Hindawi Publishing Corporation
Journal of Probability and Statistics
Volume 2011, Article ID 372512, 14 pages
doi:10.1155/2011/372512
Research Article
Joint Estimation Using Quadratic
Estimating Function
Y. Liang,1 A. Thavaneswaran,1 and B. Abraham2
1
2
Department of Statistics, University of Manitoba, 338 Machray Hall, Winnipeg, MB, Canada R3T 2N2
University of Waterloo, Waterloo, ON, Canada N2L 2G1
Correspondence should be addressed to Y. Liang, umlian33@cc.umanitoba.ca
Received 12 January 2011; Revised 10 March 2011; Accepted 11 April 2011
Academic Editor: Ricardas Zitikis
Copyright q 2011 Y. Liang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A class of martingale estimating functions is convenient and plays an important role for
inference for nonlinear time series models. However, when the information about the first
four conditional moments of the observed process becomes available, the quadratic estimating
functions are more informative. In this paper, a general framework for joint estimation of
conditional mean and variance parameters in time series models using quadratic estimating
functions is developed. Superiority of the approach is demonstrated by comparing the information
associated with the optimal quadratic estimating function with the information associated with
other estimating functions. The method is used to study the optimal quadratic estimating functions
of the parameters of autoregressive conditional duration ACD models, random coefficient
autoregressive RCA models, doubly stochastic models and regression models with ARCH errors.
Closed-form expressions for the information gain are also discussed in some detail.
1. Introduction
Godambe 1 was the first to study the inference for discrete time stochastic processes using
estimating function method. Thavaneswaran and Abraham 2 had studied the nonlinear
time series estimation problems using linear estimating functions. Naik-Nimbalkar and
Rajashi 3 and Thavaneswaran and Heyde 4 studied the filtering and prediction problems
using linear estimating functions in the Bayesian context. Chandra and Taniguchi 5,
Merkouris 6, and Ghahramani and Thavaneswaran 7 among others have studied the
estimation problems using estimating functions. In this paper, we study the linear and
quadratic martingale estimating functions and show that the quadratic estimating functions
are more informative when the conditional mean and variance of the observed process
depend on the same parameter of interest.
2
Journal of Probability and Statistics
This paper is organized as follows. The rest of Section 1 presents the basics of
estimating functions and information associated with estimating functions. Section 2 presents
the general model for the multiparameter case and the form of the optimal quadratic
estimating function. In Section 3, the theory is applied to four different models.
Suppose that {yt , t 1, . . . , n} is a realization of a discrete time stochastic process,
and its distribution depends on a vector parameter θ belonging to an open subset Θ of the
p-dimensional Euclidean space. Let Ω, F, Pθ denote the underlying probability space, and
y
let Ft be the σ-field generated by {y1 , . . . , yt , t ≥ 1}. Let ht ht y1 , . . . , yt , θ, 1 ≤ t ≤ n be
specified q-dimensional vectors that are martingales. We consider the class M of zero mean
and square integrable p-dimensional martingale estimating functions of the form
M
gn θ : gn θ n
at−1 ht ,
1.1
t1
where at−1 are p × q matrices depending on y1 , . . . , yt−1 , 1 ≤ t ≤ n. The estimating functions
gn θ are further assumed to be almost surely differentiable with respect to the components
y
y
of θ and such that E∂gn θ/∂θ | Fn−1 and Egn θgn θ | Fn−1 are nonsingular for all
θ ∈ Θ and for each n ≥ 1. The expectations are always taken with respect to Pθ . Estimators of
θ can be obtained by solving the estimating equation gn θ 0. Furthermore, the p × p matrix
y
Egn θgn θ | Fn−1 is assumed to be positive definite for all θ ∈ Θ. Then, in the class of all
zero mean and square integrable martingale estimating functions M, the optimal estimating
function g∗n θ which maximizes, in the partial order of nonnegative definite matrices, the
information matrix
Ign θ −1 ∂gn θ
∂gn θ
y
y
y
| Fn−1
| Fn−1
E gn θgn θ | Fn−1
E
E
∂θ
∂θ
1.2
is given by
g∗n θ n
n −1
∂ht
y
y
E ht ht | Ft−1
| Ft−1
E
a∗t−1 ht ht ,
∂θ
t1
t1
y
1.3
and the corresponding optimal information reduces to Eg∗n θg∗n θ | Fn−1 .
