# The Profile Likelihood Estimation for Single-Index ARCH(p)-M Model

Authors: Qiang Xiong; Yuan Li; XingFa Zhang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

We propose a class of single-index ARCH(p)-M models and investigate estimators of the parametric and nonparametric components. We first estimate the nonparametric component using local linear smoothing technique and then construct an estimator of parametric component by using profile quasimaximum likelihood method. Under regularity conditions, the asymptotic properties of our estimators are established.


---

## Body

## 1. Introduction

As an extension of autoregressive conditional heteroskedastic (ARCH) model, Engle et al. [1] introduced conditional variance into the conditional mean return equation and proposed the generalized ARCH-in mean (GARCH-M) model. In the past decades, (G)ARCH-M model has been widely studied by many researchers. It is useful to describe the relationship between the risk and return in finance. Consider a special conditional mean of ARCH-M model with the form y
t
=
δ
h
t
+
ε
t, where y
t denotes a return series, ε
t denotes an error series, and h
t denotes a conditional return volatility series. The above equality gives straightforward linear relationship between risk and return: high risk causes high return. The volatility coefficient δ can be addressed as relative risk aversion parameter in Das and Sarkar [2] and price of volatility in Chou et al. [3]. Many empirical studies have been done based on the above conditional mean. However, some researchers found that δ is time varying rather than constant. To capture the variation of the volatility coefficient, Chou et al. [3] proposed a time-varying parameter GARCH-M model. Zhang [4] introduced a functional coefficient GARCH-M model in which the volatility coefficient was treated as an unknown univariate function. Zhang's [4] model enables us to study the relationship between risk aversion and certain variable. However, Chou et al. [3] suggested that the risk aversion could be simultaneously influenced by some macroeconomic variables such as inflation rate, interest rate, and consumption price index. Consequently, it is reasonable to regard volatility coefficient as a function of several economic variables. In this paper, we will consider the following single-index ARCH(p)-M model:
(1)
y
t
=
m
(
β
τ
x
t
)
h
t
+
ε
t
,
ε
t
=
e
t
h
t
,
h
t
=
ω
0
+
ω
1
y
t
-
1
2
+
ω
2
y
t
-
2
2
+
⋯
+
ω
p
y
t
-
p
2
,
where ω
=
(
ω
0
,
ω
1
,
…
,
ω
p
)
τ and β
=
(
β
1
,
β
2
,
…
,
β
q
)
τ are p
×
1 and q
×
1 vectors of unknown parameters, respectively. The risk aversion m
(
·
) is an unknown smooth function of some macroeconomic variables x
t
=
(
x
t
1
,
x
t
2
,
…
,
x
t
q
)
τ. {
e
t
}
t
=
1
∞ is an independent and identically distributed (
i
.
i
.
d
.
)
(
0,1
) sequence. {
y
s
} is independent of {
e
t
} for any t
>
s and {
e
t
} is independent of {
h
t
,
x
t
}.  E
(
ε
t
2
∣
Ϝ
t
-
1
)
=
h
t is the conditional variance, Ϝ
t
-
1 denotes the sigma field generated by the information available up to time t
-
1. Let θ
=
(
ω
τ
,
β
τ
)
τ and the superscript τ denotes the transpose of a matrix or a vector.In (1), motivated by Xia and Li [5] and Xue and Pang [6], we assume a single-index form for the volatility coefficient to avoid the so-called curse of dimensionality problem. Namely, we treat the volatility coefficient as an unknown smooth function m
(
·
) which is a function of linear combination of certain explanatory variables. Following Ling [7] and Christensen et al. [8], we make a modification for the conditional variance h
t. Namely, we use the squared time-lagged observable returns series in the conditional variance equation rather than the squared unobservable errors series. Such a modification is helpful to model estimation in that h
t is deterministic once the parameter ω is known. For simplicity and estimability of model, we consider a ARCH(p) process for the conditional variance h
t. When h
t is observable, model (1) is reduced to the single-index coefficient regression models introduced by Xia and Li [5]. Model (1) is extended from Zhang's [4] model by generalizing the univariate function coefficient to multivariate function coefficient.The paper is arranged as follows. In Section2, we describe the estimation methods of nonparametric and parametric components and study their asymptotic properties. The concluding remark is given in Section 3. All technicalities are put in the Appendix.


