34 Handbook of Discrete-Valued Time Series
where the K
ti
(α) are independent over t and i. This can be viewed as a dynamic
system so that K(α) �
t
Y
t−1
is a sum where each countable unit at time t − 1 may
be absent, present, or split into more than one new unit at time t,and
t
consists
of the new units at time t.Also K(α) � y can be considered as a compounding or
branching operator, and the time series model can be considered as a branching
process model with immigration.
• Time series based on random coefcient thinning: this is random binomial thin-
ning, where the chance of survival to the next time is a random variable that
depends on t.
Y
t−1
Y
t
= A
t
◦ Y
t−1
+
t
= I
ti
(A
t
) +
t
, (2.3)
i=1
A beta-binomial thinning operator based on the construction in Section 2.4 ts
within this class.
Because all of the above models have a conditional expectation that is linear in the previous
observation, they have been called integer-autoregressive models of order 1, abbreviated
INAR(1). The models are not truly autoregressive in the sense of linear in the previous
observations (because such an operation would not preserve the integer domain).
2.3.1 Analogues of Gaussian AR(p)
An extension of (2.3) to a higher-order Markov time series model is given in Section 2.4 for
one special case. Otherwise, binomial thinning is a special case of generalized thinning. We
next extend (2.2) to higher-order Markov:
p p
Y
t−j
Y
t
= K(α
j
) �
t
Y
t−j
+
t
= K
tji
(α
j
) +
t
, (2.4)
j=1 j=1 i=1
where 0 ≤ α
j
≤ 1forj = 1, ..., p and the K
tji
(α
j
) are independent over t, j and i,and
t
is the
innovation at time t. This is called GINAR(p) in (Gauthier and Latour 1994; Latour 1997,
1998). It can also be interpreted as a branching process model with immigration, where
aunitattime t has independent branching at times t + 1, ..., t + p. The most common
form of INAR(p) in the statistical literature involves the binomial thinning operator; see
Du and Li (1991). For the binomial thinning operator, Alzaid and Al-Osh (1990) dene
INAR(p) in a different way from the above with a conditional multinomial distribution
for (α
1
◦ Y
t
, ..., α
p
◦ Y
t
). Because the survival/continuation interpretation for (2.1) does
not extend to second and higher orders, it is better to consider (2.4) with more general
thinning operators; if the K
tji
are Bernoulli random variables, this can still be interpreted as
a branching process model with immigration (with limited branching).
More specically, for a GINAR(2) model based on compounding, unit i at time t
con-
tributes K
t
+1,i
(α
1
) units to the next time and K
t
+2,i
(α
2
) units in two time steps. That is,
at time t, the total count comes from branching of units at times t − 1and t − 2plusthe
innovation count.
35 Markov Models for Count Time Series
It will be shown below that the GINAR(p) model has an overdispersion property if
{K(α)} satises Var[K(α)]= σ
2
= a
K
α(1 − α) where a
K
≥ 1. A sufcient condition
K(α)
for this is the self-generalizability of {K(α)} (dened below in Section 2.3.3).
Let R
t
(y
t−1
, ..., y
t−p
) =
p
j=1
y
i=
t−
1
j
K
tji
(α
j
). Then its conditional mean and variance are
p
p
1
α
j
y
t−j
and
1
y
t−j
Var[K(α
j
)], respectively. That is, this GINAR(p) model has linear
j= j=
conditional expectation and variance, given previous observations. The mean is the same
for all {K(α)} that have E[K(α)]=α, but the conditional variance depends on the family of
{K(α)}. With a self-generalized family, the conditional variance is a
K
p
1
y
t−j
α
j
(1 − α
j
) so
j=
that different families of {K(α)} lead to differing amounts of conditional heteroscedasticity,
and a larger value of a
K
leads to more heteroscedasticity.
The condition for stationarity of (2.4) is
j
p
=1
α
j
< 1. In this case, in stationary state, the
equations for the mean and variance lead to
p
μ
μ
Y
= μ
Y
α
j
+ μ
, μ
Y
=
(1 − α
1
−···−α
p
)
, (2.5)
j=1
and
p
p
σ
2
Y
= σ
2
Y
α
j
2
+ 2
ρ
k−j
α
j
α
k
+ μ
Y
σ
2
K(α
j
)
+ σ
2
, (2.6)
j=1 1≤j<k≤p j=1
where ρ
is the autocorrelation at lag . If the innovation is overdispersed relative to
Poisson (that is, σ
2
/μ
≥ 1), then we show that the stationary distribution of Y is also
overdispersed. From (2.5) and (2.6), and assuming σ
2
K(α)
= a
K
α(1 − α),
σ
Y
2
a
K
j
p
=1
α
j
1 − α
j
+ σ
2
1 −
j
p
=1
α
j
/μ
μ
Y
=
1 −
p
j=1
α
2
j
− 2
1≤j<k≤p
ρ
k−j
α
j
α
k
p
p
j=1
α
j
1 − α
j
+ 1 −
j=1
α
j
≥
1 −
p
j=1
α
2
j
− 2
1≤j<k≤p
ρ
k−j
α
j
α
k
1 −
j
p
=1
α
2
j
=
1 −
p
j=1
α
2
j
− 2
1≤j<k≤p
ρ
k−j
α
j
α
k
≥ 1,
because ρ
≥0 and the denominator is positive. The inequality is strict for p > 1withρ
1
> 0.
