155 Estimating Equation Approaches for Integer-Valued Time Series Models
The optimal EFs based on the martingale differences m
t
(θ) and M
t
(θ) are respectively
n
g
m
∗
(θ) =−
∂μ
t
(θ)
m
t
and
t=1
∂θ m
t
n
∂σ
2
t
(θ)
M
t
∗
g
M
(θ) =− .
∂θ M
t
t=1
The information associated with g
m
∗
(θ) and g
M
∗
(θ) are respectively
n
∂μ
t
(θ)
∂μ
t
(θ) 1
I
g
∗
(θ) =
∂θ
∂θ
m
t
and
m
t=1
n
∂σ
2
t
(θ)
∂σ
2
t
(θ)
1
I
g
M
∗
(θ) =
∂θ
∂θ
M
t
.
t=1
Theorem 7.2 describes the results for combined EFs based on the martingale differences
m
t
(θ) and M
t
(θ) and provides the resulting form of the recursive estimator of θ based on a
rst-order Taylor approximation (Liang et al., 2011; Thavaneswaran et al., 2015.)
Theorem 7.2 In the class of all combined EFs
n
G
C
=
g
C
(θ) : g
C
(θ) = [a
t−1
(θ)m
t
(θ) + b
t−1
(θ)M
t
(θ)]
,
t=1
(a) The optimal EF is given by
n
g
C
∗
(θ) =
a
t
∗
−1
(θ)m
t
(θ) + b
∗
t−1
(θ)M
t
(θ)
,
t=1
where
a
∗
t−1
= ρ
t
2
−
∂μ
t
(θ)
1
+
∂σ
t
2
(θ)
η
t
and (7.17)
∂θ m
t
∂θ
b
∗
t−1
= ρ
2
t
∂
∂
μ
θ
t
η
t
−
∂
∂
σ
θ
t
2
M
1
t
, (7.18)
−1
m,M
2
t
m,M
t
where ρ
2
t
=
1 −
m
t
M
t
and η
t
=
m
t
M
t
;
156 Handbook of Discrete-Valued Time Series
(b) The information I
g
∗
(θ) is given by
C
I
g
∗
(θ) =
n
ρ
2
∂μ
t
(θ)
∂μ
t
(θ) 1
+
∂σ
t
2
(θ)
∂σ
t
2
(θ)
1
C
t
∂θ
∂θ
m
t
∂θ
∂θ
M
t
t=1
∂μ
t
(θ)
∂σ
2
t
(θ)
∂σ
2
t
(θ)
∂μ
t
(θ)
− +
η
t
;
∂θ
∂θ
∂θ
∂θ
(c) The gain in information over the linear EF is
n
∂μ
t
(θ)
∂μ
t
(θ)
m, M
2
∂σ
2
t
(θ)
∂σ
2
t
(θ)
1
I
g
C
∗
(θ) − I
g
m
∗
(θ) = ρ
t
2
∂θ
∂θ
m
2
M
t
t
+
∂θ
∂θ
M
t
t=1
t
∂μ
t
(θ)
∂σ
t
2
(θ)
∂σ
t
2
(θ)
∂μ
t
(θ)
− +
η
t
;
∂θ
∂θ
∂θ
∂θ
(d) The gain in information over the quadratic EF is
n
I
g
∗
(θ) − I
g
∗
(θ) =
ρ
t
2
∂μ
t
(θ)
∂μ
t
(θ) 1
+
∂σ
2
t
(θ)
∂σ
2
t
(θ) m, M
t
2
C
M
∂θ
∂θ
m
t
∂θ
∂θ
m
t
M
t
−m, M
2
t=1
t
∂μ
t
(θ)
∂σ
t
2
(θ)
∂σ
t
2
(θ)
∂μ
t
(θ)
− +
η
t
;
∂θ
∂θ
∂θ
∂θ
(e) The recursive estimate for θ is given by
θ
t
=
θ
t−1
+ K
t
a
∗
t−1
(
θ
t−1
)m
t
(
θ
t−1
) + b
∗
t−1
(
θ
t−1
)M
t
(
θ
t−1
)
, (7.19)
K
t
= K
t−1
I
p
−
a
∗
t−1
(
θ
t−1
)
∂m
t
(
θ
t−1
)
+
∂a
t
∗
−1
(
θ
t−1
)
m
t
(
θ
t−1
)
∂θ
∂θ
+ b
t
∗
−1
(
θ
t−1
)
∂M
t
∂
(
θ
θ
t−1
)
+
∂b
t
∗
−
∂
1
(
θ
θ
t−1
)
M
t
(
θ
t−1
)
K
t−1
−1
, (7.20)
where I
p
is the p × p identity matrix and a
∗
t−1
and b
∗
t−1
can be calculated by substituting
θ
t−1
for θ in Equations (7.17) and (7.18), respectively;
(f) For the scalar parameter case, the recursive estimate of θ is given by
θ
t
=
θ
t−1
+ K
t
[a
∗
θ
t−1
)m
t
(
θ
t−1
) + b
∗
θ
t−1
)M
t
(
θ
t−1
)], where
t−1
(
t−1
(
157 Estimating Equation Approaches for Integer-Valued Time Series Models
∗
∂m
t
(
θ
t−1
)
∂a
t
∗
−1
(
θ
t−1
)
K
t
= K
t−1
1 −
