Statistical Analysis Of Heteroscedastic Models

Measuring the volatility of financial risk is an important field in finance.Volatilityin the article is the variance of asset return,which varies with time going,and this is alsocalled heteroscedasticity in Econometrics .Many high-frequency financial time seriesappear heteroscedastic .There are two methods of measuring volatility:One is ARCHmodels,including ARCH,GARCH and other extended models,the other one is SVmodel .These two models have been widely applied in modeling and research ofeconomic field ,especially of financial markets. In this paper,ARCH model ,GARCH model and SV model are introduced indetail and their poperties and characteristics are analyzed.In fact ,there are manysuccessful studies in ARCH model areas.The theoretical autocorrelation function forGARCH(1,1) model is derived and approximated and found to be exponentiallydecreasing.Then we show that contemporaneous aggregation of indepentment univariateGARCH(1,1) processes yields a weak GARCH(2,2) process ,subsequently,in two important special cases,we analyze the dependence of the parameters in theaggregate on the parameters in the underlying models,and then show that the parametersafter aggregation depend on the variance ratio and the kurtosis parameters. For stochastic volatility, we have established the relationship between SV modeland ARMA model.By using stochastic differential equation ,we show the relationshipbetween ARCH models and SV models .At last,the comparative research between theSV models and the GARCH models is carried out about their abilities to describe andcopy those facts.