If the innovation is Poisson with σ
2
/μ
= 1and a
K
= 1 for binomial thinning, then one
still has σ
2
Y
/μ
Y
> 1forp > 1, so that the stationary distribution cannot be Poisson. A Markov
model of order p with stationary Poisson marginal distributions and Poisson innovations
is developed in Section 2.4.
With p =1and α
1
=α, the above becomes D =σ
Y
2
/μ
Y
=(1 + α)
−1
[a
K
α + σ
2
/μ
].
For a GINAR(p) stationary time series without a self-generalized family {K(α)}, no general
overdispersion property can be proved.
36 Handbook of Discrete-Valued Time Series
2.3.2 Analogues of Gaussian Autoregressive Moving Average
To dene analogues of Gaussian moving-average (MA) and ARMA-like models, let
{
t
: ..., −1, 0, 1, ...}be a sequence of independent and identically distributed random vari-
ables with support on N
0
;
t
is an innovation random variable at time t. In a general context,
the extension of moving average of order q becomes q-dependent where observations more
than q apart are independent.
The model, denoted as INMA(1), is
t−1
Y
t
=
t
+ K(α
) �
t−1
=
t
+ K
ti
(α
), (2.7)
i=1
with independent K
ti
(α
) over t, i, and the model denoted as INMA(q)is
q
t−j
Y
t
=
t
+ K
tji
(α
j
), (2.8)
j=1 i=1
with independent K
tji
(α
j
) over t, j, i. The model denoted as INARMA(1, q), with a construc-
tion analogous to the Poisson ARMA(1, q) in McKenzie (1986), is the following:
q−1 q−1
t−j
Y
t
= W
t−q
+ K(α
j
) �
t−j
= W
t−q
+ K
tji
(α
j
),
j=0 j=0 i=1
(2.9)
W
s−1
W
s
= K(α) � W
s−1
+
s
= K
s
(α) +
s
,
=1
with independent K
tji
(α
j
), K
s
(α) over t, j, i, s, .If α = 0, then W
s
=
s
and Y
t
=
t−q
+
q
j=
−
0
1
K(α
j
) �
t−j
is q-dependent (but not exactly the same as (2.8)).
2.3.3 Classes of Generalized Thinning Operators
In this subsection, some classes of generalized thinning operators and known results about
the stationary distribution for GINAR series with p = 1 are summarized. The following
denitions are needed:
Denition 2.1 (Generalized discrete self-decomposability and innovation).
(a) A nonnegative integer-valued random variable Y is generalized discrete self-
decomposable (GDSD) with respect to {K(α)} if and only if (iff)
Y
=
d
K(α) � Y + (α) for each α ∈[0, 1].
In this case, (α) has pgf G
Y
(s)/G
Y
(G
K
(s; α)).
37 Markov Models for Count Time Series
(b) Under expectation thinning compounding and a GDSD marginal (with pgf
G
Y
(s)), the stationary time series model is (2.2), where the innovation
t
has pgf
G
Y
(s)/G
Y
(G
K
(s; α)).
Denition 2.2 (Self-generalized {K
K
(
(
α
α
)
)
]
}). Consider a family of K(α) ∼ F
K
(·; α) with
E[K(α)]=α and pgf G
K
(s; α) = E[s , α ∈[0, 1]. Then {F
K
(·; α)} is self-generalized iff
G
K
(G
K
(s; α); α
) = G
K
(s; αα
), ∀ α, α
∈ (0, 1).
For binomial thinning, the class of possible margins is called the discrete self-
decomposable (DSD) class. Note that unless Y is Poisson and {K(α)} corresponds to
binomial thinning, the distribution of the innovation is in a different parametric family
than F
Y
.
The terminology of self-generalizability is used in Zhu and Joe (2010b), and the concept
is called a semigroup operator in Van Harn and Steutel (1993). Zhu and Joe (2010a) show
that (1) Var[K(α)]=σ
2
= a
K
α(1 − α), where a
K
≥ 1 for a self-generalized family {K(α)}
K(α)
and (2) that generalized thinning operators without self-generalizability lack some closure
properties. Also self-generalizability is a nice property for embedding into a continuous-
time process.
For NB, Zhu and Joe (2010b) show that NB(θ, ξ) is GDSD for three self-generalizable
thinning operators that are given below. For NB(θ, ξ), with parametrization as given in
Section 2.2, the pgf is G
NB
(s; θ, ξ) =[π/{1 − (1 − π)s}]
θ
,for s > 0, θ > 0and ξ > 0.
