A Markov-Chain Sampling Algorithm And Empirical Analysis For ARMA-GARCH-M Models

Markov Chain Monte Carlo (MCMC) algorithms have achieved a considerable following in the statistics and econometrics literature in the last ten years. There has been considerable research on so-called generalized autoregressive conditional heteroskedastic (GARCH) models for dealing with these methods since the remarkable works of Chib and Greenberg (1994). Compared to MLE procedures, MCMC algorithms are more stable and the problems such as searching the multiple maximal are avoided.In order to allow the mean value of returns to depend upon volatility in the market, Engle,Lilien and Robins(1987) introduced the GARCH-M models. Here we developed the general ARMA(p,q)-GARCH(r,s)-M(k) models, which maybe become increasingly important for estimating volatility returns and exogenous shocks for finance data. After we present the posterior distribution of the model and the full conditional distributions of all the parameters of the model, we develop a hybrid Metropolis-Hastings algorithm for estimating the parameters of ARMA-GARCH-M models based on the works of Bayesian Chib and Greenberg (1994) and Nakatsuma (2000). Here we simplified the estimations in MA and GARCH block. Using the data of Shanghai security market index (1998/01/01－2001/12/31), we estimate an AR-GARCH-M model for the return of Shanghai market, and analyse the integration property of the data and the risk effect of the market. In the second part of our paper (chapter 5), we only use the classical statistical methods to analyse the risk characteristic of Chinese stock markets by using two GARCH (1,1)-M models. We find that the IGARCH (1,1)-M model have almost the same efficacy with the EGARCH (1,1)-M model in Shanghai market and the former is a little better than the latter in Shenzhen market. Then we forecast the volatility of the two index's returns.These results are very important to describe the features between the risk compensates and volatilities in Chinese stock markets.