MCMC Algorithm For The Doubly-Truncated Model And The Long-time Memory Of Stock Market

MCMC algorithms were developed by Chib and Greenberg(1994) for the untruncated ARMA model. The classical algorithms such as MLE were widely used to study time series models before, but they are limited for the complicated time series models. Compared to MLE procedures, MCMC algorithms are more stable and the problems such as searching the multiple maximal are avoided, and the rate of convergence is also faster than the classical methods. In recent years MCMC methods have been widely attended by the statistician, and have become an important subject in time series analysis. There are a lot of papers concerning about MCMC algorithms in time series models. Nakatsuma, T. (2000a) and Goldman, E. et al (2000b) developed MCMC procedures for ARMA models and ARMA-GARCH models separately, Goldman, E. and Tsurumi, S. (2001) developed MCMC algorithms for doubly-truncated ARMA-GRACH models, these results all showed the advantages of MCMC algorithms. In this paper we develop a MCMC procedure for the doubly-truncated ARMA-GARCH-M model which maybe become increasingly important for estimating volatility returns and exogenous shocks for finance data.Because the practical financial data are always limitary, for example the monetary authorities manage to limit the exchange rate to control the market risk, it is very significant to study doubly-truncated models. On the other hand, using OLS to estimate the parameter of the doubly-truncated data is prone to neglect the heteroskedastic characteristic of stochastic errors, consequently result in the basis estimation of parameter. Therefore we develop a hybrid Metropolis-Hastings algorithms to estimate the parameter of doubly-truncated ARMA-GRACH-M models. In this paper, we construct doubly-truncated ARMA-GRACH-M models first. After giving the posterior distribution and proposal distribution of parameters we develop the iterative processes of the M-H algorithm. To illustrate the validity of M-H method, we consider an example with simulated data and finally doubly-truncated ARMA-GRACH-M models of exchange rate series are estimated.The other part of this paper is to discuss the long-time memory of return rate of shanghai and shenzhen stock market, and we use the ARFIMA(p,d,q) models. We discuss the estimation methods of parameters in ARFIMA(p,d,q) model, especially the estimation of the parameter d. We use Hurst Exponent method to estimate the parameter d, and study the classical R/S method, biased modified R/S method and unbiased modified R/S method. In particular, we analyze the return rate of shanghai and shenzhen stock market using those R/S methods and give the comparative results. The results indicate that Chinese stock market has shown long-term memory. On this condition, we give the optimum ranks and the estimations of all parameters in ARFIMA(p,d,q) model, accordingly we educe the optimum ARFIMA models for the return of shanghai and shenzhen stock market.