Some Properties Of Kernel Estimation Of Value At Risk For ρ-mixing Financial Time Series

Today, VaR is widely recognized and applied risk measure, VaR method is to use astructured methodology to think about risk. Facing to risks, to be calculated VaR value of theprocess, Financial institutions may use VaR method to avoid the loss to the establishment ofa risk management to monitor the market, it has a positive meaning.Recently,chen(2005, [24]) discussed the VaR kernel estimator and properties, the non-parametric and parametric estimations comparative analysis and estimation errors forα-mixing sequences. However, the article did not give the estimated VaR convergence speedof the asymptotic normal. This paper, inspired by chen(2005, [24]), discusses VaR kernelestimation convergence speed for the strictly stationaryρ-mixing sequences, and gives therelevant Bahadur expression and the convergence speed of the asymptotic normal by usingthe moment inequality tools of Shanchao Yang((1997, [22]). Finally, we make the empiricalanalysis of the Shanghai and Shenzhen stock index.In this paper, the strictly stationaryρ-mixing sequences {X_t : t≥1}, given p∈(0, 1),VaR value in 1 - p Confidence levels is:ν_p = -inf{u : F(u)≥p},The distribution Xt is F(x) in it. We consider the following of the equation:F|^_n,h(x) = pIts solution isν_p,h which is the kernel estimation ofν_p, Fn,h(x)is the kernel estimation ofF(x)(from the definition of (2-1)).This paper gives the following assumptions:Condition1: The sequence{X_t : t≥1}is strictly stationary andρ-mixing,ρ(n) =O(n^-λ) (λ> 1), t≥1 here, X_t is continuously distributed with F(x) and f(x) as itsdistribution and density functions respectively .Condition2: f(ν_p) > 0,p∈(0, 1), f(·) has continuous thirdly derivative in a neighbor- hood in the B(ν_p) ofν_p; the thirdly partial derivatives of Fk, which is the joint distributionfunction of (X₁,X_k+1), k≥1, are all bounded in B(ν_p) uniformly with respect to k.Condition3: K(·) is a univariate probability density function,has continuous boundedsecond derivative and satisfies the following moment conditions:Condition4: The smoothing bandwidth h satisfies h�'0, nh^3-β�'∞for any (?)β> 0,and nh⁴log² n�'0 as n�'∞.From the above assumptions we can get to several conclusions:Theorem 2.1: (strong convergence properties)Let conditions 1 ? 4hold, thenTheorem 2.2: (Bahadur expression)Let conditions 1 ? 4hold,thenTheorem 2.3: (uniformly asymptotic normal properties) Let conditions 1 ? 4hold,ifwhere , andΦ(·) is the standardize normaldistribution function.Compared with chen(2005, [24]), the conditions and results of this paper have severaldifferent points: firstly, this paper discussesρ-mixing dependent sequence, but the text of[24] discussedα-mixing dependent sequence; secondly, theν_p,h convergence rate in theorem2.1 is n^-1/2 log^1/2(n) which tends to 0 is a little fasterν_p,h convergence rate which wasn^-1/2 log(n) in theorem 1 of the [24] text; thirdly, Bahadur expression ofν_p,h is givenin theorem 2.2 of this paper ,but the [24] text was given; fourthly, and only asymptoticnormal properties in theorem 3 of the [24] text was given,ν_p,h convergence rate of uniformlyasymptotic normal properties in this theorem 2.3 of this paper is given.This paper first introduces risk and risk management, as well as risk measure VaR andsome of the estimated VaR methods and the course of VaR development; secondly givesseveralρ-mixing sequences assumptions, and provides the main conclusions about the con-vergence and asymptotic normal properties; then presents the prepared lemma for the mainlytheory, given to the results of proof; finally, according the VaR kernel estimation theory wemakes Shanghai and Shenzhen index of empirical studies.