The Theory And Empirical Research Of GARCH Models After The Introduction Of Higher Order Moments

Markowitz proposed the mean-variance analysis framework to create a measure of financial and Prevention of the first of its kind.With the theory of financial econometrics and modeling technology, based on the mean-variance analysis of financial risks have gradually can not meet investor demand. A lot of research shows that:the distribution of financial asset returns and the tail characteristics of the peak negative skewness make capital gains may be greater than the decline may rise; excessive kurtosis makes the existence of extreme events increase the likelihood of occurrence.Only consider the risk of second moments, while ignoring the third moment (skewness risk), fourth moment is clearly underestimated the risk, easy to losses to investors.Furthermore, the risk of sustained higher moments, consistency was confirmed.lf you only consider the higher order moments static risk, it ignores the risk of time-varying higher order moments, cannot get a comprehensive measure of financial risk.Engle (1982) and Bollerslev (1986) proposed the generalized autoregressive conditional Heteroskedasticity (GARCH) model allows the variance with time-varying sequence, for the price at the time of the study of further expansion. Harvey (1999) proposed autoregressive conditional volatility, skewness model (GARCHS) used to describe the time series of second moment and third moment of the dynamic characteristics, Jondeau (2003), Leon (2005) proposed autoregressive conditional fluctuations in skewness and kurtosis model (GARCHSK) to describe the time series for the second moment, third moment and the fourth moment of the dynamic characteristics.These models followed the Bollerslev (1986) (GARCH) structure, is the promotion of the GARCH model to the third moment and the fourth moment. In theory, because of financial assets, income distribution has heavy tail characteristics of the peak, then the income distribution is particularly important assumptions.The traditional assumption is a normal distribution, in order to consider the risk of higher moments, normal distribution is clearly not meet the research needs.This paper considers Jondeau and Rockinger (2001) proposed Gram-Charlier distribution and Hansen (1994) generalized t distribution, and analysis their specific features.However, the assumption only requires the choice of the distribution of asymmetry and fat tails of the distribution, donot give a good choice.In this paper, provide Gram-Charlier distribution and generalized t distribution of Hansen's analysis and comparison.Generalized t distribution of Hansen's strongly rely on its parameters, and because Gram-Charlier distribution is more easily estimated, we take it to carry out this empirical study, using eviws software model parameters maximum likelihood estimation.As the model of maximum likelihood function is highly nonlinear function, we estimate the simple models firstly, then the complex models, by the way, we take the estimated parameters of simple model as the initial values of complex models again.In this paper, the risk of higher order moments are introduced to the GARCH model, added the skewness equation and kurtosis equation.On the Shanghai and the Shenzhen Stock Composite Index, GARCHSK model that is superior to GARCH model, The measure of skewness risk and kurtosis risk has also been proved and the existence of higher moments of risk is significant.In order to study the existence of leverage in the stock market, leveraging effect parameter is added in GARCHSK model, then we get NAGARCHSK model.The empirical results show the existence of the Shanghai and Shenzhen Stock leverage, investors should guard against the risk in order to avert losses.The gradual deepening of theoretical research, makes the model more complete, accurate, from the GARCH model to NAGARCHSK model, model tends to complete, and gives the corresponding comparative analysis.