Preface
Statisticians continually face new modeling challenges as data collection becomes more
widespread and the type of data more varied. Modeling techniques are constantly expand-
ing to accommodate a wider range of data structures. Traditionally, time series modeling
has been applied to data that are continuously valued. Unfortunately, many continuous
models do not adequately describe discrete data. The most common violation lies with the
case of counts. Examples include cases when the time series might contain daily counts
of individuals infected with a rare disease, the number of trades of a stock in a minute,
the yearly number of hurricanes that make landfall in Florida, etc. For these examples,
the counts tend to be small and may include zeros—discreteness of the observations can-
not be adequately modeled with a continuous distribution. During the past 30+ years,
much progress has been made for modeling a wide range of count time series. Having
reached a reasonable level of maturity, this seems an ideal time to assemble a handbook
on state-of-the-art methods for modeling time series of counts, including frequentist and
Bayesian approaches and methodology for discrete-valued spatio-temporal data and mul-
tivariate data. While the focus of this handbook is on time series of counts, some of the
techniques discussed can be extended to other types of discrete-valued time series, such as
binary-valued or categorical time series.
The main thrust of classical time series methodology considers modeling stationary time
series. This assumes that some form of preprocessing of the original time series, which might
include the application of transformations and various lters, has been applied to reduce
the series to a stationary one. Linear time series models, the mainstay of stationary time
series modeling, are motivated by the celebrated Wold’s decomposition. This decomposi-
tion says that every nondeterministic stationary time series admits a linear representation.
The autoregressive moving-average (ARMA) models are a special family of parsimonious
short memory models for linear stationary time series. Box and Jenkins’ groundbreaking
1970 text provided a paradigm for identifying, tting, and forecasting ARMA models that
were accessible to practitioners.
For stationary Gaussian processes, which are completely determined by their covariance
functions, ARMA models provide a useful description. For non-Gaussian series, ARMA
models can only be assured to capture an approximation of the covariance structure of the
process. Some dependence structures are invisible to the covariance function and hence
are suboptimally described in the ARMA paradigm. Nonlinear models are available in this
case and have received considerable attention since the 1980s.
Discrete-valued time series, and time series of counts in particular, present modeling
challenges not encountered for continuous-valued responses. First, representing a count
time series in an additive Wold-type form may not make sense. Second, the covariance
function, which is a measure of linear dependence, is not always a useful measure of
dependence for a time series of counts. As a result, one is led to consider nonlinear models.
Perhaps the most common and exible modeling formulation for count data is based on
a state-space formulation. In this setup, the counts are assumed to be independent and,
say, Poisson distributed, given a latent series λ
t
> 0attime t. Serial dependence and the
effect of possible covariates are incorporated into λ
t
. Borrowing from the generalized linear
models (GLM) literature, the log-link function is used so that log(λ
t
) is modeled as a linear
xi
xii Preface
regression model with correlated Gaussian errors. Zeger (1988) was an early proponent of
such models and applied them to a time series of monthly polio incidences in the United
States.
In the late 1970s, Jacobs and Lewis proposed mixing tactics now referred to as discrete
ARMA (DARMA) methods. A DARMA process can have any prescribed marginal distri-
bution while possessing the covariance structure of an ARMA series. Unfortunately, this
class of models generated series that remained constant in time for long runs and were sub-
sequently dismissed as unrealistic. In the 1980s, integer ARMA (INARMA) methods were
introduced by McKenzie. These methods used probabilistic thinning techniques to keep
the support set of the series integer valued and are still in use today. Covariance structures
produced by DARMA and INARMA models are not as general as ARMA; for example,
they are always nonnegative.
From a Bayesian perspective, discrete-valued time series have experienced tremendous
growth since the emergence of the dynamic generalized linear model in West et al. in the
mid-1980s. This work was followed by Fahrmeir et al. in the late 1980s and 1990s. In the
late 1990s, Waller et al., Wikle, and others proposed hierarchical spatio-temporal models
for discrete-valued data. In both time and space–time, this has become an area of signicant
research as data collection and sampling methods, including Markov chain Monte Carlo
and approximation algorithms, improve with technological and computational advances.
This handbook addresses a plethora of diverse topics on modeling discrete-valued time
series, and in particular time series of counts. The reader will nd both frequentist and
Bayesian methods that consider issues of model development, residual diagnostics, appli-
cations in business, changepoint analyses, binary series, etc. Theoretical, methodological
and practical issues are pursued.
This handbook is arranged as follows: Section I contains eight chapters that essentially
narrate history and some of the current methods for modeling and analysis of univariate
count series. Section II is a short interlude into diagnostics and applications, containing
only three chapters. Section III moves to binary and categorical time series. Section IV is
our guide to modern methods for discrete-valued spatio-temporal data. Section V con-
cludes with several chapters on multivariate and long-memory count series. Several topics
are notably absent, for example, chapters devoted to copula and empirical likelihood
methods. We hope that future editions of the handbook will remedy this shortcoming.
The list of contributors to this handbook is distinguished, and all chapters were solicited
from topical experts. Considerable effort was undertaken to make the chapters ow
together rather than be disjoint works. Taken as a whole, themes emerge. Many of the
models used in this handbook can be categorized as either parameter driven or observation
driven, a nomenclature that is originally due to Cox (1981). Essentially, a parameter-driven
model is one that has a state-space formulation, where the state variable is viewed as the
parameter. An observation-driven model expresses the current value of the time series as
a function of past observations and an independent noise term. It turns out that likelihood
methods are difcult to implement for the state-space models considered in this book since
the evaluation of the likelihood requires the computation of a large-dimensional integral.
Hence, for such models, one often turns to approximate likelihood methods, estimating
equations, simulation methods, or a fully Bayesian implementation for model tting. On
the other hand, for observation-driven models, it is often straightforward to use likelihood
methods and the theory has been well-developed in a large number of cases. There are sev-
eral chapters devoted to these topics in this handbook, which describe the advantages and
limitations of the various modeling approaches.
Preface xiii
The technical level of the individual chapters is modest and should be accessible to
masters level students with an elementary class in statistical time series. We have made an
effort to make the chapters accessible and nontechnical, keeping probabilistic technicalities
to a minimum.
Preparation of this handbook has been easeful toil on the editors. The timeliness of
chapter authors is especially appreciated, as is the cohesiveness of the editorial team. We are
also indebted to Rob Calver of Chapman & Hall for unwavering support during the prepa-
ration of this handbook. The comments and critiques of numerous unnamed reviewers
supported our efforts.
References
Cox, D. R. (1981). Statistical analysis of time series: Some recent developments. Scandinavian Journal
of Statistics, 8:93–115.
Zeger, S. L. (1988). A regression model for time series of counts. Biometrika, 75(4):621–629.
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