贝叶斯分位数回归模型及其在金融风险管理中的应用

发布时间：2018-09-08 21:43

【摘要】：分位数回归源于历史上的l1估计问题。作为一种半参数统计方法,分位数回归能够克服数据的尖峰厚尾特点以及数据的结构突变等问题,有着独特的优势。近年来分位数回归模型逐渐成为学术界的热点之一,吸引了大量学者进行相关理论和应用研究,并被广泛应用于经济金融、生物医疗等领域。分位数回归模型的一个发展方向是与贝叶斯估计的结合,贝叶斯统计通过引入先验信息与数据集结合,进行后验推断,统计结果更丰富更具有解释能力。贝叶斯统计在应用上的一个难点是后验分布往往极其复杂,甚至后验分布不存在解析形式,这非常不利于人们利用后验分布进行统计推断。近年来随着计算机性能的不断发展,MCMC方法通过随机数抽样得到估计值,跳过后验分布的具体形式,解决了贝叶斯后验估计困难的问题,在抽样次数足够大的情况下,这样的估计是有效的。在金融风险的测量与建模领域,VaR是风险测度中一个极其重要的度量指标,通常的VaR计算方法需要对金融市场收益率进行某种概率分布的假定,然后依据概率分布估计VaR值,分位数回归理论则不需要概率分布的假定,而可以利用回归方程直接对VaR值进行估计,可以方便的得到各个置信水平下的VaR值。本文首先回顾了经典的分位数回归理论,介绍了贝叶斯分析的基本框架以及MCMC方法的内涵;然后通过引入非对称拉普拉斯分布和广义逆高斯分布,将分位数回归纳入贝叶斯推理的框架,介绍了一种基于局部变量混合和Gibbs抽样的算法,同时展示了在某些特殊先验的条件下,贝叶斯分位数回归与Lasso方法的等价性;最后探讨了分位数回归在中国股市的实证研究,贝叶斯分位数回归模型在VaR计算中的同样有着非常简洁有效的应用。
[Abstract]:The quantile regression results from the historical L 1 estimation problem. As a semi-parametric statistical method, quartile regression can overcome the characteristics of peak and thick tail of data and the structural change of data, which has a unique advantage. In recent years, quantile regression model has gradually become one of the hot topics in academia, attracting a large number of scholars to carry out relevant theoretical and applied research, and has been widely used in the fields of economics and finance, biomedicine and so on. One of the development directions of quantile regression model is to combine with Bayesian estimation. Bayesian statistics combines prior information with data set and carries out posteriori inference. The statistical results are richer and more interpretive. One of the difficulties in the application of Bayesian statistics is that the posteriori distribution is often extremely complex, and even the posteriori distribution does not have an analytical form, which is very unfavorable for people to use the posteriori distribution to infer statistics. In recent years, with the continuous development of computer performance, the MCMC method obtains the estimated value by random number sampling, and solves the problem of Bayesian posteriori estimation by the concrete form of the skip posteriori distribution. When the sampling times are large enough, the problem of Bayesian posteriori estimation is solved. Such estimates are valid. In the field of financial risk measurement and modeling, VaR is an extremely important measure index in risk measurement. The usual VaR calculation methods need to assume some probability distribution of the financial market return rate, and then estimate the VaR value according to the probability distribution. The quantile regression theory does not need the assumption of probability distribution, but can use regression equation to estimate the VaR value directly, and get the VaR value of each confidence level conveniently. In this paper, the classical quantile regression theory is reviewed, the basic framework of Bayesian analysis and the connotation of MCMC method are introduced, and then the asymmetric Laplace distribution and generalized inverse Gao Si distribution are introduced. The quantile regression is introduced into the framework of Bayesian reasoning, and an algorithm based on local variable mixing and Gibbs sampling is introduced. At the same time, the equivalence between Bayesian quantile regression and Lasso method is shown under some special priori conditions. Finally, the empirical study of quantile regression in Chinese stock market is discussed. Bayesian quantile regression model also has a very simple and effective application in VaR calculation.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224;F831

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