Bayesian Estimation Of Stochastic Volatility Models By Integrated Nested Laplace Approximation Method

The volatility is a characteristic of financial time series in China.Since the stochastic volatility(SV)model has been put forward,it can describe the char-acteristics of financial volatility effectively,and it has been widely used in the study of financial volatility.The volatility set in the stochastic volatility model is random variable that cannot be directly observed,and the likelihood function is very complex.Therefore,it is difficult to estimate the parameters in the s-tochastic volatility model by maximum likelihood method.However,Bayesian method can effectively estimate t,he parameters in the SV model.The accuracy is good with Bayesian estimation by the Markov chain Monte Carlo(MCMC)method.However,the running time of MCMC method is so long that its practi-cal application is limited.This paper introduces a efficient method—integrated nested Laplacian approximation(INLA)method to achieve Bayesian inference for the SV model of Chinese stock market.Through the empirical analysis of the Shanghai Composite Index and the Shenzhen Stock Exchange Index,after comparing results of the INLA method with the MCMC method in terms of pa-rameter estimation results,DIC values and programs running time,we find that the INLA method can quickly and accurately achieve the Bayesian inference of the stochastic volatility model.So this method can expand the application space of the SV model in the financial field.This article applies the SV model based on the INLA method to the empirical analysis of stock market in China.Innovatively,we use DIC to compare the inference effects of the MCMC method with the INLA method.The main content of this paper include the following three parts:1.Introduction of Volatility and Stochastic Volatility ModelThis paper introduces the concept of volatility,the main characteristic of volatility of financial time series and the basic statistics of financial time series.Then we introduce AR(1)model,standard stochastic volatility model and thick tail stochastic volatility model.The likelihood function expression of stochastic volatility model is given.2.Introduction of Bayesian Estimation Method for Stochastic Volatility ModelThis paper uses MCMC method and INLA method to make Bayesian infer?ence.This part introduces Bayesian inference,MCMC method and Gibbs sam-pling method.Gibbs sampling method of standard SV model is given.And we introduce Laplace approximation.We discuss in detail the approximation of the distribution of parameter posterior distribution and latent variable posterior distribution by the INLA method.3.Empirical Analysis of Shanghai Composite Index and Shenzhen Stock Ex-change IndexWe use Bayesian estimation by INLA method and MCMC method to estimate the parameters in the standard SV model,SV-T model of Shanghai Composite Index.Correspondingly,we use same methods to study the standard SV model,SV-T model of Shenzhen Stock Exchange Index.We compare MCMC method with INLA method in the estimation of parameters,DIC values and running time of programs.