【作者】 赵越；

【导师】 宋立新；

【作者基本信息】 大连理工大学 ， 金融数学与保险精算， 2010， 博士

【摘要】 近二十年来,结合了参数模型和非参数模型的优点、把两种模型相结合的半参数模型,得到了学者们的广泛关注.本文首先针对部分线性模型进行了统计推断,在数据受污染情况下,使用Huber提出的M估计得到了模型参数部分的稳健估计,得到的估计是相合的并且是渐近正态的.模拟结果显示,当误差为正态分布时,M估计与最小二乘估计相比同样具有较高的精度；当误差为柯西分布时,M估计明显优于最小二乘估计,具有稳健性.随后我们提出一种新的半参数模型形式,称为部分非线性模型,不但囊括典型的部分线性模型,也将应用中常见的非线性模型关系涵盖进来.相比最一般的部分非线性模型,我们的模型可以在一个统一的框架下进行统计推断,非常方便.我们基于筛方法对这类部分非线性模型进行了参数估计,分别用分段线性函数与B-样条逼近模型的非参数部分,得到模型的最小二乘估计与极大似然估计.两类估计都具有强相合性与渐近正态性,模拟结果也支持了所得到的渐近理论.针对金融中的汇率与石油价格,我们首次将工程中的加速应力模型引入进来,建立了美元兑欧元汇率、石油价格以及美元兑人民币汇率三者之间的部分非线性模型关系,模拟效果很好.随后我们考察了全球股票市场之间的联系,针对日经指数、道琼斯指数、美元兑日元汇率三个变量进行建模,拟合效果较好.实证结果说明部分非线性模型可应用于金融中的这些建模问题.最后,针对金融时间序列的趋势项估计问题,我们基于copula函数构造了一个准则函数,能够衡量各种估计方法的优劣.更多还原

【Abstract】 In recent 20 years, much research has been focused on the semiparametric models. The advantage of such semiparametric models is that they reduce the risk of misspecification relative to a fully parametric model and avoid some serious drawbacks of fully nonparametric methods. Firstly we discuss the statistical inference for partially linear models. Using Huber’s M-estimator, we establish the robust estimator for the parametric part. The resulting estimator is shown to be consistent and asymptotically normally distributed. Simulation results show that the M-estimator performs as well as least square estimator under normally distributed errors. Especially, when the random errors are distributed as Cauchy distribution, the M-estimator is superior to the least square estimator. Secondly we propose a new kind of semiparametric models, called partially nonlinear models, which not only nest the partially linear models as a special case, but also contain many nonlinear functions widely used in applications. Compared to general partially nonlinear models, our models are convenient to make statistical inferences. We develop the sieve method to estimate the nonparametric part, which was approximated by the piecewise linear functions and B-spline functions, and then get the least square estimator and the maximum likelihood estimator, respectively. Both the two resulting estimators are consistent and asymptotically normally distributed. The simulation results support our theory procedures. By introducing the acceleration stress model, we establish the partially nonlinear model to simulate the relation among the exchange rate of USD/EUR, USD/RMB and oil price. And then we study the inner relations of the global stock market, the model is applied to simulate the relationship among the Dow Jones Industrial index, Nikkei index and USD/JPY exchange rate. The partially nonlinear models are found to be an appropriate setting in such cases in finance. Finally, we construct a criterion function using copula function, which can select the best one among the methods on trend estimation in financial time series.更多还原