A High Order Arch Model

Since its advent, the ARCH model has been used for depicting of and forecasting in financial time series extensively, due to its specification of conditional heteroscedasticity which can depict the cluster and fat tailor phenomena of the distribution of the timing financial series. But, all the model presume special distribution, thus they can' t describe the financial timing series of its cluster and fat tail sufficiently. It can be proved that there still have fat tailor in the residual of the ARCH model. The ARCH model specifies that the heteroscedasticity of the timing financial series is a linear function of the lagged squared residual. In this paper we preserve the consumption of conditional heteroscedasticity but think that the conditional heteroscedasticity of the timing financial series is the quadruple times function of the lagged residual. We called this model the high times model. The parameter can be estimated by the method of maximum likelihood, and they would be consistent to the true value. The estimating procedure is similar to that of ARCH model. In the end of this paper the S&P indexes of stokes have been estimated by one order ARCH model and one order high times ARCH model discussed above. Our conclusion is to add a quadruple times lagged residual to the variance function of the ARCH model can decrease the kurtosis of the residual and improve its ability of simulation and that the number of data used in one order high times ARCH model is not less than that in one order ARCH model.