## 2. Methodology and Results

In this paper, we will consider the model (1). For the sake of model identifiability, the commonly used assumptions are that ω
i
>
0
(
i
=
0,1
,
2
,
…
,
p
),  ∥
β
∥
=
1, and β
1
>
0.When we analyze the asymptotic properties of the estimator ofβ, we need to calculate the derivative of m
(
β
τ
x
t
) at the point β. However, ∥
β
∥
=
1 means that the true value of β is on the boundary of the unit sphere and it causes that m
(
β
τ
x
t
) does not have a derivative at the point β. Hence β needs to be reparameterized. We apply the “remove-one-component” method, introduced by Yu and Ruppert [9], on the parameter β. By removing β
1, let β
(
1
)
=
(
β
2
,
β
3
,
…
,
β
q
)
τ be a (
q
-
1
) dimensional parameter vector and we have β
=
:
β
(
β
(
1
)
)
=
(
β
1
,
β
(
1
)
τ
)
τ
,
β
1
=
(
1
-
∥
β
(
1
)
∥
2
)
1
/
2. The new parameter β
(
1
) satisfies the constraint ∥
β
(
1
)
∥
<
1. Then, m
(
β
τ
x
t
) is differentiable with respect to β
1. The Jacobian matrix of β with respect to β
1 is obtained by
(2)
J
β
(
1
)
=
∂
β
∂
β
(
1
)
=
(
-
(
1
-
∥
β
(
1
)
∥
2
)
-
1
/
2
β
(
1
)
τ
I
q
-
1
)
,
where I
q
-
1 is a (
q
-
1
) dimensional unit matrix.In this way, the parameter spaceΘ
=
𝔴
×
𝔅
(
1
), where
(3)
𝔴
=
{
ω
=
(
ω
0
,
ω
1
,
…
,
ω
p
)
τ
:
ω
i
>
0
,
i
=
0,1
,
2
,
…
,
p
}
,
𝔅
(
1
)
=
{
(
(
1
-
∥
β
(
1
)
∥
2
)
1
/
2
,
β
(
1
)
τ
)
τ
β
(
1
)
=
(
β
2
,
β
3
,
…
,
β
q
)
τ
:
β
=
(
(
1
-
∥
β
(
1
)
∥
2
)
1
/
2
,
β
(
1
)
τ
)
τ
,
∥
β
(
1
)
∥
<
1
}
.
Let θ
0
=
(
ω
0
τ
,
β
0
(
1
)
τ
)
τ and an inner point of Θ, denotes the true parameter point, where
(4)
ω
0
=
(
ω
00
,
ω
10
,
…
,
ω
p
0
)
τ
,
β
0
=
(
β
10
,
β
0
(
1
)
τ
)
τ
,
β
0
(
1
)
=
(
β
20
,
…
,
β
q
0
)
τ
,
β
10
=
(
1
-
∥
β
0
(
1
)
∥
2
)
1
/
2
.


