基于贝叶斯分位回归理论的截面相依面板协整研究
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摘要
非平稳面板数据研究是目前计量经济学领域中的前沿问题,其中,面板单位根和协整研究,作为时间序列单位根与传统协整理论在面板数据中的发展和延伸,更具有重要意义。由于全球国际趋势和国际经济周期等共同驱动的影响,宏观经济、管理或金融面板数据尤其是国家(地区或个体单元)的面板数据之间通常存在截面相依特征,因此,考虑截面相依假设条件的面板协整更加符合实际应用背景,也是面板数据研究中亟待解决的一个热点问题。与传统的面板协整不同,本文针对具有截面相依条件的面板协整进行研究,在贝叶斯理论框架中,假设各个截面个体具有截面相依特征,结合贝叶斯分位回归估计方法,提出了面板数据的贝叶斯分位协整模型。贝叶斯分位协整模型可以充分发挥贝叶斯方法考虑了参数不确定性风险的优势,并且体现了分位回归方法不仅可以刻画响应变量的中心趋势,还可以刻画变量尾部行为的优点,从而为更全面地刻画响应变量与协变量的长期均衡关系提供了方法和工具支撑,在理论上扩展面板协整的研究方法和研究视角,在实践上为经济管理问题的定量分析和决策提供技术支持和有力依据。
针对面板数据之间通常存在截面相依性,首先应用动态公共因子结构刻画面板数据的截面相依特征,结合贝叶斯决策理论,提出一类考虑了截面相依假设条件的协整模型,利用贝叶斯分位回归方法,通过把非对称Laplace分布表示成指数分布和正态分布的线性组合,获得了条件分位函数后验估计量的解析表达形式,并设计Kalman滤波与Gibbs抽样算法对模型参数进行估计和协整检验。同时,Monte Carlo仿真实验结果表明,贝叶斯分位协整可以更加全面地对变量间的协整关系进行判断。
经济金融变量因为战争,政府政策以及自然灾害等因素的影响,往往表现出结构突变性,这种结构性变化的发生会影响传统线性协整检验的判断。放松线性假设条件,本文提出一类考虑了结构变化特征的面板协整模型—平滑变结构面板协整模型,利用傅立叶级数展开形式来刻画变结构特征,并采用去除截面均值的方法消除面板数据的截面相依性,以避免参数过多的问题,进而结合贝叶斯分位回归方法得到相应条件分位函数后验估计量的解析形式,并设计MCMC抽样算法对模型进行参数后验估计和协整检验。仿真实验结果表明,贝叶斯分位变结构协整能够有效全面地刻画各个分位水平下的变结构长期关系。
与变结构协整不同,门限协整主要研究协整回归模型是线性,而其相应的误差修正项是非对称时的情形。针对传统门限协整模型由于似然函数具有多峰、不连续特征,导致冗余参数识别存在困难,最优化计算相对复杂的问题,本文从贝叶斯的角度出发,提出面板数据的贝叶斯分位门限协整模型,通过去除截面均值以消除面板数据间潜在的相依性,并对参数的先验分布进行灵敏度分析以选择合适的参数先验,结合贝叶斯分位回归方法对面板门限协整模型进行参数估计,得到条件分位函数后验估计量的解析表达式,同时,利用MCMC算法对协整模型的参数进行估计,计算出协整检验的后验概率以进行更加全面的门限协整检验。
将上述考虑了面板数据截面相依特征的贝叶斯分位协整方法应用到原油与股票市场的关系研究中,并与传统面板协整方法进行比较,发现贝叶斯分位协整方法对原油与股票市场之间联动性关系的刻画更加全面,验证了贝叶斯分位协整方法的可行性和有效性,说明贝叶斯分位方法能够提供全方面的便捷的模型参数估计和协整检验信息。
Recently, nonstationary panel data is an important topic in areas of econometrics. An approach in this direction is panel unit root and cointegration, which has attracted considerable interest and is more valuable. Panel unit root and cointegration is known as the extension of unit root and cointegration in time series. In the globalised economy, co-movements of economies are often observed. Global international trends or international business cycles for instance, can cause cross-section dependence in panel time series of macroeconomics, management and finance. Therefore, panel cointegration tests with cross-section dependence may be more valid and realistic. In contrast to classic panel cointegraiton, however, this paper proposes Bayesian quantile panel cointegrating under the assumption of cross-section dependence. In the framework of Bayesian econometrics, cross-section dependence is addressed and Bayesian quantile regression method is used. A distinctive feature of Bayesian quantile cointegrating regression is that it is the combination of Bayesian quantile regression method and cointegration. It considers the uncertainty of parameter, and can capture systematic influences of covariates on the location, scale and shape of the conditional distribution of the response. Thus, Bayesian quantile cointegrating regression could analyze the long run relationship between response variable and covariates. In theory, Bayesian quantile panel cointegration will develop research ideas and methods for panel cointegration. Meanwhile, it will also provide technology and evidence for analysis and decision making in economic and management applications.
