基于Copula理论的金融时间序列相依性研究
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摘要
经济全球化和金融国际化导致了金融市场之间的联系越来越紧密,彼此之间的关系更加复杂。准确刻画金融变量之间的相依结构是研究金融风险管理、投资组合以及资产定价等问题的基础,这对于探索金融经济系统的运行机制以及科学地进行金融决策具有重要的理论和现实意义。但是传统的相关性分析方法大多不能刻画变量间的非线性、非对称等特征,现有的多元联合分布函数也远不能满足实际分析的需要。
Copula理论能描述金融时间序列间的非线性、非对称和再生性等相关性特征,还能构建许多的多元联合分布函数。本文运用Copula理论研究金融时间序列的相依结构,首先系统地论述了Copula函数及其性质,特别是嵌套的Copula及多重相关性,并对由Copula函数导出的相依性指标进行深入的探讨,论证了尾部相关系数与缓慢变化函数共同刻画尾部相关性的联合生成函数法优于常用的上、下尾部相关系数,进一步介绍了动态Copula模型的最新研究进展。其主要工作与创新体现在:
①论文讨论了运用Copula模型的一个关键:边缘分布函数对数据的拟合和选取。基于Copula理论的相关性分析方法克服了传统方法的局限性,但必须正确选择边缘分布函数,论文主要讨论核密度函数、极值模型和波动模型等边缘分布函数。采用极值POT模型对中小企业板股票的极端风险进行实证研究,发现指数回归模型对于小样本POT模型阈值的定量选取是一种有效的方法,避免了样本Hill图在选取阈值时的不确定性;POT模型能更好地描述金融变量分布的厚尾和非对称等特征,因此对极端风险的估计比正态分布更有效,正态分布在股票上涨时期高估风险,而在股市下跌时期低估风险,并且很难通过检验。
②论文进一步分析了Copula函数的选取。Copula函数的选取是正确运用Copula理论的另一关键。在深入探讨分别以核密度函数、极值模型和波动模型为边缘分布函数的Copula模型的基础上,采用Copula-POT模型对房地产与金融行业的股票收益率的相关性进行了实证研究。结果表明房地产与金融行业的股票收益率在股市低迷时期具有较强的相关性。研究发现在选取Copula函数时应同时考虑不同参数结构的Copula函数,以选取恰当的Copula函数描述其相关性;双参数结构的Copula函数的拟合度普遍高于单参数结构的Copula函数。选取经验分布为边缘分布函数,对常相关Copula模型与时变相关Copula模型的拟合效果进行比较,结果表明房地产与金融行业股票收益率的相依结构为时变Symmetrized Joe-Clayton copula,时变相关Copula模型比常相关Copula模型更能刻画金融时间序列的相关性。
③对Copula模型能否准确刻画金融时间序列的相依结构进行深入研究。为了正确选择边缘分布函数和Copula函数,文中采用非参数核密度函数和半参数POT模型作为边缘分布,选取相应的Copula函数对沪深股市相关性进行了实证分析。研究结果表明两市在市场低迷时期的尾部相关性稍高于活跃时期的尾部相关性。研究发现非参数法对数据的尾部拟合有时要优于半参数法,半参数法的整体拟合有时要优于非参数法;虽然不同边缘分布的Copula相依结构相同;但由于边缘分布的类型不同,导致Copula模型中的参数估计值不同,从而刻画的相关程度不同。因此,Copula模型准确刻画金融时间序列的相依性,取决于边缘分布函数对数据的拟合程度。
Nowadays, the relationships between the financial markets are getting increasingly close and complex because of the rapid development of financial markets, deepening of financial innovation, economic globalization and financial internationalization. How to describe the dependence structure between financial variables accurately is the basis of studying issues such as financial risk management, portfolio and asset pricing. Correlation analysis has drawn more and more attention in the modern financial analysis. That the research of the dependence structure has an important theoretical and practical significance to explore the mechanism of financial system and make financial decisions scientifically. However, most of the traditional ways of correlation analysis couldn’t describe the nonlinear and asymmetrical features. The existing multivariate joint distribution function couldn’t meet the needs of is multivariate practical analysis neither.
The emergence and development of the Copula theory has greatly extended the multivariable correlation analysis to a new stage and become the hotspot of the current financial research. This paper employs Copula theory to study the dependence of financial time series. Firstly, we systematically expound the Copula function and its characteristics, especially for the nested Copula and multiple correlations. Then, we discuss the dependency of indicator derived from the Copula function in depth, and demonstrate that the joint generating function method which describes the tail correlation by combining the coefficient of tail dependence and slowly changing function is better than the common tail correlation coefficient. At last, we introduced the latest dynamic of Copula model. The main work and innovation are as following: This paper discusses the key issue of applying the Copula model: the data fitting and selection of Marginal distribution function. The Copula theory method based on correlation analysis overcomes the limitations of traditional methods, but the choice of marginal distribution function must be correct. This paper mainly discusses the kernel density function, extreme model and fluctuation margin distribution function model.