### 2.1. Estimation Method

In this section, we give details of model estimation. We first estimate the nonparametric component by local linear smoothing technique and then construct an estimate of parametric component by the profile quasimaximum likelihood method.For any fixedz
∈
S
β
=
{
β
τ
x
:
x
∈
S
}, where S is a compact support of the density function of random variable x, we have
(5)
σ
(
z
;
β
)
=
:
E
(
y
t
∣
β
τ
x
t
=
z
)
=
E
(
m
(
β
τ
x
t
)
h
t
+
e
t
h
t
∣
β
τ
x
t
=
z
)
=
m
(
z
)
E
(
h
t
∣
β
τ
x
t
=
z
)
+
E
(
e
t
h
t
∣
β
τ
x
t
=
z
)
=
m
(
z
)
E
(
ω
0
+
∑
k
=
1
p
‍
ω
k
y
t
-
k
2
∣
β
τ
x
t
=
z
)
=
m
(
z
)
[
ω
0
+
∑
k
=
1
p
‍
ω
k
E
(
y
t
-
k
2
∣
β
τ
x
t
=
z
)
]
=
:
m
(
z
)
[
ω
0
+
∑
k
=
1
p
‍
ω
k
σ
k
(
z
;
β
)
]
.
Let
(6)
m
(
z
;
θ
)
=
σ
(
z
;
β
)
ω
0
+
∑
k
=
1
p
ω
k
σ
k
(
z
;
β
)
.
When the true values for parameter θ are known, say θ
0, we know m
(
β
0
τ
x
t
;
θ
0
)
=
m
(
β
0
τ
x
t
). Using local linear smoothing, we obtain the estimates of σ
(
z
;
β
), σ
k
(
z
;
β
), say σ
^
(
z
;
β
), σ
^
k
(
z
;
β
). Then we define the estimate of the nonparametric component m
(
z
;
θ
) in (6) given by
(7)
m
^
(
z
;
θ
)
=
σ
^
(
z
;
β
)
ω
0
+
∑
k
=
1
p
ω
k
σ
^
k
(
z
;
β
)
,
where
(8)
σ
^
(
z
;
β
)
=
∑
t
=
1
n
d
t
(
z
;
β
)
y
t
∑
t
=
1
n
d
t
(
z
;
β
)
,
σ
^
k
(
z
;
β
)
=
∑
t
=
1
n
d
t
(
z
;
β
)
y
t
-
k
2
∑
t
=
1
n
d
t
(
z
;
β
)
,
d
t
(
z
;
β
)
=
K
h
(
β
τ
x
t
-
z
)
{
S
n
,
2
(
z
;
β
)
-
(
β
τ
x
t
-
z
h
)
S
n
