An Alternative Estimation for Functional Coefficient ARCH-M Model

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.


Introduction
ARCH-M model (Engle et al. [2]) has been widely studied in last decades due to its various applications.Specially, ARCH-M model gives a way to study the relationship between return and the volatility in finance (for instances, see [3] [4]).Let t y denote the excess return of a market and t h denote the corresponding conditional vola- tility at time t.A frequently applied conditional mean in ARCH-M models is t t t y h δ ε = + with t ε being an error term.The above equality gives a straightforward linear relationship between volatility and return: high volatility (risk) causes high return.The volatility coefficient δ can be addressed as relative risk aversion para- meter in Das and Sarkar [5] and price of volatility in Chou et al. [6].Many empirical studies have been done based on the above conditional mean.However, some researchers found δ nonconstant and counter-cyclical [7]- [9].To capture the variation of the volatility coefficient δ , Chou et al. [6] studied a time-varying parameter GARCH-M.In their GARCH-M model, the volatility coefficient was assumed to follow a random walk, namely with t v being an error term.Based on Chou et al. [6], it makes sense to study the ARCH-M model with a time-varying volatility coefficient.Motivated by the functional coefficient model, Zhang et al. [1] consider a class of functional coefficient (G) X. F. Zhang, Q. Xiong ARCH-M models.For simplicity, we focus on the functional coefficient ARCH-M model of the form ( ) ( ) ( ) , , , p a a a It is shown in Zhang et al. [1] that the above estimation is consistent.However, there is no concrete convergence rate.Moreover, it can be seen that in the above estimation, ( ) ˆt m U depends on ( ) t h θ and hence depends on θ .However, there is no simple or explicite expression between them, which will make the calculation a bit time-consuming.In this article, a new simple estimator is given for model (1), which is shown to be consistent and convergence rate is also obtained.
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.

( )
f u to be the probability density function of t u .Let A be a compact subset of R with nonempty interior and satisfies ( ) , .[14]).Then we can define a estimator for ( ) For convenience of notation, we put

Assumptions
The following assumptions will be adopted to show some asymptotic results.Throughout this paper, we let , M m denote certain positive constants, which may take different values at different places.

Assumption 1. The kernel function ( )
. k is a bounded density with a bounded support [ ] , where A is a compact subset of R with nonempty interior.Further, there are constants m and M such that ( ) Assumption 3. The considered parameter space Θ is a bounded metric space.The process ( ) { } ( (7) has an unique minimum point at 0 θ ∈ Θ .Assumption 6. ( ) ( ) , where n b is the bandwidth such that 0 n b → and for some 2, 0, 2.5, s Assumptions 1 -3 are frequently adopted in the literature.Assumptions 4 -5 have been analogously adopted by Yang [16].In Assumption 6, the boundness is regular.When the bandwidth n b suffices the described conditions and the processes { } 1 , n t t t y U = satisfies certain mixing conditions, the uniform convergence holds for local linear regression method (Fan and Yao [14], Theorem 6.5).

Asymptotic Results
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.

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

Proof of Lemma 2
Proof.We only consider the case of 2 k = , other cases can be obtained with similar and easier arguments.From ( 5)-( 6), From (A.9),