First, common factor is used to handle the cross-section dependence in panel data. In the context of Bayesian theorem and Bayesian decision theorem, the analytic posterior estimator of conditional quantile function is obtained by a linear mixture representation of asymmetric Laplace distribution. To get the estimate and conduct cointegration test, the Kalman filter and Gibbs sampling algorithms are utilized to simulate the posterior marginal distribution of quantile cointegrating parameters, which resolved the difficulties of the high dimension numerical integral. Monte Carlo simulation also indicates that Bayesian quanitle panel cointegration can conduct more comprehensive analysis cointegration relationship between variables.
It is known that the phenomenon of structural breaks is a common place in macroeconomic series. Omitting the effect of structural breaks can thus cause a deceptive inference in time series and panel data testing. This paper proposes panel cointegration model with smooth structural break, which implement the Fourier expansion to explain structural breaks in panel time series. The panel data are demeaned to control potential cross-section dependence, and Bayesian quantile regression method is employed to obtain the analytic posterior estimator of conditional quantile function. MCMC algorithm is then designed to estimate parameters and conduct cointegration test. We also conduct a small Monte Carlo study to illustrate the effect of Bayesian quantile panel cointegration with smooth structural breaks. The results show that, as expected, the Bayesian quantile cointegration methods are more effective and comprehensive to analyze the structural break long run relationship except for some special cases.
In contrast to cointegraiton with structural breaks, threshold cointegration aims to explore the asymmetry or asymmetric adjustment behaviors, where the cointgrating regression is linear and the error correction term is asymmetric. In the existing threshold cointegration methods, the jagged and potentially multimodal nature of the likelihood function of threshold model complicates optimization and also makes the identification of unknown nuisance parameters more difficult. This paper proposes Bayesian quantile threshold cointegration model and conduct quantile cointegration tests. Especially, the demeaning approach is used to control the potential cross sectional dependence. The choice of priors is discussed to get robust posterior estimator of conditional quantile function. Based on the posterior conditional distributions of the parameters, MCMC samplers are designed to estimate the parameters and compute the posterior probability for threshold cointegration tests.
Finally, the above mentioned Bayesian quantile panel cointegration methods are applied to the analysis of crude oil and stock markets. Comparing with traditional panel cointegration methods, Bayesian quantile cointegraiton methods can describe the relationship between crude oil and stock markets more comprehensively. Therefore, the usefulness of Bayesian quantile cointegration methods is also demonstrated. It shows that Bayesian quantile cointegration performs well and can provide fully information on parameter estimation and cointegration tests.
针对面板数据之间通常存在截面相依性,首先应用动态公共因子结构刻画面板数据的截面相依特征,结合贝叶斯决策理论,提出一类考虑了截面相依假设条件的协整模型,利用贝叶斯分位回归方法,通过把非对称Laplace分布表示成指数分布和正态分布的线性组合,获得了条件分位函数后验估计量的解析表达形式,并设计Kalman滤波与Gibbs抽样算法对模型参数进行估计和协整检验。同时,Monte Carlo仿真实验结果表明,贝叶斯分位协整可以更加全面地对变量间的协整关系进行判断。
经济金融变量因为战争,政府政策以及自然灾害等因素的影响,往往表现出结构突变性,这种结构性变化的发生会影响传统线性协整检验的判断。放松线性假设条件,本文提出一类考虑了结构变化特征的面板协整模型—平滑变结构面板协整模型,利用傅立叶级数展开形式来刻画变结构特征,并采用去除截面均值的方法消除面板数据的截面相依性,以避免参数过多的问题,进而结合贝叶斯分位回归方法得到相应条件分位函数后验估计量的解析形式,并设计MCMC抽样算法对模型进行参数后验估计和协整检验。仿真实验结果表明,贝叶斯分位变结构协整能够有效全面地刻画各个分位水平下的变结构长期关系。
与变结构协整不同,门限协整主要研究协整回归模型是线性,而其相应的误差修正项是非对称时的情形。针对传统门限协整模型由于似然函数具有多峰、不连续特征,导致冗余参数识别存在困难,最优化计算相对复杂的问题,本文从贝叶斯的角度出发,提出面板数据的贝叶斯分位门限协整模型,通过去除截面均值以消除面板数据间潜在的相依性,并对参数的先验分布进行灵敏度分析以选择合适的参数先验,结合贝叶斯分位回归方法对面板门限协整模型进行参数估计,得到条件分位函数后验估计量的解析表达式,同时,利用MCMC算法对协整模型的参数进行估计,计算出协整检验的后验概率以进行更加全面的门限协整检验。
将上述考虑了面板数据截面相依特征的贝叶斯分位协整方法应用到原油与股票市场的关系研究中,并与传统面板协整方法进行比较,发现贝叶斯分位协整方法对原油与股票市场之间联动性关系的刻画更加全面,验证了贝叶斯分位协整方法的可行性和有效性,说明贝叶斯分位方法能够提供全方面的便捷的模型参数估计和协整检验信息。