,
1
(
z
;
β
)
}
,
S
n
,
j
(
z
;
β
)
=
1
n
∑
t
=
1
n
‍
K
h
(
β
τ
x
t
-
z
)
(
β
τ
x
t
-
z
h
)
j
,
j
=
0,1
,
2
,
…
.
Let K
(
·
) denote a kernel density function and h denote the bandwidth, satisfying K
h
(
·
)
=
(
1
/
h
)
K
(
·
/
h
).Based on (7), we employ the quasimaximum likelihood approach to estimate parameter θ. For each θ
∈
Θ, let
(9)
L
(
θ
)
=
E
{
[
log
⁡
h
t
(
θ
)
+
ε
t
2
(
θ
)
h
t
(
θ
)
]
π
(
x
t
)
}
,
(10)
L
n
(
θ
)
=
1
n
∑
t
=
1
n
‍
{
[
log
⁡
h
t
(
θ
)
+
ε
t
2
(
θ
)
h
t
(
θ
)
]
π
(
x
t
)
}
,
(11)
L
^
n
(
θ
)
=
1
n
∑
t
=
1
n
‍
{
[
log
⁡
h
t
(
θ
)
+
ε
^
t
2
(
θ
)
h
t
(
θ
)
]
π
(
x
t
)
}
,
where
(12)
h
t
(
θ
)
=
ω
0
+
ω
1
y
t
-
1
2
+
ω
2
y
t
-
2
2
+
⋯
+
ω
p
y
t
-
p
2
,
h
t
=
h
t
(
θ
0
)
,
ε
t
(
θ
)
=
y
t
-
m
(
β
τ
x
t
;
θ
)
h
t
(
θ
)
,
ε
t
=
ε
t
(
θ
0
)
=
e
t
h
t
,
ε
^
t
(
θ
)
=
y
t
-
m
^
(
β
τ
x
t
;
θ
)
h
t
(
θ
)
,
and π
(
x
t
) is a nonnegative weight function whose compact support is contained in S. We can get the estimate of the parametric component θ; that is,
(13)
θ
^
n
=
arg
min
⁡
θ
∈
Θ
L
^
n
(
θ
)
.Remark 1.
(a) The weight functionπ
(
β
τ
x
t
) is substituted by π
(
x
t
), in that
(14)
∥
β
τ
x
t
-
β
τ
x
∥
≤
∥
β
∥
∥
x
t
-
x
∥
=
∥
x
t
-
x
∥
.
(b) In terms of the expression of h
t
(
θ
), we can get that
(15)
∂
h
t
(
θ
)
∂
θ
=
(
1
,
y
t
-
1
2
,
y
t
-
2
2
,
…
,
y
t
-
p
2
,
O
1
×
(
q
-
1
)
)
τ
,
∂
2
h
t
(
θ
)
∂
θ
∂
θ
τ
=
O
(
p
+
q
)
×
(
p
+
q
)
,
where O
s
×
r denotes a s
×
r zero matrix.