This paper employs POT model to estimate extreme risk in SME stock market for the empirically study. The result shows that the exponential regression model is an effective way to the quantities’threshold selection of small POT model, and it avoids the threshold selection uncertainty of Hill estimator. POT model can better describe the fat-tail and asymmetrical distribution features, so it is prior to normal distribution in estimating the extreme risk. The normal distribution will over-estimate risk within the rise period of stock market, under-estimate risk within the decline period and fail the test.
This paper further analyzes how to select the Copula functions. The appropriate selection of Copula functions is another key point to properly apply the Copula theory. After thoroughly discussing the Copula models whose marginal distribution functions are based on kernel density function, extreme value theory and volatility models, we empirically analysed the correlation of stock returns in real estate market and financial industry. The result shows that strong correlation exists between these two markets during the downturn period. We also find out that it is important to take into account Copula functions with varied parameter structure, in order to properly model the correlation. In fact, Copula functions with two parameters usually work better than single-parameter Copula functions. We continue to discuss the dependence with the constant copula models and the time-varying copula models based on the experience distributions. The results indicate that the depence structure of stock returns in real estate market and financial industry is Symmetrized Joe-Clayton copula, and the time-varying copula models are prior to the constant ones in simulating the correlation between the financial time series .
Lastly, the capability of Copula models to accurately characterize dependencies between financial time series is studied in depth as well. In order to correctly select the marginal distribution function and henceforth the Copula functions, we employ a non-parametric kernel density function as well as a semi-parametric POT model as the marginal distribution, and then explore the correlation between the Shanghai and Shenzhen Stock markets by the proposed Copula functions. The results show that the tail dependence in downturn is slightly higher than that of the boom. In addition, evidence tells that non-parametric method to fit the tail of the data is sometimes better than the semi-parametric method, while in terms of the overall goodness of fit the semi-parametric methods sometimes beats the non-parametric methods. The different choices of marginal distribution functions result in varied parameter estimates of the Copula models, and consequently different degree of estimated correlations. Therefore, the accuracy of Copula models to describe the dependencies in financial time series, to a large extent, depends on how well the marginal distribution functions fit the data.
Copula理论能描述金融时间序列间的非线性、非对称和再生性等相关性特征,还能构建许多的多元联合分布函数。本文运用Copula理论研究金融时间序列的相依结构,首先系统地论述了Copula函数及其性质,特别是嵌套的Copula及多重相关性,并对由Copula函数导出的相依性指标进行深入的探讨,论证了尾部相关系数与缓慢变化函数共同刻画尾部相关性的联合生成函数法优于常用的上、下尾部相关系数,进一步介绍了动态Copula模型的最新研究进展。其主要工作与创新体现在:
①论文讨论了运用Copula模型的一个关键:边缘分布函数对数据的拟合和选取。基于Copula理论的相关性分析方法克服了传统方法的局限性,但必须正确选择边缘分布函数,论文主要讨论核密度函数、极值模型和波动模型等边缘分布函数。采用极值POT模型对中小企业板股票的极端风险进行实证研究,发现指数回归模型对于小样本POT模型阈值的定量选取是一种有效的方法,避免了样本Hill图在选取阈值时的不确定性;POT模型能更好地描述金融变量分布的厚尾和非对称等特征,因此对极端风险的估计比正态分布更有效,正态分布在股票上涨时期高估风险,而在股市下跌时期低估风险,并且很难通过检验。
②论文进一步分析了Copula函数的选取。Copula函数的选取是正确运用Copula理论的另一关键。在深入探讨分别以核密度函数、极值模型和波动模型为边缘分布函数的Copula模型的基础上,采用Copula-POT模型对房地产与金融行业的股票收益率的相关性进行了实证研究。结果表明房地产与金融行业的股票收益率在股市低迷时期具有较强的相关性。研究发现在选取Copula函数时应同时考虑不同参数结构的Copula函数,以选取恰当的Copula函数描述其相关性;双参数结构的Copula函数的拟合度普遍高于单参数结构的Copula函数。选取经验分布为边缘分布函数,对常相关Copula模型与时变相关Copula模型的拟合效果进行比较,结果表明房地产与金融行业股票收益率的相依结构为时变Symmetrized Joe-Clayton copula,时变相关Copula模型比常相关Copula模型更能刻画金融时间序列的相关性。
③对Copula模型能否准确刻画金融时间序列的相依结构进行深入研究。为了正确选择边缘分布函数和Copula函数,文中采用非参数核密度函数和半参数POT模型作为边缘分布,选取相应的Copula函数对沪深股市相关性进行了实证分析。研究结果表明两市在市场低迷时期的尾部相关性稍高于活跃时期的尾部相关性。研究发现非参数法对数据的尾部拟合有时要优于半参数法,半参数法的整体拟合有时要优于非参数法;虽然不同边缘分布的Copula相依结构相同;但由于边缘分布的类型不同,导致Copula模型中的参数估计值不同,从而刻画的相关程度不同。因此,Copula模型准确刻画金融时间序列的相依性,取决于边缘分布函数对数据的拟合程度。
Nowadays, the relationships between the financial markets are getting increasingly close and complex because of the rapid development of financial markets, deepening of financial innovation, economic globalization and financial internationalization. How to describe the dependence structure between financial variables accurately is the basis of studying issues such as financial risk management, portfolio and asset pricing. Correlation analysis has drawn more and more attention in the modern financial analysis. That the research of the dependence structure has an important theoretical and practical significance to explore the mechanism of financial system and make financial decisions scientifically. However, most of the traditional ways of correlation analysis couldn’t describe the nonlinear and asymmetrical features. The existing multivariate joint distribution function couldn’t meet the needs of is multivariate practical analysis neither.