Recently, nonstationary panel data is an important topic in areas of econometrics. An approach in this direction is panel unit root and cointegration, which has attracted considerable interest and is more valuable. Panel unit root and cointegration is known as the extension of unit root and cointegration in time series. In the globalised economy, co-movements of economies are often observed. Global international trends or international business cycles for instance, can cause cross-section dependence in panel time series of macroeconomics, management and finance. Therefore, panel cointegration tests with cross-section dependence may be more valid and realistic. In contrast to classic panel cointegraiton, however, this paper proposes Bayesian quantile panel cointegrating under the assumption of cross-section dependence. In the framework of Bayesian econometrics, cross-section dependence is addressed and Bayesian quantile regression method is used. A distinctive feature of Bayesian quantile cointegrating regression is that it is the combination of Bayesian quantile regression method and cointegration. It considers the uncertainty of parameter, and can capture systematic influences of covariates on the location, scale and shape of the conditional distribution of the response. Thus, Bayesian quantile cointegrating regression could analyze the long run relationship between response variable and covariates. In theory, Bayesian quantile panel cointegration will develop research ideas and methods for panel cointegration. Meanwhile, it will also provide technology and evidence for analysis and decision making in economic and management applications.
First, common factor is used to handle the cross-section dependence in panel data. In the context of Bayesian theorem and Bayesian decision theorem, the analytic posterior estimator of conditional quantile function is obtained by a linear mixture representation of asymmetric Laplace distribution. To get the estimate and conduct cointegration test, the Kalman filter and Gibbs sampling algorithms are utilized to simulate the posterior marginal distribution of quantile cointegrating parameters, which resolved the difficulties of the high dimension numerical integral. Monte Carlo simulation also indicates that Bayesian quanitle panel cointegration can conduct more comprehensive analysis cointegration relationship between variables.
It is known that the phenomenon of structural breaks is a common place in macroeconomic series. Omitting the effect of structural breaks can thus cause a deceptive inference in time series and panel data testing. This paper proposes panel cointegration model with smooth structural break, which implement the Fourier expansion to explain structural breaks in panel time series. The panel data are demeaned to control potential cross-section dependence, and Bayesian quantile regression method is employed to obtain the analytic posterior estimator of conditional quantile function. MCMC algorithm is then designed to estimate parameters and conduct cointegration test. We also conduct a small Monte Carlo study to illustrate the effect of Bayesian quantile panel cointegration with smooth structural breaks. The results show that, as expected, the Bayesian quantile cointegration methods are more effective and comprehensive to analyze the structural break long run relationship except for some special cases.
In contrast to cointegraiton with structural breaks, threshold cointegration aims to explore the asymmetry or asymmetric adjustment behaviors, where the cointgrating regression is linear and the error correction term is asymmetric. In the existing threshold cointegration methods, the jagged and potentially multimodal nature of the likelihood function of threshold model complicates optimization and also makes the identification of unknown nuisance parameters more difficult. This paper proposes Bayesian quantile threshold cointegration model and conduct quantile cointegration tests. Especially, the demeaning approach is used to control the potential cross sectional dependence. The choice of priors is discussed to get robust posterior estimator of conditional quantile function. Based on the posterior conditional distributions of the parameters, MCMC samplers are designed to estimate the parameters and compute the posterior probability for threshold cointegration tests.
Finally, the above mentioned Bayesian quantile panel cointegration methods are applied to the analysis of crude oil and stock markets. Comparing with traditional panel cointegration methods, Bayesian quantile cointegraiton methods can describe the relationship between crude oil and stock markets more comprehensively. Therefore, the usefulness of Bayesian quantile cointegration methods is also demonstrated. It shows that Bayesian quantile cointegration performs well and can provide fully information on parameter estimation and cointegration tests.
引文
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