150 Handbook of Discrete-Valued Time Series
Example 7.1
Suppose that the conditional mean, variance, skewness, and kurtosis of y
t
are speci-
λ
t
, σ
2
−1/2
−1
ed as μ
t
=
t
= λ
t
, γ
t
= λ
t
,and κ
t
= λ
t
, and suppose that μ
t
is modeled
by (7.5). These moments match the rst four moments of y
t
generated by what Ferland
et al. (2006) referred to as a Poisson INGARCH process, which assumes that y
t
|F
t−1
∼
Poisson(λ
t
), so that λ
t
is the conditional variance as well as the conditional mean. While it
seems a misnomer to use the term INGARCH for modeling the conditional mean and not
the conditional variance as GARCH models do, the form of (7.5) is similar to the normal-
GARCH model (Bollerslev, 1986), where y
t
|F
t−1
∼ N(0, σ
2
t
) for all t, and the model for
the conditional variance σ
2
t
follows the right side of (7.5), subject to the same conditions
on the parameters. For conformity with the literature, we use the term INGARCH in this
chapter. The moments of the Poisson INGARCH random variable y
t
are easily derived
from the probability generating function G
y
(s) = E(exp(sy
t
)|F
t−1
) = exp[λ
t
(s − 1)].
Implementation of the EF approach does not require that at each time t, y
t
has a condi-
tional Poisson distribution, but only requires specication of the conditional moments of
y
t
|F
t−1
for each t. Such moment specications are also sufcient for the other INGARCH
models described in this chapter. �
Example 7.2
Suppose the conditional mean, variance, skewness, and kurtosis of y
t
given F
t−1
are
μ
t
= λ
∗
t
/(1 − τ) = λ
t
, σ
t
2
= λ
∗
t
/(1 − τ)
3
= τ
∗2
λ
t
, γ
t
= (1 + 2τ)/
λ
∗
t
(1 − τ),and κ
t
= (1 +
8τ + 6τ
2
)/[λ
∗
t
(1 − τ)], corresponding to moments from the GP INGARCH process (Zhu,
2012a), where τ
∗
= 1/(1−τ). This process is dened as y
t
|F
t−1
∼ GP(λ
t
∗
, τ), λ
∗
t
= (1−τ)λ
t
,
max(−1, −λ
∗
t
/4)< τ < 1, and the conditional mean is again modeled by (7.5). The GP
distribution for y
t
conditional on F
t−1
is
P(y
t
λ
t
(λ
t
+ τk)
k−1
exp[−(λ
t
+ τk)]/k!, k = 0, 1, 2, ...
(7.7)
= k|F
t−1
) =
0, k > m if τ < 0,
where m is the largest positive integer for which λ
t
+ τm > 0when τ < 0. To derive
the conditional moments of the GP distribution shown above, we can use the recursive
relation for the rth raw moment μ(r), that is, (1 − τ)μ(r) = λ
t
μ(r − 1) + λ
t
∂
∂
μ
λ
(
t
r)
+ τ
∂μ
∂τ
(r)
,
where λ
t
> 0and max(−1, −λ
t
/m)< τ < 1. �
Example 7.3
For p
t
= 1/(1 + λ
t
) and q
t
= 1 − p
t
, suppose the conditional mean, variance, skewness,
and kurtosis of y
t
given F
t−1
are μ
t
= rq
t
/p
t
= rλ
t
, σ
2
= rq
t
/p
t
2
, γ
t
= (2 − p
t
)/(rq
t
)
1/2
,
t
and κ
t
= (p
t
2
− 6p
t
+ 6)/rq
t
, which correspond to the moments of a negative binomial
INGARCH process, where y
t
|F
t−1
∼ NB(r, λ
t
), the conditional mean is modeled by (7.5)
as before, and the probability generating function is given by G
y
(s) = p
r
t
/(1 − q
t
s)
r
,and
the conditional probability mass function (pmf) of y
t
has the form
P(y
t
= k|F
t−1
) =
k + r − 1
p
r
t
q
k
t
, k = 0, 1, 2, ... . � (7.8)
r − 1
7.3.2 Models for Counts with Excess Zeros
In several applications, observed counts over time may show an excess of zeros, and the
usual Poisson or negative binomial models are inadequate. One example in the area of
public health could involve surveillance of a rare disease over time, where the observed
151 Estimating Equation Approaches for Integer-Valued Time Series Models
counts typically show zero ination. Yang (2012) studied statistical modeling for time series
with excess zeros. We consider two examples.
