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This article provides an alternative approach to estimate the functional coefficient ARCH-M model given by Zhang, Wong and Li (2016) [1]. The new method has improvement in both computational and theoretical parts. It is found that the computation cost is saved and certain convergence rate for parameter estimation has been obtained.

ARCH-M model (Engle et al. [

Based on Chou et al. [

Here

this article, the superscript

For model (1), we need to estimate

Zhang et al. [

Firstly, given

Next, getting the estimator

Thirdly, calculating residuals

with respect to

It is shown in Zhang et al. [

The article is arranged as follows. In Section 2, we explain the idea about estimation approach. Section 3 lists the necessary assumptions to show the convergence results followed in Section 4. We conclude the paper in Section 5. Proofs of lemmas are put in the Appendix.

For model (1), we need to estimate

where,

Denote

For convenience of notation, we put

Further, define

where

In the above estimation procedure, we follow the ideas from Christensen et al. [

Remark 1. From (4), it can be seen that there is a simple specification between

The following assumptions will be adopted to show some asymptotic results. Throughout this paper, we let

Assumption 1. The kernel function

Assumption 2. The process

Assumption 3. The considered parameter space

Assumption 4.

Assumption 5. The function

Assumption 6.

Remark 2. Assumptions 1 - 3 are frequently adopted in the literature. Assumptions 4 - 5 have been analogously adopted by Yang [

Theorem 1. Suppose that Assumptions 1 - 6 hold. Then for any

Theorem 1 shows our estimators are consistent. The following Theorem 2 further gives certain convergence rate.

Theorem 2. Suppose that Assumptions 1 - 6 hold. Then for any

In order to prove Theorem 1 and 2, we need the following lemmas whose proofs can be found in the Appendix.

Lemma 1. For

Lemma 2. For

Proof of Theorem 1. From (7)-(8), it is not difficult to get

Here, for each

holds for certain finite M. Put

According to (A.18) and (A.19), (13)-(15), for certain M, it follows

Note

(12) and (17) give

which implies the consistency of

where

Proof of Theorem 2. According to (10) and (12), it follows

where,

From Theorem 1 and Lemmas 1 - 2,

In the above second equality, the first

From (A.9),

By the martingale central limit theorem (see, for example, Theorem 35.12 in Billingsley [

According to (19)-(23), it follows that

Moreover,

Conjecture. According to (19)-(25), if one can show

where

In this paper, a new approach is proposed to estimate the functional coefficient ARCH-M model. The proposed estimators are more efficient and, under regularity conditions, they are shown to be consistent. Certain convergence rate is also given.

Besides that the proof of conjecture in Section 4 needs further development, it is meaningful to further consider a GARCH type conditional variance in model (1). However, such an improvement is not trivial because the estimation method adopted in this paper can not be applied to the GARCH case. An alternative approach needs further development.

We thank the Editor and the referee for their comments. Research of X. Zhang and Q. Xiong is funded by National Natural Science Foundation of China (Grant No. 11401123, 11271095) and the Foundation for Fostering the Scientific and Technical Innovation of Guangzhou University. These supports are greatly appreciated.

Xingfa Zhang,Qiang Xiong, (2016) An Alternative Estimation for Functional Coefficient ARCH-M Model. Theoretical Economics Letters,06,647-657. doi: 10.4236/tel.2016.64070

Proof of Lemma 1

Proof. We only show the case of

written as

other cases. Then it is easy to have

Hence,

According to Assumption.6, it is easy to obtain the following equalities:

Note that

Proof of Lemma 2

Proof. We only consider the case of

Further,

Let

Then,

We can further have

From (A.9),

Here,

Note

In terms of (A.4)-(A.5),

Without loss of generality, there exists a

The last inequality comes from the fact

From Lemma 1, it follows that

(A.17)-(A.20) gives

Note that

(A.20)-(A.22) implies

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