阈值协整及其对我国的应用研究
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摘要
宏观经济中的时间序列数据大多数是单位根过程,标准的协整理论和方法因而得到快速发展并成为现代时间序列经济研究的最常用方法。在标准协整框架下,协整向量刻画了变量之间的长期均衡关系,误差校正模型描述了协整变量的短期关系及其调节效应,因此,协整向量和误差校正模型相辅相成、互为补充,联合描述了变量的长期关系和短期变化。Granger正是对这一理论所作的原创性贡献而获得2003年诺贝尔经济学奖。但是,标准的协整理论至少包含三个严格假定:第一,长期均衡是线性的;第二,向长期均衡的调节是对称的;第三,向长期均衡调节的速度是不变的。然而,现实经济中变量之间的相互关系常常具有非线性特征,一旦使用标准协整方法分析具有非线性特征的经济变量之间的关系,得出的结论很可能是错误的。阈值协整正是针对上述标准协整理论的缺陷而进行的扩展,将非平稳和非线性结合起来,研究实际经济问题中的非线性问题,因而,阈值协整理论得到西方学者的青睐并成为国际计量经济学的前沿热点领域之一。
阈值协整的基本思想是在标准协整框架下,引入TR(STR)模型去刻画调节效应或长期协整关系。但是,阈值协整是最近才产生的计量经济学前沿热点领域,还有许多问题急待完善和发展。本文针对实际经济问题的需要,扩展和改进现有的非线性阈值协整方法,使之能够更好地应用于实际经济问题研究。本文的方法论贡献主要体现在以下四个方面:(1)Kapetanios, Shin(2004)在ECM中使用指数函数为转移函数刻画非线性调节效应,本文将转移函数扩展为未知,并设定检验阈值协整和转移函数形式的检验程序和构造相应的统计量。仿真结果显示,本文的改进有限样本性质很好。(2)Hansen,Seo(2002)在阈值协整向量未知条件下,在ECM中使用TR模型刻画非线性调节特征,因此,这种模型设定刻画的调节效应是急剧变化的,这与许多实际经济背景不一致。本文在协整向量和转移函数未知条件下,使用STR模型描述非线性调节效应,由此而刻画的调节效应是连续的。进一步,本文构造了相应的检验程序和估计方法对这一扩展模型进行估计和检验,仿真实验显示,本文的方法估计得到的参数具有一致性,构造的统计量具有较好的有限样本性质。(3)Choi, Saikkonen(2004)允许解释变量内生和随机误差项序列相关,研究在协整向量具有阈值效应时,基于动态最小二乘估计量检验阈值协整的方法。进一步,一旦检验结果表明协整向量具有非线性特征,Choi,Saikkonen(2005)提出使用动态非线性最小二乘估计量估计阈值协整向量,并进而对动态非线性最小二乘估计量的残差进行分块,使用分块残差构造统计量检验阈值协整。本文将上述基于动态非线性最小二乘估计量的一系列估计、检验方法扩展为完全修正的最小二乘法。我们的数学推导和仿真实验表明动态非线性最小二乘估计量和完全修正的最小二乘估计量具有相同的极限分布,但在有限样本下,完全修正的最小二乘估计量优于动态非线性最小二乘估计量。(4)本文将阈值协整扩展至非平稳的面板数据上,使用TR模型刻画异质面板协整向量,并基于Westerlund,Edgerton(2005)的思想构造检验面板阈值协整的统计量。本文的数学推导表明,面板阈值协整检验统计量的极限分布为正态分布,并不依赖未知参数,进一步,仿真实验发现,阈值面板协整统计量有较好的有限样本性质。
为对本文所介绍的方法提供较完整的应用案例,本文首先分别对泰勒规则在我国货币政策中的应用,我国菲利普斯曲线的机制转移以及我国农民医疗卫生支出的影响因素三个问题进行研究。其中,对泰勒规则的研究使用的是基于ECM调节效应的阈值协整,对菲利普斯曲线研究使用的是定义于协整向量的阈值协整方法,对农民医疗卫生支出的研究使用的是面板协整。最后,作为较完整的专题分析,本文使用阈值协整和面板数据机制转移协整对我国城乡收入差距与经济增长的关系进行研究。我们首先基于我国的城乡二元经济结构特征定义度量我国城乡收入差距的泰尔指数,然后根据我国城乡收入差距和经济增长的数据变化特征以及我国的实际经济结构转型背景,分别设定能够反映城乡收入差距对经济增长的效应因收入差距水平和经济发展阶段的不同而有机制变化特征的面板协整模型和阈值协整模型。模型估计结果表明:我国的城乡收入差距与经济增长之间存在非线性协整关系:城乡收入差距对经济增长的长期效应取决于城乡收入差距水平和经济发展阶段,并有地区差异。这一结论说明,改革初期的城乡收入差距促进了经济增长,而现阶段城乡收入差距的扩大对经济增长产生阻滞作用。并且,这种长期效应抑制了短期经济增长并对城乡收入差距的扩大产生刺激效应。
In macroeconomy, most of time series data are unit root process; therefore the theory of cointegration has undergone considerable development and various econometric and statistical methods have been developed for the analysis of cointegrated time series and many of them are now routinely applied in empirical studies. Under the standard cointegration frame, cointegration vector characterizes the long-run equilibrium relationship, and the ECM characterizes the short-run variety and adjustment effect, so the cointegration vector and ECM are supplement each other. Granger gained Nobel Prize of economics in 2003 for the contribution to cointegration. But standard cointegration theory imposes at least three strong restrictions on the underlying economic behavior:(1)the long-run equilibrium is unque;(2)the adjustment toward the equilibrium is symmetric; (3)the equilibrium correction is a constant proportion of the previous equilibrium