Example 7.4
When count time series are observed with an excess of zeros in applications, we may
assume a specication of the conditional mean, variance, skewness, and kurtosis of y
t
given F
t−1
as μ
t
= (1−ω)λ
t
, σ
2
t
= (1−ω)(1+ωλ
t
)λ
t
, γ
t
= ((1−ω)λ
t
)
−1/2
(1+ωλ
t
)
−3/2
(1+
ωλ
t
(3 +2ωλ
t
+λ
t
)),and κ
t
=[(1 −ω)(1 +ωλ
t
)
2
λ
t
]
−1
[1 +ωλ
t
(7 +λ
t
(−6 +12ω +λ
t
(1 −
6(1 − ω)ω)))],where0 < ω < 1, and we assume that the conditional mean is modeled
by (7.5). The moments correspond to the rst four moments of a zero-inated Poisson
INGARCH process (Zhu, 2012b) given by y
t
|F
t−1
∼ ZIP(λ
t
, ω) dened by
P(y
t
= k|F
t−1
) = ω
k,0
+ (1 − ω)λ
t
k
exp(−λ
t
)/k!, (7.9)
for k = 0, 1, 2, ....When ω = 0, the ZIP INGARCH model reduces to the Poisson
INGARCH model. The probability generating function of a ZIP random variable is given
by G
y
(s) = ω + (1 − ω) exp[λ
t
(s − 1)]. The conditional variance exceeds the conditional
mean, so the ZIP-INGARCH model can handle overdispersion. As mentioned earlier, in
this and the following models, we only require specication of the conditional moments
of y
t
|F
t−1
for each t (and no marginal distributional assumptions). �
Example 7.5
An alternate specication for count time series with an excess of zeros corresponds to the
conditional moment specications:
(1 − ω)rq
t
μ
t
= ,
p
t
σ
2
t
=
(1 − ω)rq
t
(1 + rωq
t
)
,
2
p
t
rωq
t
(3 − rq
t
+ 2ωrq
t
) + 2 − p
t
γ
t
= ,
[(1 − ω)rq
t
]
1/2
[1 − ωrq
t
]
3/2
1
2 2
κ
t
=
ωrq
t
11 − 4p
t
− 6rq
t
+ r q
t
(1 − ω)rq
t
(1 − ωrq
t
)
2
+ 6ω
2
r
2
q
t
2
(2 − rq
t
) + 6r
3
ω
3
q
t
3
+ 6 − 6p
t
+ p
t
2
.
These are the rst four moments of a zero-inated negative binomial INGARCH pro-
cess (Zhu, 2012b). Here, y
t
|F
t−1
∼ ZINB(λ
t
, α, ω), the conditional mean is again given
by (7.5), and α ≥ 0 is the dispersion parameter. The conditional ZINB distribution is
dened by
(k + λ
1
t
−c
/α)
1
λ
1
t
−c
/α
αλ
t
c
k
P(y
t
= k|F
t−1
) = ω
k,0
+ (1 − ω)
k!(λ
t
/α)
1 + αλ
c
1 + αλ
c
t
,
1−c
for k = 0, 1, 2, ...,andwhere c is an index that assumes the values 0 or 1 and identies
the form of the underlying negative binomial distribution. The probability generating
function is G
y
(s) =ω +(1 − ω)p
t
r
/(1 −q
t
s)
r
. In comparison to the negative binomial dis-
tribution shown in (7.8), we have p
t
= 1/α + λ
c
t
, q
t
= 1 − p
t
,and r = λ
t
/α. Note that in
the limit as α →∞, the ZINB-INGARCH model reduces to the ZIP-INGARCH model,
and when ω = 0, the model reduces to the NB-INGARCH model. �
152 Handbook of Discrete-Valued Time Series
7.3.3 Models in the GAS Framework
Recently, Creal et al. (2013) proposed a novel observation-driven modeling strategy for
time series, that is, the GAS model. Following their approach, we propose an extension of
the GAS model for an integer-valued time series {y
t
} with specied rst four conditional
moments, and describe the use of estimating equations. Let f
t
=f
t
(θ) denote a vector-
valued time-varying function of an unknown vector-valued parameter θ, and suppose
that the evolution of f
t
is determined by an autoregressive updating equation with an
innovation s
t
, which is a suitably chosen martingale difference vector:
P
Q
f
t
= ω + A
i
s
t−i
+ B
j
f
t−j
. (7.10)
i=1 j=1
Suppose that ω = ω(θ), A
i
= A
i
(θ),and B
j
= B
j
(θ). For instance, f
t
could represent the
conditional mean μ
t
of y
t
or its conditional variance σ
2
t
.