### 2.2. Asymptotic Properties

In order to obtain the asymptotic properties of the estimators of parametric and nonparametric component, we first introduce some regularity conditions as follows.(A1)
The probability density functionf
(
β
τ
x
) of the random variable β
τ
x is bounded and satisfies inf
⁡
β
τ
x
∈
S
β
⁡
f
(
β
τ
x
)
>
0, where S
β
=
{
β
τ
x
:
x
∈
S
} and S is a compact support set of the probability density function p
(
x
) of x. f
(
·
) and p
(
·
) have two bounded and continuous derivatives.
(A2)
The parameter spaceΘ is a bounded metric space, satisfying θ
L
≤
θ
≤
θ
U.
(A3)
For anyz
∈
S
β, θ
∈
Θ, k
=
1,2
,
…
,
p, σ
(
z
;
β
) and σ
k
(
z
;
β
) are uniformly bounded and have continuous second-order derivatives, satisfying Lipschitz condition of order 1.
(A4)
For anyz
∈
S
β, θ
∈
Θ, m
(
z
;
θ
), ∥
∂
m
(
z
;
θ
)
/
∂
θ
∥ and ∥
∂
2
m
(
z
;
θ
)
/
∂
θ
∂
θ
τ
∥ are uniformly bounded.
(A5)
For eachθ
∈
Θ, x
∈
S, j
=
1,2, E
(
h
t
j
(
θ
)
∣
x
t
=
x
) is bounded away from infinity.
(A6)
The process{
x
t
,
y
t
} is stationary and ergodic. The process {
e
t
,
x
t
,
y
t
} is α-mixing with the mixing coefficients satisfying the condition α
(
k
)
≤
a
k
-
b for some positive constants a, b
>
r
/
(
r
-
2
) and r
>
2. e
t satisfies E
(
|
e
t
|
4
+
δ
)
<
∞ for some δ
>
0.
(A7)
The functionL
(
θ
) defined in (9) is continuous and has unique minimum θ
0.
(A8)
The kernelK
(
·
) is a bounded and symmetric probability density function with a bounded support and derivative and satisfies
(16)
∫
-
∞
∞
‍
u
2
K
(
u
)
d
u
≠
0
,
∫
-
∞
∞
‍
|
u
|
l
K
(
u
)
d
u
<
∞
,
l
=
1,2
,
…
.Remark 2.
Condition (A1) means that the probability density function of the random variableβ
τ
x
t is positive, which is used to ensure that the denominators of nonparametric estimators are bounded away from 0 with high probability. Conditions (A2) and (A7) have been analogously adopted by Yang [10]. Conditions (A3)–(A6) are essential conditions for the asymptotic properties of the estimators. Condition (A6) is a standard assumption in time series literature. Condition (A8) is commonly assumed smoothness condition for the consistency of kernel based estimators.In Section2.1, we have described two-step estimation method and obtain the final estimate of the nonparametric component, that is, m
^
(
z
;
θ
^
n
). We now establish the asymptotic results for the estimates of θ and m
(
·
). And the related rates of convergence are given.Theorem 3.
If conditions (A1)–(A8) are satisfied, then asn
→
∞ and h
→
0 such that n
h
→
0, it holds that
(17)
sup
⁡
θ
∈
Θ
|
σ
^
(
z
;
β
)
-
σ
(
z
;
β
)
|
=
O
p
{
h
2
+
(
log
⁡
n
n
h
)
1
/
2
}
,
sup
⁡
θ
∈
Θ
|
σ
^
k
(
z
;
β
)
-
σ
k
(
z
;
β
)
|
=
O
p
{
h
2
+
(
log
⁡
n
n
h
)
1
/
2
}
,
22222222222222222222222222222
k
=
1,2
,
…
,
p
.Theorem 4.
Under the conditions of Theorem3, it holds that
(18)
sup
⁡
θ
∈
Θ
|
m
^
(
z
;
θ
)
-
m
(
z
;
θ
)
|
=
O
p
{
h
2
+
(
log
⁡
n
n
h
)
1
/
2
}
.Theorem 5.
Under the conditions of Theorem3, consistency for our estimates holds: θ
^
n
→
p
θ
0, m
^
(
z
;
θ
^
n
)
→
p
m
(
z
;
θ
0
).Theorem 6.
Under the conditions of Theorem3, if n
h
2
→
∞ and n
h
4
→
0, n
(
θ
^
n
-
θ
0
) has an asymptotic normal distribution with mean zero and covariance matrix Σ
I
-
1
Σ
Σ
I
-
1, where Σ
I and Σ are given in (A.101) and (A.105) in Appendix, respectively.


