| Volatility agglomeration effect is one of the main characteristics of financial time series volatility.Accurately describing volatility effect is helpful to the pricing and risk management of financial assets.Since Bollerslev extended the ARCH model to the GARCH model in 1986,many scholars have applied the GARCH model to the study of financial time series.However,in order to avoid the disadvantage that quasi maximum likelihood estimation is too sensitive to extreme returns,Xiao and Koenker proposed the Linear GARCH model(LGARCH)in 2009using the idea of quantile estimation,assuming that conditional standard deviations rather than conditional variances have a linear structure.The idea of linear GARCH-type error is introduced into this model and proves that this structure produces more robust inferences than assuming conditional variance with a linear structure.Therefore,this paper carries out some extension research on the linear GARCH-type error models.Firstly,aiming at the choice of order p in practical application of LDAR(p)model,a moving average model with linear GARCH-type error is proposed,which is denoted as MA-LGARCH model.This model can be regarded as a special LDAR(∞)model,which considers p=∞in mean value equation and conditional standard deviation equation at the same time,and there are only 4 parameters to be estimated.In this paper,the asymptotic normality of the quasi maximum likelihood estimation of the model is proved under the weak moment condition.Sim-ulation results show that the parameter estimators perform effectively with limited sample sizes.Empirical studies based on domestic and foreign data also show that the MA-LGARCH model can effectively improve the fitting effect of data.Secondly,with the development of electronic information technology,it has become eas-ier to obtain intraday high-frequency data in the financial market.Studies have shown that LGARCH model is more robust than other GARCH-type error models,and these high-frequency data usually contain a lot of useful information,which is valuable for improving model estima-tion.Therefore,this paper introduces intraday high-frequency data into the QMLE and QMELE estimation methods of the daily frequency LGARCH model,which is expected to improve the estimation effect of the model.The simulation results show that the estimations of QMLE based on normal distribution and QMELE based on Laplace distribution perform well in the finite sample.The empirical studies based on SSE 50 index show that compared with the es-timation method which only uses daily frequency data,the estimation accuracy of the QMLE and QMELE methods of the daily frequency LGARCH model can be significantly improved by using intraday high-frequency data.Finally,in the linear GARCH-type error models studied above,we do not assume that the error termε_tfollows a specific distribution,but requires it to be independent and identically distributed,with a mean of 0 and a variance of 1.However,in real life,the specific distribution of error term is of great significance for calculating financial risk.Therefore,we generalize the Hausman test proposed by Zhu Ke in 2015 to the linear GARCH-type error models to examine the specific distribution of error term in the model,and consider introducing high-frequency data into the Hausman test.The simulation results show that the Hausman test statistic under the daily frequency data is valid.The empirical results show that the introduction of intraday high-frequency data for Hausman test can provide more information about the distribution of error term. |
