417 Models for Multivariate Count Time Series
Application of composite likelihood approach implies the usage, for example, of bivari-
ate marginal log-likelihood functions over all pairs instead of the usage of the multivariate
likelihood. As an illustration, consider a trivariate probability function P(x, y, z; θ), where
θ is a vector of parameters to estimate. The log-likelihood to be maximized is of the form
(θ) =
n
i=1
log P(x
i
, y
i
, z
i
; θ) while the composite log-likelihood is
n
c
(θ) = [log P(x
i
, y
i
; θ) + log P(x
i
, z
i
; θ) + log P(y
i
, z
i
; θ)] ,
i=1
that is, we replace the trivariate distribution by the product of the bivariate ones. The price
to be paid is some loss of efciency which can be very large for some models. On the other
hand, the optimization problem is usually easier. Simulations for the diagonal trivariate
MINAR(1) model have shown small efciency loss (Pedeli and Karlis, 2013a). Therefore, the
composite likelihood method makes feasible the application of some multivariate models
for time series and this is worth further exploration. Alternatively, one may employ an
expectation-maximization (EM) algorithm making use of the latent structure imposed by
the convolution (Pedeli and Karlis, 2013a).
Finally, Bayesian estimation of the BINAR model with bivariate Poisson innovations is
described in Sofronas (2012).
19.3.4 Other Models in This Category
Ristic et al. (2012) developed a simple bivariate integer-valued time series model with posi-
tively correlated geometric marginals based on the negative binomial thinning mechanism.
Bulla et al. (2011) described a model based on another operator called the signed binomial
operator allowing tting integer-valued data in Z. Recently, Scotto et al. (2014) derived a
model for correlated binomial data and Nastic et al. (2014) discussed a model with Poisson
marginals with same means.
Quoreshi (2006, 2008) described properties of a bivariate moving-average model
(BINMA). The BINMA(q
1
, q
2
) model takes the form
y
1t
= u
1t
+ a
11
◦ u
1,t−1
+···+a
1q
1
u
1,t−q
1
y
2t
= u
2t
+ a
21
◦ u
2,t−1
+···+a
2q
2
u
2,t−q
2
,
where u
it
are innovation terms following some positive discrete distribution. Estimation
using conditional least squares, feasible least squares, and generalized method of moments
is described. Cross-correlation between the series is implied by assuming dependent inno-
vations terms. Extensions to the multivariate case are described in Quoreshi (2008). Similar
to the MINAR model, the multivariate vector integer MA model can allow for mixing
between the series. In both the bivariate and the multivariate models, no parametric
assumptions for the innovations were given. A BINMA model has been studied in Brännäs
and Nordström (2000).
418 Handbook of Discrete-Valued Time Series
19.4 Parameter-Driven Models
Parameter-driven models have also been considered for count time series. The main idea is
that the serial correlation imposed is due to some correlated latent processes on the param-
eter space. They offer useful properties since the serial correlation of the latent process
drives the serial correlation properties for the count model. On the other hand, estimation
is usually much harder. We describe such multivariate models in the following.
19.4.1 Latent Factor Model
Jung et al. (2011) presented a factor model which in fact belongs to the family of parameter-
driven models. The model was applied to the number of trades of ve stocks belonging to
two different sectors within a 5 min interval for a period of 61 days. The model assumed
that the number of trades y
it
for the ith stock at time t follows a Poisson distribution with
mean θ
it
, while
log θ
it
= µ
i
+ γλ
t
+ δ
i
τ
s
i
t
+ φ
i
ω
it
,
where µ
i
is a mean specic to the ith stock, which perhaps may relate to some covariates
specic to the stock as well, λ
t
is a latent common market factor, τ
t
= (τ
1t
, τ
s
i
t
)
is a latent
vector of industry-specic factors and ω
t
= (ω
1t
, ..., ω
Jt
) are latent stock-specic factors.
The model assumes an AR(1) specication for the latent factors, namely,
λ
t
| λ
t−1
∼ N
κ
λ
+ ν
λ
λ
t−1
, σ
2
λ
τ
st
| τ
s,t−1
∼ N
κ
τ
s
+ ν
τ
s
ν
s,t−1
, σ
2
τ
s
ω
it
| ω
i,t−1
∼ N
κ
ω
i
+ ν
ω
i
ω
i,t−1
, σ
ω
2
i
.
