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【Wanghui】Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1) models

作者:     日期:2015-04-08    来源:

<Statistics and Probability Letters> 91st volume in 2014 publishes the cooperation paper by Professor Hui Wang in our school and Doctor Jiazhu Pan, <Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1) models>.
Modern financial theory deems that risk assessing and pricing are both connected with the measurement of the fluctuation of financial asset revenues. Effective recognition of fluctuation rate directly influences asset pricing, capital allocation and risk management. Owing to this, ARCH/GARCH model has been widely used since it was brought up. A great many positive researches show that financial data have the attributes of Leptokurtosis and Fat-Tail and leverage effect. On the other hand, instability is another important characteristic of economic and financial time series. A stable model for an instable time series will lead to faulty model type and terrible fluctuation rate anticipation. That’s why the stability of TGARCH model should be inspected.
NM-QMLE is more effective for Leptokurtosis and Fat-Tail data than QMLE. This paper puts forward the thought to use NM-QMLE without restrictions to estimate instable TGARCH model and testifies that the estimate corresponds to parameters except for position parameter under some regularity conditions. Since no position parameter has been restricted and the estimate corresponds in stable conditions, direct estimate can be made in TGARCH model no matter the model is stable or not.
ARCH/GARCH shows the uncertainty, accumulation and fat tail in fluctuation rate of financial income ratio, which has been on spotlight since put forward and been widely used in finance and economics. While the standard GARCH is symmetrical, surveys show that Heteroskedasticity varies with the positive or negative errors, which is called leverage effect or dissymmetry. Papers bring up TGARCH to better portray the dissymmetry of financial data. For the estimate of such models, G-QMLE is often used and testified corresponding in certain circumstances. A great many positive researches show that financial data have the attributes of Leptokurtosis and Fat-Tail. The real distribution of information is far from normal distribution, so G-QMLE is inefficient, while Gaussian mixture distribution is preferred for it can better portray Leptokurtosis and Fat-Tail and biasness. On the other hand, numerous positive surveys show the fluctuation rate of financial revenue ratio is instable or even explosive, which makes the estimate and foundation of the instability fluctuation model necessary.
Taking the factors above into consideration, this essay puts forward NM-QMLE, based on TGARCH(1,1), testifying the estimate has the attributes of correspondence and asymptotic normality. Since it’s hard to demonstrate the efficiency of the estimate, we do simulation research on it and the results show the fatter of the tail or the larger of the deviation of information, NM-QMLE is more effective than G-QMLE. At last, the paper builds up the model and analyzes the 5-year Italian CDS data of the whole year of 2011. The results turn out the distribution of information is more like a Gaussian mixture distribution. Traditional G-QMLE will lead to erroneous model and bad simulation, the model taken by this essay is more effective.