The function g∗n θ is also called the “quasi-score” and has properties similar to those
of a score function in the sense that Eg∗n θ 0 and Eg∗n θg∗n θ −E∂g∗n θ/∂θ . This is
a more general result in the sense that for its validity, we do not need to assume that the true
underlying distribution belongs to the exponential family of distributions. The maximum
correlation between the optimal estimating function and the true unknown score justifies the
terminology “quasi-score” for g∗n θ. Moreover, it follows from Lindsay 8, page 916 that if
we solve an unbiased estimating equation gn θ 0 to get an estimator, then the asymptotic
variance of the resulting estimator is the inverse of the information Ign . Hence, the estimator
obtained from a more informative estimating equation is asymptotically more efficient.
Journal of Probability and Statistics
3
2. General Model and Method
Consider a discrete time stochastic process {yt , t 1, 2, . . .} with conditional moments
y
μt θ E yt | Ft−1 ,
y
σt2 θ Var yt | Ft−1 ,
γt θ κt θ 3
1
y
,
E
−
μ
|
F
y
θ
t
t
t−1
σt3 θ
2.1
4
1
y
E
−
μ
|
F
y
θ
t
t
t−1 − 3.
σt4 θ
That is, we assume that the skewness and the excess kurtosis of the standardized variable
yt do not contain any additional parameters. In order to estimate the parameter θ based
on the observations y1 , . . . , yn , we consider two classes of martingale differences {mt θ yt − μt θ, t 1, . . . , n} and {st θ m2t θ − σt2 θ, t 1, . . . , n} such that
2
y
y
mt E m2t | Ft−1 E yt − μt | Ft−1 σt2 ,
4
2
y
y
st E s2t | Ft−1 E yt − μt σt4 − 2σt2 yt − μt | Ft−1 σt4 κt 2,
2.2
3
y
y
m, st E mt st | Ft−1 E yt − μt − σt2 yt − μt | Ft−1 σt3 γt .
The optimal estimating functions based on the martingale differences mt
and st are g∗M θ − nt1 ∂μt /∂θmt /mt and g∗S θ − nt1 ∂σt2 /∂θst /st , respectively. Then, the information associated with g∗M θ and g∗S θ are Ig∗M θ n
n
2
2
∗
t1 ∂μt /∂θ∂μt /∂θ 1/mt and IgS θ t1 ∂σt /∂θ∂σt /∂θ 1/st , respectively.
Crowder 9 studied the optimal quadratic estimating function with independent observations. For the discrete time stochastic process {yt }, the following theorem provides optimality
of the quadratic estimating function for the multiparameter case.