Clearly, cross-correlation enters the model by assuming common factors, while serial cor-
relation from the latent AR processes. The likelihood is complicated, and the authors
developed an efcient importance sampling algorithm to evaluate the log-likelihood and
apply the simulated likelihood approach.
19.4.2 State Space Model
Jorgensen et al. (1999) proposed a multivariate Poisson state space model with a common
factor that can be analyzed by a standard Kalman lter. The model assumes that the multi-
ple counts y
it
, i = 1, ..., d at time t follow conditionally independent Poisson distributions,
namely,
Y
it
∼ Poisson(α
it
θ
t
)
419 Models for Multivariate Count Time Series
with α
it
= exp(x
t
α
i
), where x
t
is a vector of time-varying covariates including a constant
and further
σ
2
θ
t
|θ
t−1
∼ Gamma
b
t
θ
t−1
,
,
θ
t−1
where Gamma(a, b
2
) is a gamma distribution with mean a and coefcient of variation b.
Kalman ltering is used for the latent process.
Finally, Lee et al. (2005) proposed a model starting from a bivariate zero-inated Poisson
regression with covariates, adding random effects that are correlated in time.
19.5 Observation-Driven Models
In observation-driven models, serial correlation comes from the fact that current obser-
vations relate directly to the previous ones. The Poisson autoregression model dened in
Fokianos et al. (2009) constitutes an important member of this class for univariate series.
The model has a feedback mechanism and is dened as
Y
t
|F
t−1
∼ Poisson(λ
t
), λ
t
= d + aλ
t−1
+ bY
t−1
,
for t ≥ 1, where the parameters d, a, b are assumed to be positive. In addition, assume
that Y
0
and λ
0
are xed and F
t−1
is the information up to time t − 1. The model is called
INGARCH, but the name perhaps is very ambitious. Properties of the model can be seen in
Fokianos et al. (2009). The model has found a lot of work after this (Trostheim, 2012; Davis
and Liu, 2015). The Poisson INGARCH is incapable of modeling negative serial depen-
dence in the observations which is, however, possible by the self-excited threshold Poisson
autoregression model (see Wang et al., 2014). Extensions to higher-order INGARCH(p,q)
(see Weiß, 2009), nonlinear relationships, and other distributional assumptions have been
also proposed in the literature. This type of model offers a much richer autocorrelation
structure than models based on thinning, like long memory properties, for example. Their
estimation is more computational demanding, the same is true for deriving their properties.
Extension to higher dimensions has also been proposed. Liu (2012) proposed a bivariate
Poisson integer-valued GARCH (BINGARCH) model. This model is capable of modeling
the time dependence between two time series of counts. Consider two time series Y
1t
and
Y
2t
. We assume that
Y
t
= (Y
1t
, Y
2t
)|F
t−1
∼ BP2(λ
1t
, λ
2t
, φ),
where BP2(λ
1
, λ
2
, φ) denotes a bivariate Poisson distribution with marginal means λ
1
and λ
2
, respectively, and covariance equal to φ. This is a reparameterized version of
the distribution in (19.1). Furthermore, we assume for the general BIV. INGARCH(m,q)
model that
m
q
λ
t
= δ + A
i
λ
t−i
+ B
j
Y
t−j
,
i=1 j=1
420 Handbook of Discrete-Valued Time Series
λ
t
= (λ
1t
, λ
2t
)
, δ > 0 is 2-vector and A
i
and B
j
are 2 × 2 matrices with nonnegative entries.
Conditions to ensure the positivity of λ are given.
For example, the BIV.INGARCH(1,1) model takes the form
λ
t
= δ + Aλ
t−1
+ BY
t−1
or equivalently
λ
1t
=
δ
1
+
a
11
a
12
λ
1,t−1
+
b
11
b
12
Y
1,t−1
.
λ
2t
δ
2
a
21
a
22
λ
2,t−1
b
21
b
22
Y
2,t−1
Stability properties for this model are not simple. Liu (2012) used the iterated random
functions approach that allows us to derive the stability properties under a contracting
constraint on the coefcient matrices. Inference procedures are also presented and applied
to real data in the area of trafc accident analysis.