## 2.1. Estimation Method

In this section, we give details of model estimation. We first estimate the nonparametric component by local linear smoothing technique and then construct an estimate of parametric component by the profile quasimaximum likelihood method.For any fixedz
∈
S
β
=
{
β
τ
x
:
x
∈
S
}, where S is a compact support of the density function of random variable x, we have
(5)
σ
(
z
;
β
)
=
:
E
(
y
t
∣
β
τ
x
t
=
z
)
=
E
(
m
(
β
τ
x
t
)
h
t
+
e
t
h
t
∣
β
τ
x
t
=
z
)
=
m
(
z
)
E
(
h
t
∣
β
τ
x
t
=
z
)
+
E
(
e
t
h
t
∣
β
τ
x
t
=
z
)
=
m
(
z
)
E
(
ω
0
+
∑
k
=
1
p
‍
ω
k
y
t
-
k
2
∣
β
τ
x
t
=
z
)
=
m
(
z
)
[
ω
0
+
∑
k
=
1
p
‍
ω
k
E
(
y
t
-
k
2
∣
β
τ
x
t
=
z
)
]
=
:
m
(
z
)
[
ω
0
+
∑
k
=
1
p
‍
ω
k
σ
k
(
z
;
β
)
]
.
Let
(6)
m
(
z
;
θ
)
=
σ
(
z
;
β
)
ω
0
+
∑
k
=
1
p
ω
k
σ
k
(
z
;
β
)
.
When the true values for parameter θ are known, say θ
0, we know m
(
β
0
τ
x
t
;
θ
0
)
=
m
(
β
0
τ
x
t
). Using local linear smoothing, we obtain the estimates of σ
(
z
;
β
), σ
k
(
z
;
β
), say σ
^
(
z
;
β
), σ
^
k
(
z
;
β
). Then we define the estimate of the nonparametric component m
(
z
;
θ
) in (6) given by
(7)
m
^
(
z
;
θ
)
=
σ
^
(
z
;
β
)
ω
0
+
∑
k
=
1
p
ω
k
σ
^
k
(
z
;
β
)
,
where
(8)
σ
^
(
z
;
β
)
=
∑
t
=
1
n
d
t
(
z
;
β
)
y
t
∑
t
=
1
n
d
t
(
z
;
β
)
,
σ
^
k
(
z
;
β
)
=
∑
t
=
1
n
d
t
(
z
;
β
)
y
t
-
k
2
∑
t
=
1
n
d
t
(
z
;
β
)
,
d
t
(
z
;
β
)
=
K
h
(
β
τ
x
t
-
z
)
{
S
n
,
2
(
z
;
β
)
-
(
β
τ
x
t
-
z
h
)
S
n
,
1
(
z
;
β
)
}
,
S
n
,
j
(
z
;
β
)
=
1
n
∑
t
=
1
n
‍
K
h
(
β
τ
x
t
-
z
)
(
β
τ
x
t
-
z
h
)
j
,
j
=
0,1
,
2
,
…
.
Let K
(
·
) denote a kernel density function and h denote the bandwidth, satisfying K
h
(
·
)
=
(
1
/
h
)
K
(
·
/
h
).Based on (7), we employ the quasimaximum likelihood approach to estimate parameter θ. For each θ
∈
Θ, let
(9)
L
(
θ
)
=
E
{
[
log
⁡
h
t
(
θ
)
+
ε
t
2
(
θ
)
h
t
(
θ
)
]
π
(
x
t
)
}
,
(10)
L
n
(
θ
)
=
1
n
∑
t
=
1
n
‍
{
[
log
⁡
h
t
(
θ
)
+
ε
t
2
(
θ
)
h
t
(
θ
)
]
π
(
x
t
)
}
,
(11)
L
^
n
(
θ
)
=
1
n
∑
t
=
1
n
‍
{
[
log
⁡
h
t
(
θ
)
+
ε
^
t
2
(
θ
)
h
t
(
θ
)
]
π
(
x
t
)
}
,
where
(12)
h
t
(
θ
)
=
ω
0
+
ω
1
y
t
-
1
2
+
ω
2
y
t
-
2
2
+
⋯
+
ω
p
y
t
-
p
2
,
h
t
=
h
t
(
θ
0
)
,
ε
t
(
θ
)
=
y
t
-
m
(
β
τ
x
t
;
θ
)
h
t
(
θ
)
,
ε
t
=
ε
t
(
θ
0
)
=
e
t
h
t
,
ε
^
t
(
θ
)
=
y
t
-
m
^
(
β
τ
x
t
;
θ
)
h
t
(
θ
)
,
and π
(
x
t
) is a nonnegative weight function whose compact support is contained in S. We can get the estimate of the parametric component θ; that is,
(13)
θ
^
n
=
arg
min
⁡
θ
∈
Θ
L
^
n
(
θ
)
.Remark 1.
(a) The weight functionπ
(
β
τ
x
t
) is substituted by π
(
x
t
), in that
(14)
∥
β
τ
x
t
-
β
τ
x
∥
≤
∥
β
∥
∥
x
t
-
x
∥
=
∥
x
t
-
x
∥
.
(b) In terms of the expression of h
t
(
θ
), we can get that
(15)
∂
h
t
(
θ
)
∂
θ
=
(
1
,
y
t
-
1
2
,
y
t
-
2
2
,
…
,
y
t
-
p
2
,
O
1
×
(
q
-
1
)
)
τ
,
∂
2
h
t
(
θ
)
∂
θ
∂
θ
τ
=
O
(
p
+
q
)
×
(
p
+
q
)
,
where O
s
×
r denotes a s
×
r zero matrix.