Theorem 2.1. For the general model in 2.1, in the class of all quadratic estimating functions of the
form GQ {gQ θ : gQ θ nt1 at−1 mt bt−1 st },
a the optimal estimating function is given by g∗Q θ a∗t−1
m, s2t
1−
mt st
b∗t−1 1−
n
∗
t1 at−1 mt
b∗t−1 st , where
−1 m, s2t
mt st
∂σ 2 m, st
∂μt 1
−
t
∂θ mt
∂θ mt st
−1 ∂σt2
∂μt m, st
1
−
∂θ mt st
∂θ st
,
2.3
;
4
Journal of Probability and Statistics
b the information IgQ∗ θ is given by
Ig∗Q θ n
t1
m, s2t
1−
mt st
−1
∂σt2 ∂σt2 1
∂μt ∂μt 1
∂μt ∂σt2 ∂σt2 ∂μt
m, st
;
−
∂θ ∂θ mt ∂θ ∂θ st
∂θ ∂θ ∂θ ∂θ mt st
2.4
c the gain in information Ig∗Q θ − Ig∗M θ is given by
−1
n
∂σt2 ∂σt2 1
∂μt ∂μt m, s2t
∂μt ∂σt2 ∂σt2 ∂μt
m, s2t
m, st
;
1−
−
∂θ ∂θ m2t st ∂θ ∂θ st
∂θ ∂θ ∂θ ∂θ mt st
mt st
t1
2.5
d the gain in information Ig∗Q θ − Ig∗S θ is given by
n
−1 ∂σ 2 ∂σt2
∂μt ∂μt 1
m, s2t
m, s2t
1−
t
∂θ ∂θ mt
∂θ ∂θ mt st − m, s2t
mt st
∂μt ∂σt2 ∂σt2 ∂μt
m, st
.
−
∂θ ∂θ ∂θ ∂θ mt st
t1
2.6
Proof. We choose two orthogonal martingale differences mt and ψt st − σt γt mt , where the
conditional variance of ψt is given by ψt mt st − m, s2t /mt σt4 κt 2 − γt2 .
That is, mt and ψt are uncorrelated with conditional variance mt and ψt , respectively.
Moreover, the optimal martingale estimating function and associated information based on
the martingale differences ψt are
g∗Ψ θ
n
t1
∂μt m, st ∂σt2
−
∂θ mt
∂θ
ψt
ψ t
−1
m, s2t
1−
mt st
t1
∂σt2 m, st
∂σt2 1
∂μt m, s2t
∂μt m, st
mt st ,
×
−
−
∂θ m2t st
∂θ mt st
∂θ mt st
∂θ st
n
∂μt m, st ∂σt2
∂μt m, st ∂σt2
1
Ig∗Ψ θ −
−
∂θ
∂θ
m
m
ψ
∂θ
∂θ
t
t
t1
t
n
n
t1
×
m, s2t
1−
mt st
−1
∂σt2 ∂σt2 1
∂μt ∂μt m, s2t
−
∂θ ∂θ m2t st
∂θ ∂θ st
∂μt ∂σt2 ∂σt2 ∂μt
∂θ ∂θ ∂θ ∂θ
m, st
mt st
.
2.7
Journal of Probability and Statistics
5
Then, the quadratic estimating function based on mt and ψt becomes
n
−1
m, s2t
1−
mt st
t1
∂σt2 m, st
∂μt m, st ∂σt2 1
∂μt 1
×
−
−
mt st
∂θ mt ∂θ mt st
∂θ mt st ∂θ st
g∗Q θ 2.8
and satisfies the sufficient condition for optimality
∂gQ θ
y
y
E
| Ft−1 Cov gQ θ, g∗Q θ | Ft−1 K,
∂θ
2.9
∀gQ θ ∈ GQ ,
where K is a constant matrix. Hence, g∗Q θ is optimal in the class GQ , and part a follows.
Since mt and ψt are orthogonal, the information Ig∗Q θ Ig∗M θ Ig∗Ψ θ and part b follow.
∗
θi nor gS∗ θ is fully informative, that
Hence, for each component θi , i 1, . . . , p, neither gM
∗ θ and I ∗ θ ≥ I ∗ θ .
is, IgQ∗ θi ≥ IgM
i
gQ i
gS i
Corollary 2.2. When the conditional skewness γ and kurtosis κ are constants, the optimal quadratic
estimating function and associated information, based on the martingale differences mt yt − μt and
st m2t − σt2 , are given by
g∗Q θ
γ2
1−
κ
2
Ig∗Q θ −1
n
1
3
t1 σt
γ2
1−
κ
2
−1 γ ∂σt2
∂μt
−σt
∂θ κ 2 ∂θ
1
mt κ
2
n
γ 1
Ig∗M θ Ig∗S θ −
κ 2 t1 σt3
∂μt
1 ∂σt2
γ
−
∂θ σt ∂θ
∂μt ∂σt2 ∂σt2 ∂μt
∂θ ∂θ ∂θ ∂θ
st ,
.
