The Time Series Analysis Of Index Returns In Chinese Stock Market

The volatility of stock returns have important theoretical and practicality meaning in the applications ofrisk management and asset pricing. Volatility clusters (ie. Volatility may be high for certain periods and low for other periods) are commonly seen in finicial time series. Engle (1982) promote the Au-toregressive Conditional Heteroskedastic (ARCH) model to capture the serial dependence. Bolleslev (1986) popularized the ARCH model to be Generilized Autoregressive Conditional Heteroskedastic (GARCH) model. Also, finicial time series, which marginal density distribution have higher excess kurtosis, typically exhibit the feature of heavy tail. Although we can obtain heavier tail marginal distribution than innovations' distribution by the GARCH process, the normal GARCH model can't interpret all the excess kurtosis unfortunately. A popular candidate is the GARCH-t model, which used the t distribution innovations assumption. The discussion of the GARCH process excess kurtosis make some sences only if the freedom of t innovation distribution greater than 4, that is to say, we must constrain the freedom of t innovations distribution greater than 4, otherwise, the ability of capture the tails behavior for the model will be cut down.In this paper, we considered the Normal Scale Mixture (NSM) distribution and proved an useful property about the kurtosis of NSM, which can lead to the conclution that NSM distribution have enough excess kurtosis (only if select appropriate parameters' value). We also proposed the theoretical frame of GARCH model based on the NSM distribution innovations distribution and give the parameters estimation procedure using EM algorithm, then introduced the SEM algorithm to obtain asymptotic variance-covariance matrices given by Meng & Rubin. Finally, modelling for Shanghai Securities Index returns time series of Chinese stock market, and comparing GARCH-NSM model with other popular models in model specification tests using Hong & Li (2002, 2005) nonparametric model specification tests.Our study shows that, the likelihood value and goodness-of-fit seems to be improved by add the autoregressive items to the random walk model in certain extent, whereas can't be improve the model specification errors essentially. Introducing the GARCH effects in the model can signficantly improve the goodness-of-fit and reduce the model specification errors, but remain be overwhelmingly rejected by nonparametric model specification tests, which means normal GARCH model still exist model specification errors. We further find that introduce the NSM innovations distribution into GARCH model can improve the model performance ability greatly and reduce the model specification errors. More important, GARCH-NSM model pass the nonparametric model specification tests under the Significant levels 0.05, which can be used in the researches of the risk management and asset pricing.