A functional coefficient GARCH-M model

ABSTRACT

Motivated by the time varying property of the risk aversion and the functional coefficient regression model, a functional coefficient GARCH-M model is studied. The proposed GARCH-M type model gives a way to study the relationship between risk aversion and certain variable. An approach is given to estimate the model and some theoretical results are obtained. Simulations demonstrate that the method performs well. From the empirical studies, it is shown that the proposed model can better fit the considered data compared to the usual parametric models.

Keywords:

• Consistency
• Functional coefficient
• GARCH-M model
• Risk aversion

Mathematics Subject Classification:

• Primary 62J02
• Secondary 62P20

Acknowledgments

The authors are grateful to an anonymous referee for useful comments, which led to improvements in the presentation of the paper.

Funding

Heung Wong’s research was supported by research grants from the Research Committee of The Hong Kong Polytechnic University. Yuan Li’s work was partially supported by National Natural Science Foundation of China (Grant No. 11271095) and Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20124410110002).

Notes

1 According to (2.1(2.1) ), h_t is determined by observations {y_s}^{t − 1}_{s = −∞}when φ₀ is known. Such an idea has been adopted by many authors such as Ling (2004, 2007) and Zhang et al. (2011). Under certain stationarity conditions, var(ϵ_t) = E(h_t) = E[v(y_t, φ₀)]/(1 − β₀).

2 Initial value for θ can be obtained by estimating model (1.1(1.1) ) with m( · ) being a constant δ and the sample variance of {y_t}^T_{t = 1} can be used as an initial value for h_t or h_t(θ).

3 When θ = (ω, α, β)^τ, h_t(θ) = ω + αf(y_{t − 1}) + βh_{t − 1}(θ), f(.) ⩾ 0, θ ∈ [ω_L, ω_U] × [α_L, α_U] × [β_L, β_U], θ_U = (ω_U, α_U, β_U)^τ satisfies (A.6(A.6) ).