2.10
3. Applications
3.1. Autoregressive Conditional Duration (ACD) Models
There is growing interest in the analysis of intraday financial data such as transaction and
quote data. Such data have increasingly been made available by many stock exchanges.
Unlike closing prices which are measured daily, monthly, or yearly, intra-day data or highfrequency data tend to be irregularly spaced. Furthermore, the durations between events
themselves are random variables. The autoregressive conditional duration ACD process
due to Engle and Russell 10 had been proposed to model such durations, in order to study
the dynamic structure of the adjusted durations xi , with xi ti − ti−1 , where ti is the time
of the ith transaction. The crucial assumption underlying the ACD model is that the time
dependence is described by a function ψi , where ψi is the conditional expectation of the
6
Journal of Probability and Statistics
adjusted duration between the i − 1th and the ith trades. The basic ACD model is defined
as
xi ψi εi ,
3.1
ψi E xi | Ftxi −1 ,
where εi are the iid nonnegative random variables with density function f· and unit
mean, and Ftxi −1 is the information available at the i − 1th trade. We also assume that εi
x
is independent of Ft−1
. It is clear that the types of ACD models vary according to different
distributions of εi and specifications of ψi . In this paper, we will discuss a specific class of
models which is known as ACD p, q model and given by
xt ψt εt ,
ψt ω p
aj xt−j j1
q
3.2
bj ψt−j ,
j1
maxp,q
aj bj < 1. We assume that εt ’s are iid nonnegative
where ω > 0, aj > 0, bj > 0, and j1
random variables with mean με , variance σε2 , skewness γε , and excess kurtosis κε . In order to
estimate the parameter vector θ ω, a1 , . . . , ap , b1 , . . . , bq , we use the estimating function
approach. For this model, the conditional moments are μt με ψt , σt2 σε2 ψt2 , γt γε , and
κt κε . Let mt xt − μt and st m2t − σt2 be the sequences of martingale differences such that
mt σε2 ψt2 , st σε4 κε 2ψt4 , and m, st σε3 γε ψt3 . The optimal estimating function and
associated information based on mt are given by g∗M θ −με /σε2 nt1 1/ψt2 ∂ψt /∂θmt
n
2
and Ig∗M θ μ2ε /σε2 t1 1/ψt ∂ψt /∂θ∂ψt /∂θ . The optimal estimating function and the
associated information based on st are given by g∗S θ −2/σε2 κε 2 nt1 1/ψt3 ∂ψt /∂θst
n
and Ig∗S θ 4/κε 2 t1 1/ψt2 ∂ψt /∂θ∂ψt /∂θ . Then, by Corollary 2.2 that the
optimal quadratic estimating function and associated information are given by
g∗Q θ
n
−με κε 2 2σε γε ∂ψt
με γε − 2σε ψt ∂ψt
1
mt st ,
2
∂θ
∂θ
σε κε 2 − γε2 t1
ψt2
σε ψt3
Ig∗Q θ γε2
1−
κε 2
−1 n
4με γε 1 ∂ψt ∂ψt
Ig∗M θ Ig∗S θ −
σε κε 2 t1 ψt2 ∂θ ∂θ 3.3
n
4σε2 μ2ε κε 2 − 4με σε γε 1 ∂ψt ∂ψt
,
2
2
2 ∂θ
∂θ σε κε 2 − γε
t1 ψt
the information gain in using g∗Q θ over g∗M θ is
2 n
2σε − με γε 1 ∂ψt ∂ψt
,
2
2
σε κε 2 − γε t1 ψt2 ∂θ ∂θ 3.4
Journal of Probability and Statistics
7
and the information gain in using g∗Q θ over g∗S θ is
2 n
με κε 2 − 2σε γε 1 ∂ψt ∂ψt
,
2
2
σε κε 2 − γε κε 2 t1 ψt2 ∂θ ∂θ 3.5
which are both nonnegative definite.