a
t−1
(
θ
t−1
) +
m
t
(
θ
t−1
)
∂θ ∂θ
∂M
t
(
θ
t−1
)
∂b
t
∗
−1
(
θ
t−1
)
−1
+ b
∗
t−1
(
θ
t−1
) +
M
t
(
θ
t−1
)
K
t−1
. (7.21)
∂θ ∂θ
∗
Since −E
∂g
C
(θ)
F
t−1
denotes the optimal information matrix based on the rst t observations,
∂θ
∗
it follows that K
t
−1
=−
t
s=1
∂g
C
∂
(
θ
θ
s−1
)
can be interpreted as the observed information matrix
associated with the optimal combined EF g
C
∗
(θ). The proof of this theorem is given in Thavaneswaran
et al. (2015). �
In an interesting recent paper, Fokianos et al. (2009) described estimation for linear and
nonlinear autoregression models for Poisson count time series, and used simulation studies
to compare conditional least squares estimates with maximum likelihood estimates. Sim-
ilar to Fisher (1924), we compare the information associated with the corresponding EFs,
and show that the optimal EF is more informative than the conditional least squares EF.
In the class of estimating functions of the form G ={g
m
(θ) : g
m
(θ) =
n
1
a
t−1
(θ)m
t
(θ)},
t=
the optimal EF is given by g
m
∗
(θ) =
t
n
=1
a
t
∗
−1
(θ)m
t
(θ), where a
t
∗
−1
=
−
∂μ
∂
t
θ
(θ)
m
1
.The
t
optimal EF and the conditional least squares EF belong to the class G, and the optimal value
of a
t−1
is chosen to maximize the information. Hence I
g
∗
m
− I
g
CLS
is nonnegative denite. It
follows from page 919 of Lindsay (1985) that the optimal estimates are more efcient than
the conditional least squares estimates for any class of count time series models.
Note that g
n
∗
= 0 corresponds in general to a set of nonconvex, nonlinear equations.
The formulas for (7.17) through (7.21) may be easily coded as functions in R. For each
data/model combination, use of the EF approach in practice requires soft coding of the rst
four conditional moments, derivatives of the rst two conditional moments with respect
to model parameters, and specication of initial values to start the recursive estimation.
Example 7.6
Consider a zero-inated regression model. Let {y
t
}denote a count time series with excess
zeros, and assume that the mean, variance, skewness, and kurtosis of y
t
conditional on
F
t−1
are given by
μ
t
(θ) = (1 − ω
t
)λ
t
,
σ
2
t
(θ) = (1 − ω
t
)λ
t
(1 + ω
t
λ
t
),
ω
t
(1 + 2ω
t
)λ
2
+ 3ω
t
λ
t
+ 1
γ
t
(θ) =
t
,and
((1 − ω
t
)λ
t
)
1/2
(1 + ω
t
λ
t
)
3/2
ω
t
(6ω
2
− 6ω
t
+ 1)λ
3
+ 6ω
t
(2ω
t
− 1)λ
2
+ 7ω
t
λ
t
+ 1
κ
t
(θ) =
t
t
t
,
(1 − ω
t
)λ
t
(1 + ω
t
λ
t
)
2
therefore corresponding to the moments of a ZIP(λ
t
, ω
t
) distribution with pmf
ω
t
+ (1 − ω
t
) exp(−λ
t
),if y
t
= 0,
p(y
t
|F
t−1
) =
(1 − ω
t
) exp(−λ
t
)λ
y
t
t
/y
t
!,if y
t
> 0.