Suppose a valid conditional probability distribution p(y
t
|F
t−1
, f
t
; θ) is specied (rather
than just assuming the form of the rst few moments). Let s
t
correspond to the standardized
score function, that is, s
t
= S
t
∇
t
, where
∂
∇
t
=
∂f
t
log p(y
t
|F
t−1
, f
t
; θ) and
−1
∂
S
t
=−E log p(y
t
|F
t−1
, f
t
; θ) .
∂f
t
∂f
t
Then (7.10) corresponds to the GAS(P, Q) model discussed by Creal et al. (2013). Alternate
specications have been suggested for S
t
that scale ∇
t
in addition to the inverse informa-
tion given above. These include the positive square root of the inverse information or the
identity matrix.
7.4 Parametric Inference via EFs
In Section 7.4.1, we describe the estimation of the parameter vector θ in integer-valued
time series models via linear estimating equations and give a recursive scheme for fast
optimal estimation. In Section 7.4.2, we describe a combined EF approach based on linear
and quadratic martingale differences, and show that these combined EFs are more infor-
mative when the conditional mean and variance of the observed process depend on the
same parameter.
7.4.1 Linear EFs
Consider the class M of all unbiased EFs g(m
t
(θ)) based on the martingale difference
m
t
(θ) =y
t
− μ
t
(θ). Theorem 7.1 gives the optimal linear estimating equation with corre-
sponding optimal information and the form of an approximate recursive estimator of θ
which is based on a rst order Taylor approximation.
153 Estimating Equation Approaches for Integer-Valued Time Series Models
Theorem 7.1 The optimal linear estimating equation and corresponding information are obtained
by substituting m
t
(θ) = y
t
− μ
t
(θ) into (7.2) and (7.3). The recursive estimator for θ is given by
∗
θ
t
=
θ
t−1
+ K
t
a
t−1
(
θ
t−1
)g(m
t
(
θ
t−1
)),
−1
K
t
= K
t−1
I
p
−
a
t
∗
−1
(
θ
t−1
)
∂g(m
∂
t
(
θ
θ
t−1
))
+
∂a
t
∗
−1
∂
(
θ
θ
t−1
)
g(m
t
(
θ
t−1
))
K
t−1
,(7.11)
where I
p
is the identity matrix. If g(x) = x, then
∗
∂μ
t
(θ)/∂θ
a
t−1
(θ) =
,
Var(g(m
t
(θ))|F
t−1
)
while for any other function g (such as the score function),
∗
[∂μ
t
(θ)/∂θ][∂g(m
t
(θ))/∂m
t
(θ)]
a
t−1
(θ) =
.
Var(g(m
t
(θ))|F
t−1
)
The proof is similar to that in Thavaneswaran and Heyde (1999) for the scalar parameter case. �
Corollary 7.1 is a special case for the scalar parameter case, where a
t
∗
−1
does not depend
on θ, while Corollary 7.2 discusses a nonlinear time series model.