When εt follows an exponential distribution, με 1/λ, σε2 1/λ2 , γε 2, and κε 3.
Then, Ig∗M θ nt1 1/ψt2 ∂ψt /∂θ∂ψt /∂θ , Ig∗S θ 4/5 nt1 1/ψt2 ∂ψt /∂θ∂ψt /∂θ ,
n
and Ig∗Q θ t1 1/ψt2 ∂ψt /∂θ∂ψt /∂θ , and hence Ig∗Q θ Ig∗M θ > Ig∗S θ.
3.2. Random Coefficient Autoregressive Models
In this section, we will investigate the properties of the quadratic estimating functions for the
random coefficient autoregressive RCA time series which were first introduced by Nicholls
and Quinn 11.
Consider the RCA model
yt θ bt yt−1 εt ,
3.6
where {bt } and {εt } are uncorrelated zero mean processes with unknown variance σb2 and
variance σε2 σε2 θ with unknown parameter θ, respectively. Further, we denote the
skewness and excess kurtosis of {bt } by γb , κb which are known, and of {εt } by γε θ, κε θ,
respectively. In the model 3.6, both the parameter θ and β σb2 need to be estimated. Let
θ θ, β , we will discuss the joint estimation of θ and β. In this model, the conditional
2
β σε2 θ. The parameter θ
mean is μt yt−1 θ then and the conditional variance is σt2 yt−1
appears simultaneously in the mean and variance. Let mt yt − μt and st m2t − σt2 such that
3
2
4
2
σb2 σε2 , st yt−1
σb4 κb 2 σε4 κε 2 4yt−1
σb2 σε2 , m, st yt−1
σb3 γb σε3 γε .
mt yt−1
3
Then the conditional skewness is γt m, st /σt , and the conditional excess kurtosis is
κt st /σt4 − 2.
2
Since ∂μt /∂θ yt−1 , 0 and ∂σt2 /∂θ ∂σε2 /∂θ, yt−1
, by applying Theorem 2.1, the
optimal quadratic estimating function for θ and β based on the martingale differences mt and
st is given by g∗Q θ nt1 a∗t−1 mt b∗t−1 st , where
a∗t−1
m, s2t
1−
mt st
b∗t−1 1−
−1 m, s2t
mt st
yt−1
∂σ 2 m, st
−
ε
∂θ mt st
mt
−1 y2 m, st
, t−1
mt st
yt−1 m, st
1
−
∂θ st
mt st
∂σε2
,−
2
yt−1
st
,
3.7
.
8
Journal of Probability and Statistics
Hence, the component quadratic estimating function for θ is
∗
gQ
θ
n
−1
m, s2t
1−
mt st
t1
yt−1
yt−1 m, st ∂σε2 1
∂σε2 m, st
mt st ,
×
−
−
∂θ mt st
∂θ st
mt
mt st
3.8
and the component quadratic estimating function for β is
∗
gQ
−1 2
2
n
st
yt−1 m, st mt yt−1
m, s2t
1−
.
β −
mt st
mt st
st
t1
3.9
Moreover, the information matrix of the optimal quadratic estimating function for θ and β is
given by
I θ g∗Q
Iθθ Iθβ
Iβθ Iββ
,
3.10
where
Iθθ n
t1
m, s2t
1−
mt st
Iθβ Iβθ ⎞
−1 ⎛ 2
2
2
2 y
yt−1
s
m,
∂σ
∂σ
1
t−1
t
ε
ε
⎝
⎠,
−2
∂θ
∂θ mt st
mt
st
n
t1
m, s2t
1−
mt st
Iββ n
t1
−1 yt−1 m, st
∂σε2 1
−
∂θ st
mt st
m, s2t
1−
mt st
−1
4
yt−1
.
