On The Research Of Parameter Estimation For A Class Of β-ARCH Model

With the development of the modern world, economy play a. very important role in world stage. At the mean time, the research field of social science and natural science which are associated with economy have been made a profound improvement consequently. Among these, in the field of statistics, the topic of finance and economy turn to be the most prevailed one.Owing to propose the conditional heteroscedasticity model, Engle was awarded Nobel Price of economy in 2003. For the time varying volatility is the main character of financial series data, it is no doubt that ARCH models have brought a momentous and profound influence into the field of finance. Bollerslev (2001) make a retrospection of past developments in financial econometrics and forecasts its' future challenges. He emphasis that, the interesting current directions for future research in this field is how to analysis the time-varying volatility in the high-frequency data, long-memory, heavy-tails, and large dimensional systems.Besides the conditional heteroscedasticity of financial data, another important character of the these data is heavy-tail. If our investigation work of the financial data are all based on the assumption of innovations' conditional normal distribution as before, the effect which are brought by the heavy-tail distribution would be ignored, but modern financial management science, especially the science of financial risk management, center their attentions on this kind of effect on the contrary.Therefore, the problem we should deal with is that how to make the statistical inference of the conditional heteroscedasticity model, when the distributions of innovations are unknown. Due to the interest of problem mensioned above, we makes some attempt to theparameter estimation in the such situations. In this paper, we propose two kinds of estimation methods for a class of -ARCH model. One method is quasi-likelihood estimation method, the other one is empirical likelihood estimation method.Based on the spirit of "likelihood", some statistical methods have be developed. Amonge those, the methods of quasi-likelihood and empirical likelihood can overcome the shortcomings of method of likelihood from different aspects. That is the reason why we adopt these two methods into the time series analysis. In chronicling the development in financial econometric over the past two decades time-varying volatility models, in the form of ARCH and stochastic volatility formulations , and robust methods-of-moment estimation procedures, such as GMM, stand out as milestones. The most important developments in the financial field are notably ARCH and GMM. Nowadays, econometricians are moving away for the maximum likelihood estimation. How to interplay between Fisher's concept of likelihood and Pearson's method of moments would be the future challenges. These ideas have been blended fruitfully by the non-parametric method of empirical likelihood. That is another valuable point of the research of empirical likelihood.In this paper, we consider such model. Suppose that time series {Xt} satisfy the following nonlinear time series model:where r > 0, 0 < /3 < 1, QO >0, QJ >0 (i = 1,2, - - - , q ). {i)t} are independent identically distributed random variables such that E t = 0 and E7t|r = 1- We assume further that {r)t} have a common and almost everywhere positive density and rjt is independent of cr(Xs,s < t). In this paper, the random variable series {rjt} which are associated with error seises are called noise seises. Let a = (ao,ai, - ,0,)' denote (q + 1)- dimension parameter vector, which " ' " stands for the transpose of a vector or a matrix, and we use a'? stands for the real parameter value.In the following, we mainly introduce two kinds methods of parameter estimation for the above model. One method is quasi-likelihood estimation method, the other one is maximum empirical likelihood estimation method. Besides, we give the asymptotic properties of these two kinds of parameter estimation methods.1. Quasi-likelihood parameter estimation method:...