158 Handbook of Discrete-Valued Time Series
where λ
t
is the intensity parameter of the baseline Poisson distribution and ω
t
is the zero-
ination parameter. The ZIP model for count time series is an extension of the Poisson
autoregression discussed in Chapter 4 of Kedem and Fokianos (2002).
Suppose that λ
t
and ω
t
are parametrized by λ
t
(β) and ω
t
(δ), which are exible func-
tions of the unknown parameters β and δ and exogenous explanatory variables at time
t − 1, viz., x
t−1
and z
t−1
:
λ
t
(β) = exp(x
t
−1
β) and
exp(z
t−1
δ)
ω
t
(δ) =
1 + exp(z
t−1
δ)
. (7.22)
Let θ = (β
, δ
)
, which appears in the conditional mean and variance of y
t
.Let m
t
=
y
t
− μ
t
, M
t
= m
t
2
− σ
2
t
, and refer to λ
t
(β) and ω
t
(δ) by λ
t
and ω
t
, respectively. Then
m
t
= λ
t
[1 − ω
t
][1 + ω
t
λ
t
],
M
t
= (1 − ω
t
)λ
t
(ω
t
(4ω
2
t
− 4ω
t
+ 1)λ
t
3
+ 2ω
t
(4ω
t
− 1)λ
2
t
+ 5ω
t
λ
t
+ 2),
m, M
t
= (1 − ω
t
)λ
t
(ω
t
(1 + 2ω
t
)λ
2
t
+ 3ω
t
λ
t
+ 1).
Also,
∂μ
t
(1 − ω
t
)
∂λ
t
,
=
∂β
and
∂θ
−λ
t
∂ω
t
∂δ
∂σ
2
(1 − ω
t
)(1 + 2ω
t
λ
t
)
∂λ
t
t
=
∂β
.
∂θ
λ
t
(λ
t
(1 − 2ω
t
) − 1)
∂
∂
ω
δ
t
The combined optimal EF based on m
t
and M
t
is given by
n
g
∗
C
(θ) =
a
∗
t−1
m
t
+ b
∗
t−1
M
t
,
t=1
where
a
t
∗
−1
= ρ
2
t
−(1 − ω
t
)
∂
∂
λ
β
t
1
+
(1 − ω
t
)(1 + 2ω
t
λ
t
)
∂
∂
λ
β
t
η
t
,
λ
t
∂
∂
ω
δ
t
m
t
λ
t
(λ
t
(1 − 2ω
t
) − 1)
∂
∂
ω
δ
t
b
∗
−(1 − ω
t
)
∂
∂
λ
β
t
(1 − ω
t
)(1 + 2ω
t
λ
t
)
∂
∂
λ
β
t
1
t−1
= ρ
2
t
∂ω
t
η
t
−
∂ω
t
.
λ
t
∂δ
−λ
t
(λ
t
(1 − 2ω
t
) − 1)
∂δ
M
t
The corresponding information matrix is given by
I
∗
C
(θ) = I
∗
(θ) + I
∗
M
(θ)
m
2
n
2(1 − ω
t
)
2
(1 + 2ω
t
λ
t
)
∂λ
t
λ
t
(1 − ω
t
)(λ
t
(1 − 4ω
t
) − 2)
∂λ
t
∂ω
t
=−
ρ
2
t
∂β
∂β
2
∂δ
.
