Goodness Of Fit Test Of Multivariate Skew-t Distribution And Its Application

Value at risk(VaR) is a standard risk measure in financial risk management.Portfolio is an excellent investment method to spread asset risk.Based on the volatility model of portfolio,using Monte Carlo method to simulate VaR can accurately predict the risk status of the portfolio.In the application of statistics,a common problem is whether a group of observation data obey a known distribution.In other words,whether the known distribution can be used to fit this set of data.This is the problem of goodness of fit test.If the judgment of the overall distribution is wrong,the subsequent statistica l inference based on this judgment will have a large deviation.Skew-t distribution can well describe the "peak thick tail and asymmetry" of time series,so it is widely used in different industries.However,there is no substantial progress on the goodne ss of fit test of multivariate skew-t distribution.Unit spherical uniform distribution is the basic probability distribution for constructing multivariate distribution.Therefore,the goodness of fit test of multivariate skew-t distribution can be transformed into that of spherical uniform distribution by data conversion.This paper introduces the estimation method of multivariate skew-t distribution parameters.The results of random simulation experiments show that with the increase of sample size,the estimation error gradually decreases and tends to zero.Then,a method for estimating the degrees of freedom of multivariate skew-t distribution based on Ander son-Darling(AD)test statistics is introduced.The feasibility of this method is verified by random simulation experiments.Based on the canonical form of multivariate skew-t distribution,the multivariate skew-t distribution is finally transformed into spherical uniform distribution through the data transformation process.Based on the feature test met hod of spherical uniform distribution,a new goodness of fit test method of multivariate skew-t distribution is proposed,and the transformation process is realized through the algorithm.In the subsequent empirical analysis,using the above goodness of fi t test method,the goodness of fit test of multivariate GARCH model innovation subject to multivariate skew-t distribution is carried out to test whether the model distribution can better fit the residual sequence.Through practical application,the practi cability of the proposed goodness of fit test method for multivariate skew-t distribution is verified.In the empirical analysis,we select the closing prices of six stocks in different industry sectors,and establish a multivariate GARCH-skew-t model based on the return of each asset.The parameters of GARCH model are obtained by using quasi maximum likelihood estimation,and then the parameter estimation of multivariate skew-t distribution is obtained.The degree of freedom of distribution is estimated by using Anderson darling statistics.The innovation of standardized residual sequence vector is tested by fitting goodness o f fit which obeys multivariate skew-t distribution based on characteristic test.After passing the test,Monte Carlo simulation technology is used to predict the VaR of six stocks and portfolios.Random simulation experiments show that using multivariate skew-t distribution to describe the innovation distribution of multivariate GARCH model can better describe the tail characteristics of the data,more accurately reflect the actual characteristics of the distribution of financial returns,prevent overvaluation or undervaluation of VaR to a certain extent,and have a better risk measurement effect.By comparing the average VaR of six stocks with the VaR of portfolio,it is found that portfolio investment can effectively reduce the risk of loss.