st
3.11
2
yt−1
,
3.12
3.13
In view of the parameter θ only, the conditional least squares CLS estimating
function and the associated information are directly given by gCLS θ nt1 yt−1 mt and
n 2 2 n 2
ICLS θ t1 yt−1 / t1 yt−1 mt . The optimal martingale estimating function and the
∗
∗ θ θ − nt1 yt−1 mt /mt and IgM
associated information based on mt are given by gM
n
2
t1 yt−1 /mt . Moreover, the inequality
n
t1
2
yt−1
mt
2
n y2
n
t−1
2
≥
yt−1
mt
t1
t1
3.14
∗ θ. Hence the optimal estimating function is more informative than
implies that ICLS θ ≤ IgM
the conditional least squares one. The optimal quadratic estimating function based on the
martingale differences mt and st is given by 3.8 and 3.11, respectively. It is obvious to see
Journal of Probability and Statistics
9
∗
∗
θ is larger than that of gM
θ. Therefore, we can conclude that
that the information of gQ
∗ θ ≤ I ∗ θ, and hence, the estimate obtained by solving
for the RCA model, ICLS θ ≤ IgM
gQ
the optimal quadratic estimating equation is more efficient than the CLS estimate and the
estimate obtained by solving the optimal linear estimating equation.
3.3. Doubly Stochastic Time Series Model
Random coefficient autoregressive models we discussed in the previous section are special
cases of what Tjøstheim 12 refers to as doubly stochastic time series model. In the nonlinear
case, these models are given by
y
yt θt f t, Ft−1 εt ,
3.15
where {θ bt } of 3.6 is replaced by a more general stochastic sequence {θt } and yt−1 is
y
replaced by a function of the past, Ft−1 . Suppose that {θt } is a moving average sequence of
the form
θt θ at at−1 ,
3.16
where {at } consists of square integrable independent random variables with mean zero and
y
variance σa2 . We further assume that {εt } and {at } are independent, then Eyt | Ft−1 depends
y
y
on the posterior mean ut Eat | Ft−1 , and variance vt Eat − ut 2 | Ft−1 of at . Under the
normality assumption of {εt } and {at }, and the initial condition y0 0, ut and vt satisfy the
following Kalman-like recursive algorithms see 13, page 439:
y
y
σa2 f t, Ft−1 yt − θ mt−1 f t, Ft−1
ut θ ,
y
σe2 θ f 2 t, Ft−1 σa2 vt−1
y
σa4 f 2 t, Ft−1
vt θ σa2 −
,
y
σe2 θ f 2 t, Ft−1 σa2 vt−1
3.17
where u0 0 and v0 σa2 . Hence, the conditional mean and variance of yt are given by
y
μt θ θ ut−1 θf t, Ft−1 ,
σt2 θ
σe2 θ
f
2
y
t, Ft−1
σa2
3.18
vt−1 θ ,
which can be computed recursively.
Let mt yt − μt and st m2t − σt2 , then {mt } and {st } are sequences of martingale
y
differences. We can derive that m, st 0, mt σe2 θ f 2 t, Ft−1 σa2 vt−1 θ, and st 10
Journal of Probability and Statistics
y
y
2
2σe4 θ 4f 2 t, Ft−1 σe2 θσa2 vt−1 θ 2f 4 t, Ft−1 σa2 vt−1 θ . The optimal estimating
function and associated information based on mt are given by
n ∂ut−1 θ mt
y
,
− f t, Ft−1 1 ∂θ
mt
t1
y
2
n f 2 t, F
t−1 1 ∂ut−1 θ/∂θ
∗ θ .