Heinen and Rengifo (2007) developed a model called the Autoregressive Conditional
double Poisson model for a d-dimensional vector of counts y
t
. The model relates to the one
given earlier. Their model was based on the double Poisson distribution dened in Efron
(1986):
Y
it
| F
t−1
∼ DP(λ
it
, φ
i
), i = 1, ..., d,
where λ
it
is the mean of the double Poisson distribution of the ith count at time t and φ
i
is
an overdispersion parameter. For φ
i
= 1, we get the Poisson distribution. Then the mean
vector λ
t
= (λ
1t
, ..., λ
dt
) is dened as a VARMA process
λ
t
= ω + Aµ
t−1
+ BY
t−1
,
where A and B are appropriate matrices. Stationarity is ensured as long as the eigenvalues
of (I − A − B) lie within the unit circle. The VARMA order specication can be modied.
The cross-correlation between the counts is imposed via a Gaussian copula. The model
uses a trick to avoid problems when working with discrete-valued data and copulas, by
using a continued extension argument, that is, by adding some noise to the counts to make
them continuous and working with the continuous versions. The latter adjustment may
create some noise around the model. A recent paper (Nikoloulopoulos, 2013a) discusses
the problems with such a method.
Bien et al. (2011) used copulas to create a multivariate time series model dened on Z
and not only on positive integers. They modeled a bivariate time series of bid and ask
quote changes sampled at a high frequency. The marginal models used were assumed to
follow a dynamic integer count hurdle (ICH) process, tied together with a copula, which
was constructed by properly taking into account the discreteness of the data.
Finally, Held et al. (2005) described a model with multivariate counts where at time t
counts of the other series at time t − 1 enter as covariates for the mean of each series. Also,
Brandt and Sandler (2012) used the model of Chib and Winkelmann (2001) and made it
dynamic by adding autoregressive terms in the mean of the mixing distribution.
421 Models for Multivariate Count Time Series
19.6 More Models
The earlier mentioned list of models does not limit other families of models to be derived.
Such an example is the creation of hidden markov models (HMMs) for multivariate count
time series. In the univariate case, Poisson HMMs have been used to model integer-valued
time series data. Extensions to the multivariate case are possible but multivariate discrete
distributions are needed to model the state distribution. In Orfanogiannaki and Karlis
(2013), multivariate Poisson distributions based on copulas have been considered for this
purpose.
Another approach to create multivariate time series models for counts could be through
the discretization of standard continuous models. Consider, for example, the vector autore-
gressive model based on a bivariate normal distribution. Discretizing the output can lead
to the desired time series. However, such a discretization is not unique and perhaps prob-
lems may occur while estimating the parameters, as multivariate integrals need to be
calculated.
Joe (1996) described an approach to create time series models based on additively closed
families. Working with bivariate distributions with this property, as, for example, the
bivariate Poisson distribution, one can derive such a time series model. Such models
share common elements with models based on thinning operations; see Joe (1996) for a
methodology to create an appropriate operator.
19.7 Discussion
In this chapter, we have pulled together the existing literature on multivariate integer-
valued time series modeling. Table 19.1 summarizes the models. An obstacle for such
models is the lack of, or at least the lack of familiarity with, multivariate discrete distri-
butions which are basic tools for their construction. Given greater availability of such basic
tools, we expect that more models will become available in the near future. Also, ideas for
tackling estimation problems like the composite likelihood approach can help a lot to this
direction.
Such models can have also some other interesting potential. For example, taking the
difference of the two time series from a bivariate model, we end up with a time series
TABLE 19.1
Models for multivariate count time series
Type of Model References
Models based on thinning Franke and Rao (1995); Latour (1997); Pedeli and Karlis (2011); Pedeli and Karlis
(2013a,c); Pedeli and Karlis (2013b); Karlis and Pedeli (2013); Boudreault and
Charpentier (2011); Quoreshi (2006, 2008); Ristic et al. (2012); Brännäs and
Nordström (2000); Bulla et al. (2011)
Observation-driven models Liu (2012); Heinen and Rengifo (2007); Bien et al. (2011); Held et al. (2005); Brandt
and Sandler (2012)
Parameter-driven models Jung et al. (2011); Jorgensen et al. (1999); Lee et al. (2005)
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