Corollary 7.1 (Thavaneswaran and Heyde, 1999). For the class M of all unbiased EFs
g(m
t
(θ)) based on the martingale difference m
t
(θ) = y
t
− μ
t
(θ),let μ
t
(θ) be differentiable
with respect to θ. The recursive estimator for θ is given by
K
t−1
a
t
∗
−1
g(m
t
(θ))
θ
t
=
θ
t−1
+
1 +[∂μ
t
(
θ
t−1
)/∂θ]K
t−1
a
∗
, where
t−1
K
t−1
K
t
= , (7.12)
1 +[∂μ
t
(
θ
t−1
)/∂θ]K
t−1
a
∗
t−1
and a
t
∗
−1
is a function of the observations. �
Corollary 7.2 Consider nonlinear time series models of the form
y
t
= θf(F
t−1
) + σ(F
t−1
)ε
t
, (7.13)
where {ε
t
} is an uncorrelated sequence with zero mean and unit variance and f (F
t−1
)
denotes a nonlinear function of F
t−1
, such as y
t
2
−1
. When g(x) = x in Theorem 7.1, the
recursive estimate based on the optimal linear EF
t
n
=1
a
t
∗
−1
(y
t
− θf (F
t−1
)) is given by
(Thavaneswaran and Abraham, 1988)
154 Handbook of Discrete-Valued Time Series
θ
t
=
θ
t−1
+ K
t
a
∗
t−1
[y
t
−
θ
t−1
f (F
t−1
)], where
K
t−1
K
t
=
∗
, (7.14)
1 + f (F
t−1
)K
t−1
a
t−1
a
t
∗
−1
=−f (F
t−1
)/σ
2
(F
t−1
) and K
−1
=
n
1
a
t
∗
−1
f (F
t−1
). After some algebra, (7.14) has the
t
t=
following familiar Kalman lter form:
θ
t
=
θ
t−1
+
σ
2
(F
t−
K
1
t
)
−
+
1
f
f
(
2
F
(
t
F
−
t
1
−
)
1
)K
t−1
[y
t
− θ
ˆ
t−1
f (F
t−1
)], where
K
t
= K
t−1
−
(K
t−1
f (F
t−1
))
2
. � (7.15)
σ
2
(F
t−1
) + f
2
(F
t−1
)K
t−1
The form of the recursive estimate of the xed parameter θ in (7.14) motivates use of
the EF approach for the model in the GAS framework discussed in Section 7.3.3. When the
observation at time t comes in, we update the recursive estimate at time t as the sum of the
estimate at t−1 and the product of the inverse of the information K
t
(which is the term S
t
in
the GAS formulation) and the optimal martingale difference a
t
∗
−1
[y
t
−
θ
t−1
f (F
t−1
)] (which
is ∇
t
in the GAS formulation).
Estimating equation approaches for recursive estimation of a time-varying parameter
have not been discussed in the literature. Consider the simple case where θ
t
= φθ
t−1
for a
given φ. The following recursions have a Kalman lter form when the state equation has
no error:
= φ
φK
t−1
f (F
t−1
)
y
t
−
, where
θ
t
θ
t−1
+
σ
2
(F
t−1
) + f
2
(F
t−1
)K
t−1
θ
t−1
f (F
t−1
)
K
t
= φ
2
K
t−1
−
(φK
t−1
f (F
t−1
))
2
. (7.16)
σ
2
(F
t−1
) + f
2
(F
t−1
)K
t−1
7.4.2 Combined EFs
To estimate θ based on the integer-valued data y
1
, ... , y
n
, consider two classes of martingale
differences for t = 1, ... , n,viz., {m
t
(θ) = y
t
− μ
t
(θ)} and {M
t
(θ) = m
t
(θ)
2
− σ
2
t
(θ)},
see Thavaneswaran et al. (2015). The quadratic variations of m
t
(θ), M
t
(θ), and the quadratic
covariation of m
t
(θ) and M
t
(θ) are respectively
m
t
= E(m
t
2
(θ)|F
t−1
) = σ
2
t
(θ),
M
t
= E(M
t
(θ)
2
|F
t−1
) = σ
t
4
(θ)(κ
t
(θ) − 1),and
m, M
t
= E(m
t
(θ)M
t
(θ)|F
t−1
) = σ
3
t
(θ)γ
t
(θ).
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