; see
Chapter 6. en we consider diverse models for categorical time series
in Chapter 7, namely types of Markov models, a kind of discrete ARMA
model, and again regression and hidden-Markov models, but now tailored to
categorical outcomes.
So for both count and categorical time series, a variety of models are pre-
paredheretobeusedinpractice.Onceamodelhasbeenfoundtobeadequate
for the given time series data, it can be applied to forecasting future values.