Research On Bayesian Analysis Of Autoregressive Conditional Heteroscedaticity Models And Their Application

The classical estimation methods such as the maximum likelihood estimation (MLE) have been always employed to estimate the GARCH models. The classical methods are difficult to use in numerical optimization of the objective function, which is not necessarily convex. And constraints imposed on GARCH coefficients complicate statistical inference on the coefficients as well as the optimization procedures. However, in the Bayesian approach we compute integrals of the posterior distribution in terms of nuisance parameters to estimate them instead of the maximum of the likelihood function. Moreover, estimation of the probability of an inequality is straightforward in the Bayesian approach, thus constraints on GARCH coefficients are also easily handled by using the truncated posterior distribution of the GARCH coefficients. Meanwhile, we can resolve the difficulties of the high-dimension numerical integral by using MCMC methods and WinBUGS software. Therefore, we may avoid these problems in the classical methods by using the Bayesian approach. We try to do some things in this aspect.Firstly, we analyzed the ARCH family of models systemically and their classification. We also showed the classical methods such as the MLE and BHHH algorithm and explained their disadvantages in detail.Secondly, we discussed how to carry out the Bayesian inference on the linear GARCH. We then applied Griddy-Gibbs sampler to implement the model simulation with WinBUGS, and accomplished the Bayesian estimation of the model. In the end, we analyzed validly the volatility of the SSCI.Furthermore, we explained Bayesian analysis of AR-GJR-GARCH model, and employed a Markov-chain sampling technique with the Metropolis-Hastings algorithm to generate samples from the joint posterior distribution because the analytical knowledge of conditional posterior densities is not available for its Bayesian inference. Then we used these samples to estimate the model. It proves that this method is objective and valid in analysis of the volatility of the stock market in China.Finally, we derived a type of GARCH models in which multiple structural changes in the conditional variance are modeled explicitly and the number of the changes is known. After the introduction of auxiliary variables, estimation of the model is made possible by the use of the Gibbs sampler. We applied this model to the USD/RMB foreign-exchange-rate daily return series, and found that the highly persistent artificially result from structural changes.This paper includes the conclusion and innovation points as follows. Firstly, it discusses the Bayesian inference on the linear GARCH model, AR-GJR-GARCH model and structural changing GARCH model. Secondly, it develops Griddy-Gibbs sampling, M-H sampling and Gibbs sampling using auxiliary variable respectively for those models to simulate their conditional posterior densities. And combining MCMC methods with WinBUGS software for data analysis, it designs three WinBUGS procedures to resolve the difficulties of the high-dimension numerical integral. Thirdly, it proves these methods are valid in analysis of the volatility of the stock market and foreign-exchange market in China by these methods.