440 Handbook of Discrete-Valued Time Series
Own drug
Observed
Prediction
0
20
25
10
15
5
0 20
40 60 80 100
Physician
Observed
Prediction
Challenger drug
0
20
25
10
15
5
Leader drug
0 20
40 60 80 100
Physician
30
20
10
0
Physician
Observed
Prediction
0 20 40 60 80 100
FIGURE 20.3
One-month-ahead predictions for all physicians.
possible to construct other models for multivariate time series of counts. For instance, Ma
et al. (2008) describe a multivariate Poisson-lognormal framework for modeling multivari-
ate count time series with an application to transportation safety, while Li et al. (1999)
describe a framework for tting a multivariate ZIP model.
One consideration in tting multivariate Poisson models is the computation time for
calculating the likelihood function, which can increase rapidly with increasing compo-
nent dimension m. In such cases, it is useful to explore approximations for carrying out
Bayesian inference, such as the integrated nested Laplace approximation (INLA), or the
variational Bayes approach. The INLA approach has been discussed in Rue and Martino
(2007) and Rue et al. (2009); also see Gamerman et al. (2015; Chapter 8 in this volume)
Serhiyenho et al. (2015) described an application to trafc safety using R-INLA. The vari-
ational Bayes approach has been discussed in several papers including McGrory and
Titterington (2007), Ormerod and Wand (2010), and Wand et al. (2011), among others. A
study of these approaches for modeling multivariate time series of counts and a comparison
with the MCMC approach will be useful.
20.A Appendix
Details on the complete conditional distributions for the HMDM and MDFM models are
shown below.
441 Dynamic Models for Time Series of Counts with a Marketing Application
20.A.1 Complete Conditional Distributions for the HMDM Model
The complete conditional density of β
it
for i = 1, ..., N and t = 1, ..., T is
1
f (β
it
|γ
t
, η, V
i
, W, Y) ∝ MP
m
(y
it
|λ
it
) exp −
(β
it
− γ
t
)
V
−
i
1
(β
it
− γ
t
)
.
2
The complete conditional density of γ
t
for t = 1, ..., T is
N
1
f (γ
t
|β
it
, η, V
i
, W, Y) ∝ exp
−
2
(β
it
− γ
t
)
V
−
i
1
(β
it
− γ
t
)
i=1
× exp
−
1
(γ
t
− Gγ
t−1
)
W
−1
(γ
t
− Gγ
t−1
)
2
× exp −
1
(γ
t+1
− Gγ
t
)
W
−1
(γ
t+1
− Gγ
t
)
.
2
The complete conditional density of η is
T
N
1
−1
f (η|β
it
, γ
t
, V
i
, W, Y) ∝ MP
m
(y
it
|λ
it
) exp −
(η − μ
η
)
η
(η − μ
η
)
.
2
t=1
i=1
The complete conditional density of V
i
for i = 1, ..., N is
T
1
V
−1
f (V
i
|β
it
, γ
t
, η, W, Y) ∝ |V
i
|
−1/2
exp
−
2
(β
it
− γ
t
)
i
(β
it
− γ
t
)
t=1
×|V
i
|
−n
v
/2
exp
−
1
2
tr(V
−
i
1
S
v
)
.
The complete conditional density of W is
T
1
f (W|β
it
, γ
t
, η, V
i
, Y) ∝ |W|
−1/2
exp −
(γ
t
− Gγ
t−1
)
W
−1
(γ
t
− Gγ
t−1
)
2
t=1
×|W|
−1/2
exp
−
1
(γ
0
− m
0
)
C
−
0
1
(γ
0
− m
0
)
2
×|W|
−n
w
/2
exp
−
1
tr(W
−1
S
w
)
.
2
442 Handbook of Discrete-Valued Time Series
20.A.2 Complete Conditional Distributions and Sampling Algorithms for the MDFM Model
For t = 2, ..., T and h = 1, ..., H,
N
z
i,t,h
1
W
−1
f (β
0,t,h
) ∝ π
h
[p(y
i,t
|λ
i,t,h
)]
z
i,t,h
exp
−
2
(β
0,t,h
− Gβ
0,t−1,h
−)
h
(β
0,t,h
− Gβ
0,t−1,h
)
i=1
1
W
−1
.
× exp −
2
(β
0,t+1,h
− Gβ
0,t,h
)
h
(β
0,t+1,h
− Gβ
0,t,h
)
For i = 1, ..., N and h = 1, ...H,
T
f
β
1,i,h
∝
π
h
z
i,t,h
p
y
i,t
|λ
i,t,h

z
i,t,h
exp −
1
β
1,i,h
− a
1,i,h
R
−
1,i
1
,h
β
1,i,h
− a
1,i,h
,and
2
t=1

z
i,t,h
f
β
2,i,h
∝
T
π
z
h
i,t,h
p
y
i,t
|λ
i,t,h
exp
−
1
2
β
2,i,h
− a
2,i,h
R
−
2,i
1
,h
β
2,i,h
− a
2,i,h
.
t=1
For h = 1, ..., H,
T
exp −
1
W
−1
t=2
f (W
h
) ∝
2
(β
0,t,h
− Gβ
0,t−1,h
)
h
(β
0,t,h
− Gβ
0,t−1,h
)
× exp
−
1
(α − μ
α
)
σ
−1
|W
h
|
−n
h
/2
exp
−
1
tr(W
−1
.
α
(α − μ
α
)
h
S
h
)
2 2
The complete conditional density of π is
H
N
T
f (π) ∝
π
d
h
−1+
i=1
t=1
z
i,t,h
,
h
h=1
d
1
−1+
N
T
which reduces to f (π
1
) ∝ π
1
i=1
t=1
z
i,t,1
(1 − π
1
)
d
2
−1
when H = 2.
The sampling algorithms are shown in the following. At the lth Gibbs iteration, (a) we
generate z
(l)
∼ f (z|y,
(l)
) and (b) we then generate
(l+1)
∼ π(|y, z)). Specically, in step
(a), for i = 1, ..., N and t = 1, ..., T, we simulate z
i,t
from a multinomial distribution with
probabilities
H
p
j
= π
(
j
l)
[p(y
i,t
|λ
i,t,h
)]
z
i,t,h
/ π
(
h
l)
[p(y
i,t
|λ
i,t,h
)]
z
i,t,h
h=1
for j = 1, ..., H. When H = 2, we simulate z
i,t,1
from Bernoulli distribution with weight
π
1
(l)
[p(y
i,t
|λ
i,t,h
)]
z
i,t,h
p
z
=
(l) (l)
.
π
1
[p(y
i,t
|λ
i,t,h
)]
z
i,t,h
+ π
2
[p(y
i,t
|λ
i,t,h
)]
z
i,t,h

443 Dynamic Models for Time Series of Counts with a Marketing Application
In step (b), the complete conditional densities of W
h
for h = 1, ..., H and the mixture
proportion π have closed forms; we directly sample W
h
from IW(n
∗
h
, S
h
∗
), where
∗
n
h
= n
h
+ T
T
S
∗
h
= S
h
+
β
0,t,h
− Gβ
0,t−1,h
β
0,t,h
− Gβ
0,t−1,h
.
t=2
N
T
N
T
We sample π from a Dirichlet
d
1
+
i=1 t=1
z
i,t,1
, ..., d
H
+
i=1 t=1
z
i,t,h
distribution
N
T
when H > 2andfromaBeta d
1
+
i=1 t=1
z
i,t,1
, d
2
distribution when H = 2. The draws
are made using a random walk Metropolis–Hastings algorithm.
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