## 2.2. Asymptotic Properties

In order to obtain the asymptotic properties of the estimators of parametric and nonparametric component, we first introduce some regularity conditions as follows.(A1)
The probability density functionf
(
β
τ
x
) of the random variable β
τ
x is bounded and satisfies inf
⁡
β
τ
x
∈
S
β
⁡
f
(
β
τ
x
)
>
0, where S
β
=
{
β
τ
x
:
x
∈
S
} and S is a compact support set of the probability density function p
(
x
) of x. f
(
·
) and p
(
·
) have two bounded and continuous derivatives.
(A2)
The parameter spaceΘ is a bounded metric space, satisfying θ
L
≤
θ
≤
θ
U.
(A3)
For anyz
∈
S
β, θ
∈
Θ, k
=
1,2
,
…
,
p, σ
(
z
;
β
) and σ
k
(
z
;
β
) are uniformly bounded and have continuous second-order derivatives, satisfying Lipschitz condition of order 1.
(A4)
For anyz
∈
S
β, θ
∈
Θ, m
(
z
;
θ
), ∥
∂
m
(
z
;
θ
)
/
∂
θ
∥ and ∥
∂
2
m
(
z
;
θ
)
/
∂
θ
∂
θ
τ
∥ are uniformly bounded.
(A5)
For eachθ
∈
Θ, x
∈
S, j
=
1,2, E
(
h
t
j
(
θ
)
∣
x
t
=
x
) is bounded away from infinity.
(A6)
The process{
x
t
,
y
t
} is stationary and ergodic. The process {
e
t
,
x
t
,
y
t
} is α-mixing with the mixing coefficients satisfying the condition α
(
k
)
≤
a
k
-
b for some positive constants a, b
>
r
/
(
r
-
2
) and r
>
2. e
t satisfies E
(
|
e
t
|
4
+
δ
)
<
∞ for some δ
>
0.
(A7)
The functionL
(
θ
) defined in (9) is continuous and has unique minimum θ
0.
(A8)
The kernelK
(
·
) is a bounded and symmetric probability density function with a bounded support and derivative and satisfies
(16)
∫
-
∞
∞
‍
u
2
K
(
u
)
d
u
≠
0
,
∫
-
∞
∞
‍
|
u
|
l
K
(
u
)
d
u
<
∞
,
l
=
1,2
,
…
.Remark 2.
Condition (A1) means that the probability density function of the random variableβ
τ
x
t is positive, which is used to ensure that the denominators of nonparametric estimators are bounded away from 0 with high probability. Conditions (A2) and (A7) have been analogously adopted by Yang [10]. Conditions (A3)–(A6) are essential conditions for the asymptotic properties of the estimators. Condition (A6) is a standard assumption in time series literature. Condition (A8) is commonly assumed smoothness condition for the consistency of kernel based estimators.In Section2.1, we have described two-step estimation method and obtain the final estimate of the nonparametric component, that is, m
^
(
z
;
θ
^
n
). We now establish the asymptotic results for the estimates of θ and m
(
·
). And the related rates of convergence are given.Theorem 3.
If conditions (A1)–(A8) are satisfied, then asn
→
∞ and h
→
0 such that n
h
→
0, it holds that
(17)
sup
⁡
θ
∈
Θ
|
σ
^
(
z
;
β
)
-
σ
(
z
;
β
)
|
=
O
p
{
h
2
+
(
log
⁡
n
n
h
)
1
/
2
}
,
sup
⁡
θ
∈
Θ
|
σ
^
k
(
z
;
β
)
-
σ
k
(
z
;
β
)
|
=
O
p
{
h
2
+
(
log
⁡
n
n
h
)
1
/
2
}
,
22222222222222222222222222222
k
=
1,2
,
…
,
p
.Theorem 4.
Under the conditions of Theorem3, it holds that
(18)
sup
⁡
θ
∈
Θ
|
m
^
(
z
;
θ
)
-
m
(
z
;
θ
)
|
=
O
p
{
h
2
+
(
log
⁡
n
n
h
)
1
/
2
}
.Theorem 5.
Under the conditions of Theorem3, consistency for our estimates holds: θ
^
n
→
p
θ
0, m
^
(
z
;
θ
^
n
)
→
p
m
(
z
;
θ
0
).Theorem 6.
Under the conditions of Theorem3, if n
h
2
→
∞ and n
h
4
→
0, n
(
θ
^
n
-
θ
0
) has an asymptotic normal distribution with mean zero and covariance matrix Σ
I
-
1
Σ
Σ
I
-
1, where Σ
I and Σ are given in (A.101) and (A.105) in Appendix, respectively.


## 3. Concluding Remark

In this paper, we consider a single-index ARCH(p)-M model. We applied profile likelihood approach to investigate the estimators for the parametric and nonparametric components. We first estimate the nonparametric component using local linear smoothing technique then construct an estimator of parametric component by using profile quasimaximum likelihood method. Under some regularity conditions, the asymptotic properties, namely, consistency and asymptotic normality with related rates of convergence, are derived.



---

*Source: 189426-2014-02-20.xml*