t=1
λ
t
(1 − ω
t
)(λ
t
(1 − 4ω
t
) − 2)
∂
∂
λ
β
t
∂
∂
ω
δ
t
−2λ
t
2
(λ
t
(1 − 2ω
t
) − 1)
∂
∂
ω
δ
t
159 Estimating Equation Approaches for Integer-Valued Time Series Models
The recursive estimator for θ follows from (7.19) and (7.20) as
∂
δ
t−1
))λ
t
(
−
∂β
λ
t
(β
t−1
)[y
t
−(1−ω
t
(
β
t−1
)]
λ
t
(
δ
t−1
)λ
t
(
∂
θ
t
=
θ
t−1
+ K
t
β
t−1
)(1+ω
t
(
β
t−1
))
,
ω
t
(
δ
t−1
)[y
t
−(1−ω
t
(δ
t−1
))λ
t
(β
t−1
)]
∂δ
(1−ω
t
(
δ
t−1
)λ
t
(β
t−1
))δ
t−1
))(1+ω
t
(
and
2
∂λ
t
(1−ω
t
)
1
∂β
λ
t
−
∂
∂
λ
β
t
∂
∂
ω
δ
t
K
t
= K
t−1
I
2
−
2
1 + ω
t
λ
t
∂ω
t
λ
t
−
∂λ
t
∂ω
t
∂δ
+
(y
t
−(1−ω
t
)λ
t
)
∂β ∂δ
1−ω
t
(1+ω
t
λ
t
)
2
2
−1
−
∂
∂
2
β
λ
2
t
λ
t
(1+ω
t
λ
t
)+
∂λ
t
(1+2ω
t
λ
t
)
∂λ
t
∂ω
t
∂β
λ
2
∂β ∂δ
×
t
∂
2
ω
t
2
K
t−1
.
∂
∂
λ
β
t
∂
∂
ω
δ
t
ω
t
∂δ
2
(1−ω
t
)(1+ω
t
λ
t
)−
∂
∂
ω
δ
t
(λ
t
+1−2ω
t
λ
t
)
−
(1−ω
t
)
(1−ω
t
)
2
For this example, it is also straightforward to show that the linear optimal EF and the
corresponding optimal coefcient are given by
∂λ
t
n
∂β
−
∗
λ
t
(1+ω
t
λ
t
)
g
m
(θ) =
∂ω
t
m
t
,and
t=1
∂δ
(1−ω
t
)(1+ω
t
λ
t
)
∂λ
t
−
∂β
∗
λ
t
(1+ω
t
λ
t
)
a
t−1
(θ) =
∂ω
t
.�
∂δ
(1−ω
t
)(1+ω
t
λ
t
)
Example 7.7
Consider an extended GAS(P, Q) model. As discussed in Section 7.3.3, suppose {y
t
} is an
integer-valued time series, its rst four conditional moments given F
t−1
are available,
and f
t
is modeled by (7.10), s
t
being a suitably chosen martingale difference. Suppose
the time-varying parameter f
t
corresponds to the conditional mean μ
t
(θ) = E(y
t
|F
t−1
).
Following the discussion in Theorem 7.1, it is natural to choose s
t
as a
t
∗
−1
(θ)m
t
(θ).
Suppose instead that f
t
corresponds to the conditional variance σ
2
t
(θ) = Var(y
t
|F
t−1
);a
natural choice of s
t
is b
∗
t−1
(θ)M
t
(θ).When f
t
is modeled by (7.10), the most informative
innovation is given by
s
t
= K
t
a
∗
t−1
(
θ
t−1
)m
t
(
θ
t−1
) + b
∗
t−1
(
θ
t−1
)M
t
(
θ
t−1
) , (7.23)
where K
t
is dened in (7.20) and a
t
∗
−1
and b
∗
t−1
can be calculated by substituting
θ
t−1
in
equations (7.17) and (7.18) respectively for the xed parameter θ.
When the form of the conditional distribution of y
t
given F
t−1
is available, and the
score function is easy to obtain, then the optimal choice for ∇
t
is the score function. How-
ever, in situations where we do not wish to assume an explicit form for the conditional
distribution, the optimal choice for ∇
t
is given by components of the optimal linear, or
quadratic, or combined EFs.
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