IgM
mt
t1
∗
gM
θ
3.19
Then, the inequality
⎛
⎞
y
2
2
n
n f 2 t, F
∂u
θ/∂θ
1
t−1
t−1
∂ut−1 θ
y
⎜
⎟
f 2 t, Ft−1 1 mt ⎝
⎠
∂θ
m
t
t1
t1
≥
n
y
f 2 t, Ft−1
1
t1
∂ut−1 θ
∂θ
2 2
3.20
implies that
2
y
f 2 t, Ft−1 1 ∂ut−1 θ/∂θ2
∗ θ,
ICLS θ ≤ IgM
2
n
2 t, Fy
f
∂u
θ/∂θ
1
m
t−1
t
t1
t−1
n
t1
3.21
∗
θ is more informative than the conditional
that is, the optimal linear estimating function gM
least squares estimating function gCLS θ.
The optimal estimating function and the associated information based on st are given
by
gS∗ θ
−
IgS∗ θ n
∂σe2 θ
t1
∂θ
n
∂σe2 θ
t1
∂θ
f
2
y
t, Ft−1
y
f 2 t, Ft−1
∂v
t−1 θ
∂θ
∂v
t−1 θ
∂θ
2
st
,
st
3.22
1
.
st
Hence, by Theorem 2.1, the optimal quadratic estimating function is given by
∗
gQ
θ −
n
t1
σe2 θ 1
y
f 2 t, Ft−1
σa2 vt−1 θ
⎛
⎞
y
2
2
t,
F
f
∂σ
θ/∂θ
θ/∂θ
∂v
t−1
e
t−1
∂u
y
⎜
⎟
t−1 θ
× ⎝ f t, Ft−1 1 mt st ⎠.
y
2
2
∂θ
2
σe θ f t, Ft−1 σa vt−1 θ
3.23
Journal of Probability and Statistics
11
∗ θ I ∗ θ, is given by
And the associated information, IgQ∗ θ IgM
gS
IgQ∗ θ n
2
t1 σe θ
⎛
f2
1
y
⎜
× ⎝f 2 t, Ft−1
y
t, Ft−1
1
σa2 vt−1 θ
∂ut−1 θ
∂θ
2
2 ⎞
y
∂σe2 θ/∂θ f 2 t, Ft−1 ∂vt−1 θ/∂θ
⎟
⎠.
y
σe2 θ f 2 t, Ft−1 σa2 vt−1 θ
3.24
∗
∗
It is obvious to see that the information of gQ
is larger than that of gM
and gS∗ , and hence, the
estimate obtained by solving the optimal quadratic estimating equation is more efficient than
the CLS estimate and the estimate obtained by solving the optimal linear estimating equation.
Moreover, the relations
f
∂ut θ
−
∂θ
2
y
y
t, Ft−1 σa2 1 ∂ut−1 θ/∂θ σe2 θ f 2 t, Ft−1 σa2 vt−1 θ
2
y
σe2 θ f 2 t, Ft−1 σa2 vt−1 θ
y
y
σa2 yt − f t, Ft−1 θ ut−1 θ ∂σe2 θ/∂θ f 2 t, Ft−1 ∂vt−1 θ/∂θ
−
,
2
y
2
2
2
σe θ f t, Ft−1 σa vt−1 θ
y
y
4 2
2
2
∂vt θ σa f t, Ft−1 ∂σe θ/∂θ f t, Ft−1 ∂vt−1 θ/∂θ
2
∂θ
y
σe2 θ f 2 t, Ft−1 σa2 vt−1 θ
3.25
can be applied to calculate the estimating functions and associated information recursively.
3.4. Regression Model with ARCH Errors
Consider a regression model with ARCH s errors εt of the form
yt xt β εt ,
y
y
3.26
2
2
such that Eεt | Ft−1 0, and Varεt | Ft−1 ht α0 α1 εt−1
· · · αs εt−s
. In this
2
model, the conditional mean is μt xt β, the conditional variance is σt ht , and the
conditional skewness and excess kurtosis are assumed to be constants γ and κ, respectively. It
12
Journal of Probability and Statistics
follows form Theorem 2.1 that the optimal component quadratic estimating function for the
parameter vector θ β1 , . . . , βr , α0 , . . . , αs β , α is
g∗Q β
1
κ 2
γ2
1−
κ
2
⎛⎛
−1
⎞
⎞ ⎞
⎛
n
s
s
1 ⎝⎝
1/2
1/2
−ht κ 2xt 2ht γ αj xt εt−j ⎠mt ⎝ht γxt − 2 αj xt εt−j ⎠st ⎠,
×
2
t1 ht
j1
j1
g∗Q α
−1
γ2
1−
κ
2
n
n 1
1/2
2
2
2
2
ht γ 1, εt−1 , . . . , εt−p mt −
1, εt−1 , . . . , εt−p st .
