Mean Square Error And Optimal Bandwidth Of Kernel-type Quantile Estimator For Value At Risk Under ρ-mixing Assumption

Along with the financial crisis happens more and more frequently, universality and destruc-tive impact, financial risk management has became the important part of financial affairs.How to avoid the financial risk effectively became the main topic of the financial practitioners studied. In recent years, VaR and CVaR(Conditional Value at Risk) are widely recognized and applied in risk measures.A major challenge faced by practitioners is to calculate them accurately.The estimation method of VaR can be roughly divided into parametric estimation and non-parametric estimation method.In practice,the distribution forms of the financial return series are complex,it is hard for us to confirm the specific distribution of return series. If you use parametric estimation method it may cause model error easily. Comparatively speaking, nonparametric esti-mation method do not need to assume the statistical distribution of financial return series, and it can deal the fat-tail and non-symmetric of financial return series.This article studies the Kernel-type quantile estimator of VaR:v(h,λ)=-Tn(λ), where Tn(λ) is quantile estimation, it's form as followsThis quantile estimation was first proposed by Parzen, then many scholars discussed it's properties. Simon gave it's mean square error and a bandwidth selection method in independent random variables.Wei Xianglan discussed it's Bahadur expression,strong consistence and mean square error in a-mixing series. This paper is to discuss the VaR kernel-type estimation under p-mixing sequences, and to give the mean square error by using the Bahadur expression of Shanchao Yang. Moreover, we deliver a selection method of optimal bandwidth by using the reference to a standard distribution and the specific expression of optimal bandwidth. The reference to a standard distribution method is easy and settable than the bandwidth selection method in Simon.In recent years, the research of stock market shows that there exists long-rang dependence features in the changes of yield, such as fractional Brownian motion can very well describe in the stock return volatility. The fourth part is numerical simulation,we use the optimal bandwidth given in the third part to estimate VaR in different situations, like fractional Brownian motion,Gaussian distribution and Student distribution. Then we compare the performance of the kernel-type quantile estimator of VaR with a common order statistic estimator,the simulations show that the kernel-type estimator is more accurate and it's a reasonable VaR estimator. Finally, the demonstration analysis on the data of Shanghai index and hang seng index during the period from March 26 in 2002 to March 18 in 2010 shows that the investment risk in stock market of Hong Kong is bigger than in stock market of Shanghai.