The emergence and development of the Copula theory has greatly extended the multivariable correlation analysis to a new stage and become the hotspot of the current financial research. This paper employs Copula theory to study the dependence of financial time series. Firstly, we systematically expound the Copula function and its characteristics, especially for the nested Copula and multiple correlations. Then, we discuss the dependency of indicator derived from the Copula function in depth, and demonstrate that the joint generating function method which describes the tail correlation by combining the coefficient of tail dependence and slowly changing function is better than the common tail correlation coefficient. At last, we introduced the latest dynamic of Copula model. The main work and innovation are as following: This paper discusses the key issue of applying the Copula model: the data fitting and selection of Marginal distribution function. The Copula theory method based on correlation analysis overcomes the limitations of traditional methods, but the choice of marginal distribution function must be correct. This paper mainly discusses the kernel density function, extreme model and fluctuation margin distribution function model.
This paper employs POT model to estimate extreme risk in SME stock market for the empirically study. The result shows that the exponential regression model is an effective way to the quantities’threshold selection of small POT model, and it avoids the threshold selection uncertainty of Hill estimator. POT model can better describe the fat-tail and asymmetrical distribution features, so it is prior to normal distribution in estimating the extreme risk. The normal distribution will over-estimate risk within the rise period of stock market, under-estimate risk within the decline period and fail the test.
This paper further analyzes how to select the Copula functions. The appropriate selection of Copula functions is another key point to properly apply the Copula theory. After thoroughly discussing the Copula models whose marginal distribution functions are based on kernel density function, extreme value theory and volatility models, we empirically analysed the correlation of stock returns in real estate market and financial industry. The result shows that strong correlation exists between these two markets during the downturn period. We also find out that it is important to take into account Copula functions with varied parameter structure, in order to properly model the correlation. In fact, Copula functions with two parameters usually work better than single-parameter Copula functions. We continue to discuss the dependence with the constant copula models and the time-varying copula models based on the experience distributions. The results indicate that the depence structure of stock returns in real estate market and financial industry is Symmetrized Joe-Clayton copula, and the time-varying copula models are prior to the constant ones in simulating the correlation between the financial time series .
Lastly, the capability of Copula models to accurately characterize dependencies between financial time series is studied in depth as well. In order to correctly select the marginal distribution function and henceforth the Copula functions, we employ a non-parametric kernel density function as well as a semi-parametric POT model as the marginal distribution, and then explore the correlation between the Shanghai and Shenzhen Stock markets by the proposed Copula functions. The results show that the tail dependence in downturn is slightly higher than that of the boom. In addition, evidence tells that non-parametric method to fit the tail of the data is sometimes better than the semi-parametric method, while in terms of the overall goodness of fit the semi-parametric methods sometimes beats the non-parametric methods. The different choices of marginal distribution functions result in varied parameter estimates of the Copula models, and consequently different degree of estimated correlations. Therefore, the accuracy of Copula models to describe the dependencies in financial time series, to a large extent, depends on how well the marginal distribution functions fit the data.
引文
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