×
2
t1 ht
t1
1
κ 2
3.27
Moreover, the information matrix for θ β , α is given by
γ2
1−
κ
2
I
−1 Iββ Iβα
Iαβ Iαα
3.28
,
where
Iββ n
xt xt
ht
t1
Iβα −
2
2
2
2
4 1, εt−1
, . . . , εt−s
1, εt−1 , . . . , εt−s
,
h2t κ 2
s
2
2
n
h1/2
γ
x
−
2
α
x
ε
, . . . , εt−s
1, εt−1
t
t
j
t
t−j
j1
t
h2t κ 2
t1
Iαβ Iβα
−
n
2
2
, . . . , εt−s
1, εt−1
h2t κ
t1
Iαα h1/2
t γxt
−2
s
j1
αj xt εt−j
2
2
2
n 1, ε2 , . . . , ε2
1, εt−1 , . . . , εt−s
t−s
t−1
h2t κ 2
t1
,
3.29
,
.
It is of interest to note that when {εt } are conditionally Gaussian such that γ 0, κ 0,
⎡ ⎢
E⎣
s
j1
αj xt εt−j
2
2
, . . . , εt−s
1, εt−1
h2t κ 2
⎤
⎥
⎦ 0,
3.30
Journal of Probability and Statistics
13
the optimal quadratic estimating functions for β and α based on the estimating functions
mt yt − xt β and st m2t − ht , are, respectively, given by
g∗Q β
⎛
⎛
⎞ ⎞
n
n
s
1⎝
ht xt mt ⎝ αj xt εt−j ⎠st ⎠,
−
2
t1 ht
t1
j1
n
1 2
2
g∗Q α −
1,
ε
,
.
.
.
,
ε
t−s st .
t−1
2
t1 ht
3.31
Moreover, the information matrix for θ β , α in 3.28 has Iβα Iαβ 0,
Iββ s
s
n ht xt xt 2
j1 αj xt εt−j
j1 αj xt εt−j
t1
Iαα n 1, ε2 , . . . , ε2
t−s
t−1
t1
h2t
2
2
, . . . , εt−s
1, εt−1
2h2t
,
3.32
.
4. Conclusions
In this paper, we use appropriate martingale differences and derive the general form of
the optimal quadratic estimating function for the multiparameter case with dependent
observations. We also show that the optimal quadratic estimating function is more
informative than the estimating function used in Thavaneswaran and Abraham 2.
Following Lindsay 8, we conclude that the resulting estimates are more efficient in general.
Examples based on ACD models, RCA models, doubly stochastic models, and the regression
model with ARCH errors are also discussed in some detail. For RCA models and doubly
stochastic models, we have shown the superiority of the approach over the CLS method.
References
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72, no. 2, pp. 419–428, 1985.
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equations,” Journal of Time Series Analysis, vol. 9, no. 1, pp. 99–108, 1988.
3 U. V. Naik-Nimbalkar and M. B. Rajarshi, “Filtering and smoothing via estimating functions,” Journal
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9 M. Crowder, “On linear and quadratic estimating functions,” Biometrika, vol. 74, no. 3, pp. 591–597,
1987.
10 R. F. Engle and J. R. Russell, “Autoregressive conditional duration: a new model for irregularly spaced
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Journal of Time Series Analysis, vol. 1, no. 1, pp. 37–46, 1980.
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