Bayesian Quantile Regression And Its Application In Financial Risk Management

In history,quantile regression originated from the problem of l1 estimation.As a kind of semi-parametric method,quantile regression has many special advantages.Quantile regression can put up with the high peak and fat tail phenomenon and can solve structural catastrophe problem.In recent years,quantile regression has become more and more popular,it has been applied in economic,finance,biology,medical and many other areas.And it still attracts plenty of scholars to do theoretical and application research.One direction of the quantile regression is the combination with the Bayesian analysis.Bayesian statistic combines the prior information with the data set to do posterior inference,the result of Bayesian estimation is more explanato-ry and more abundant.One difficult problem in the application of Bayesian statistic is the complication of posterior distribution,sometimes the poste-rior distribution even doesn' t exist closed form.However,benefited from the fast development of computer performance,Markov Chain Monte Carlo method has become popular in almost every area of statistic.MCMC method skipped the close form of posterior distribution,sampled from the posterior distribution,and the random numbers formed a markov chain whose stable distribution is the posterior distribution.This solved the difficulty of Bayesian estimation,and the estimation is valid on the condition that the frequency in the sampling is large enough.In the field of measuring and modeling financial risk,VaR is a very impor-tant measure.Usually the calculation of VaR needs to assume a probability distribution for financial market returns,and then estimated VaR values based on the probability distribution of.People do not need to assume that probabili-ty distribution in quantile regression theory,and can directly use the regression equation to estimate.Additionally it can be conveniently obtained for various confidence levels of VaR value,by quantile regression.This paper first reviews the classical theory of quantile regression,intro-duces the basic framework of Bayesian analysis and the contents of MCMC method;then by using Asymmetric Laplace distribution and generalized in-verse Gauss distribution,we are able to bring the quantile regression into the framework of Bayesian analysis.This paper presents a algorithm based on hy-brid of local variables and Gibbs sampling.At the same time this paper shows the equivalence of Lasso and Bayesian quantile regression with some special priors.Finally,we discuss the empirical research of the quantile regression in Chinese stock market,Bayesian quantile regression model in the calculation of VaR is very simple and effective.