Estimation Based On Distance Of Density For ARCH Models

Estimation Based on Distance of Density for ARCH ModelsARCH model have become the most popular and extensively studied financial econometric models in past twenty years. Chandra and Taniguchi (2006) proposed the minimumα-divergence estimation for ARCH models. In this paper, we propose the Estimation Based on Distance of Density for ARCH Models using techniques of the minimumα-divergence estimation for ARCH models and prove that the proposed estimator is consistent.In the first chapter, we introduce several basic concepts.Firstly, a class of ARCH(p) processes is characterized by the equations where {ε_t} is a sequence of i.i.d.(0,1) random variables with probability density g(x) andε_t is independent of X_s, s＜t. It is assumed that b₀＞0, b_i≥0, 1≤i＜p, and b₁+...+b_p＜1.Secondly, we introduce the concept ofα-divergence functionals. We propose a postulated model f_θ,θ∈H, where H is a compact subset of R_k and g is a true probability density. Theα-divergence from f_θto g is given by whereK_α(z)={4/(1-α²)}{1-z^(1+α)/2}, -1＜α＜1.The limit of (1.2.1) asα→-1 reduces to the well-known Kullback-Liebler information:α=0, the 0-divergence is the Hellinger distance,Lastly, we present the introduction of kernel density estimator.In the second chapter, we introduce three various distance of density and propose the Estimation Based on these Distances of Density for ARCH Models using techniques of the minimumα-divergence estimation for ARCH models and prove that the proposed estimator is consistent.Based on the Hellinger distance, we give three various distance of density as follow.whereΦ(x) is a bounded probability density which is even and monotone decreasing in positive side. Takingas example, we construct Estimation Based on Distance of Density for ARCH Models and prove that this new estimator is consistent.At first, we present (?)_n(x) which is the nonparametric kernel density estimator of the innovation density g(x) according the techniques of the minimumα-divergence estimation for ARCH models.Then, we give the definition of T₁ based on D₁(f_θ,g). A functional T₁ defined on (?) is determined by the requirement that for every g∈(?),Here, denote by (?) the set of all bounded probability densities with respect to Lebesque measure on R, f_t is a postulated model, t∈H, where H is a compact subset of R^k. T₁(g) may be multiple-valued to indicate any one of the possible values, chosen arbitrarily.Under assumption 2.2.1, we establishes the existence, continuity and uniqueness of T₁(g) in theorem 2.2.1.Assumption 2.2.1(â…°) f_θ(x) and g(x) are uniformly bounded continuous, and are integrable functions with respect to x.(â…±) f_θ^1/2 and g^1/2 are integrable functions on R.(â…²) g has mean 0 and variance 1.Theorem2.2.1 Suppose thatθ₁≠θ₂ implies f_θ1≠f_θ2 on a set of positive Lebesgue measure, and for almost every x, f_θ(x) is continuous inθ. Then (a) For every g∈f, there exists a value T₁(g)∈H satisfying (2.2.4).(b) If T₁(g) is unique, and if g_n∈(?) and is uniformly bounded, g_n(?)g, then T₁(g_n)→T₁(g), as n→∞.(c) T₁(f_θ)=θuniquely for everyθ∈H.Thus, we present T₁((?)_n) as the estimator of T₁(g), and under assumption 2.2.2 prove that this new estimator is consistent in Theorem 2.2.2.Assumption2.2.2(â…°) W is even, and twice continuously differentiable with compact support Sw.(â…±) W″is bounded.(â…²) c_n= O(n^-λ) with 1/4＜λ＜1/3.Theorem 2.2.2 Suppose that Assumption 2.2.1 and 2.2.2 hold, then(a)‖(?)_n-g‖(?)0, as n→∞.(b) If T₁(g) is unique, then T₁((?)_n)(?)T₁(g), as n→∞.This theorem is proved by Chandra and Taniguchi.In the last chapter, we give the estimation value (?)_n(T₁((?)_n))ofθusing random data and test the approximation between f_(?)n(x) and g(x) by graph.