Three types of thinning operators based on {K(α)} are given below in terms of the pgf,
together with Var[K(α)]; the second operator (I2) has been used by various authors in sev-
eral different parametrizations; the specication is simplest via pgfs. The different {K(α)}
families allow different degrees of conditional heteroscedasticity.
(I1) (binomial thinning) G
K
(s; α) = (1 − α) + αs,with Var[K(α)]=α(1 − α).
(I2) G
K
(s; α; γ) =
(1−α)+(α−γ)s
,0 ≤ γ ≤ 1, with Var[K(α)]=α(1−α)(1+γ)/(1−γ).
(1−αγ)−(1−α)γs
Note that γ = 0 implies G
K
(z; α) = (1 − α) + αs.
(I3) G
K
(s; α; γ) = γ
−1
[1 +γ −(1 +γ −γs)
α
],0 ≤ γ,withVar[K(α)]=α(1 −α)(1 +γ).
Note that γ → 0 implies G
K
(s; α) = (1 − α) + αs.
For NB(θ, ξ), GDSD with respect to I2(γ)holds for0 ≤ γ ≤ 1 − π = ξ/(1 + ξ),and
GDSD with respect to I3(γ)holdsfor0 ≤ γ ≤ (1 − π)/π = ξ. For GP(θ, η), the property
of DSD is shown in Zhu and Joe (2003), and it can be shown that GP(θ, η)isGDSDwith
respect to I2(γ(η)), where γ(η) increases as the overdispersion η increases. Note that the
GP distribution does not have a closed-form pgf.
2.3.4 Estimation
For parameter estimation in count time series models, a common estimation approach is
CLS. This involves the minimization of
n
2
(y
i
− E[Y
i
|y
i−1
, y
i−2
, ...])
2
for a time series of
i=
length n. For a stationary model, it is straightforward to get point estimators of μ
Y
and
some autocorrelation parameters. One problem with conditional least squares (CLS) is that
it cannot distinguish overdispersed Poisson models for
t
and Y
t
. For example, if a NB or
GP time series is assumed with one of the above generalized thinning operators, then the
overdispersion cannot be reliably estimated with an extra moment equation after CLS.
38 Handbook of Discrete-Valued Time Series
We next mention what can be done for computations of pmfs and the likelihood for
binomial thinning and generalized thinning.
1. Zhu and Joe (2006) have an iterative method for computing pmfs with binomial
thinning and a DSD stationary margin.
2. The pgf of K � y has closed form if the pgf of K(α) has closed form and the pgf of
the innovation has closed form if the pgf of Y has closed form. In this case, Zhu
and Joe (2010b) invert a characteristic function for the pgf of G
K(α)�y
G
(α)
using an
algorithm of Davies (1973) to compute the conditional pmf of Y
t
given Y
t−1
= y
t−1
.
Let ϕ
W
(s) = E
e
isW
= G
W
e
is
for a nonnegative integer random variable W and
�
π
ϕ
W
(u)e
−iuw
dene a(w) :=
1
2
− (2π)
−1
−π
Re
1−e
−iu
du. Then Pr(W < w) = a(w).The
pmf of W is
f
W
(0) = Pr(W < 1) = a(1), f
W
(w) = a(w + 1) − a(w), w = 1, 2, ....
This works for NB but not GP because the latter does not have a closed-form pgf.
2.3.5 Incorporation of Covariates
For a NB(θ, ξ) stationary INAR(1) model, the pdf of the innovation is
G
NB
(s; θ, ξ)
.
G
NB
(G
K
(s; α); θ, ξ)
For a time-varying θ
t
that depends on covariates with xed ξ (xed overdispersion index),
suppose the innovation
t
has pgf
G
NB
(s; θ
t
, ξ)
. (2.10)
G
NB
(G
K
(s; α); θ
t
, ξ)
An advantage of this assumption is that a NB stationary margin results when θ
t
is constant.
More generally, for GINAR(p) series where the stationary distribution does not have a
simple form, the simplest extension to accommodate covariates is to assume an overdis-
persed distribution for
t
and absorb a function of covariates into the mean of
t
.
Alternatively, other parameters can be made into functions of the covariates.
2.4 Operators in Convolution-Closed Class
The viewpoint in this section is to construct a stationary time series of order p based on
a joint pmf f
1···(p+1)
for (Y
t−p
, ..., Y
t
), where marginal pmfs satisfy f
1:m
= f
(1+i):(m+i)
for
i = 1, ..., p + 1 − m and m = 2, ..., p. Suppose the univariate marginal pmfs of f
1···(p+1)
are
all f
Y
= f
Y
t
. From this, one has a transition probability f
p+1|1···p
= f
Y
t
|Y
t−p
,...,Y
t−1
. For p = 1,
f
2|1
leads to a stationary Markov time series of order 1. For p = 2, f
3|12
leads to a stationary
Markov time series of order 2 if f
123
has bivariate marginal pmfs f
12
= f
23
.
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