23 Statistical Analysis of Count Time Series Models
be recovered explicitly as a function of the past responses. However, a different point of
view has been taken by Zeger (1988), who introduced regression models for time series of
counts by assuming that the observed process is driven by a latent (unobserved) process. To
be more specic, suppose that, conditional on an unobserved process {ξ
t
, t ≥ 1}, {Y
t
, t ≥ 1},
is a sequence of independent counts such that
E[Y
t
| ξ
t
]=Var[Y
t
| ξ
t
]=ξ
t
exp(d + a
1
y
t−1
). (1.26)
In (1.26) we consider a simple model for illustration, but more complex models that include
higher-order lagged values of the response and any covariates can be assumed. It can be
proved that the earlier formulation, although similar to a Poisson log-linear model, reveals
that the observed data are overdispersed. Estimation of all unknown parameters is dis-
cussed by Zeger (1988). A further detailed study of model (1.26) can be found in Davis et al.
(2000), where the authors address the problem of existence of the latent stochastic process
{ξ
t
} and derive the asymptotic distribution of the regression coefcients when the latter
exist. In the context of negative binomial regression, the latent process model (1.26) has
been extended by Davis and Wu (2009). See also Harvey and Fernandes (1989) for a state-
space approach with conjugate priors for the analysis of count time series and Jørgensen
et al. (1999) for multivariate longitudinal count data. More generally, state space models for
count time series are discussed in West and Harrison (1997), Durbin and Koopman (2001),
and Cappé et al. (2005), among others.
1.7 Other Extensions
There are several other possible directions for extending the theory and methods discussed
in this chapter. Threshold models have been considered recently by Woodard et al. (2011),
Douc et al. (2013), and Wang et al. (2014). Several questions are posed by such models such
as estimation of regression and threshold or/and delay parameters. The concept of mixture
models for the analysis of count time series data is a topic closely related to that of threshold
models; for the real-valued case, see Wong and Li (2001) among others. Finally, I would
like to bring forward the possibility of introducing local stationarity to count time series
in the sense of Dahlhaus (1997, 2000). Such models pose several problems; for instance,
the question of existence of a stationary approximation and estimation of the time-varying
parameters by nonparametric likelihood inference.
Acknowledgments
Many thanks to the editors for all their efforts and for inviting me to contribute to this vol-
ume. Special thanks are due to V. Christou, N. Papamichael, D. Tjøstheim, and a referee for
several useful comments and suggestions. Support was provided by the Cyprus Research
Promotion Foundation grant TEXNOLOGIA/THEPIS/0609(BE)/02.
24 Handbook of Discrete-Valued Time Series
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