error. Those indicate that the cointegration relationship and adjustment effect of standard cointegration are linear, which is according to economic theory and actual economic status quo, need not be the case. Once ignoring nonlinear of actual, your results maybe wrong when you instead of nonlinear model with linear model. Threshold cointegration expends standard cointegration with nonlinear aiming to the shortcomings of standard cointegration. The goal of threshold cointegration is to study the nonlinearity and nonstationarity in economics; therefore, threshold cointegration became a central part of modern time series econometrics.
The basic idea of threshold cointegration is using the TR (or STR) model to characterize the long-run equilibrium and adjustment effect in cointegration frame. But threshold cointegration is an advanced econometrics theory developed recently, there are much problems needing to solution. Base on the need of actual economic status quo, this paper expands and ameliorate the disadvantages of existing threshold cointegration. So the threshold cointegration developed in this paper can better describe actual economic. The contributions of this paper in theory are those: (1) Kapetanios, Shin (2004) use exponential function describe nonlinear adjustment effect in ECM. In this paper, we describe nonlinear adjustment effect in ECM with unknown transition function (exponential function or logistic function), and develop test procedures and test statistic to test transition function. The Monte Carlo simulation result shows its small-sample wonderful performances. (2) Under the condition of threshold cointegration vector is unknown, Hansen,Seo(2002) characterizes nonlinear adjustment effect with two-regime TR model in ECM, therefore the adjustment effect change sharply, which is too restrictive and not fit of many actual economic phenomena. This paper develops an ECM,in which the threshold cointegration vector and smooth transition function are unknown, to characterize nonlinear adjustment effect, therefore adjustment is continuous. Furthermore, we propose test procedures and estimation method to test and estimatethreshold cointegration vector and adjustment effect of the ECM. The simulation result shows that estimated parameters are consistent and test statistic have good small sample performances. (3) Choi, Saikkonen(2004) follow the common practice in threshold cointegration and permit both serial and contemporaneous correlations between the regressions and the error term of the model. In order to allow for this feature, they estimate the model with DNLS and propose LM statistic to test linearity. Once the test results show that the model is nonlinear (threshold cointegration), Choi,Saikkonen(2005) propose DNLS to estimate cointegration vector and employ KPSS statistic, calculating with subsamples of the DNLS residuals, to test threshold cointegration. This paper extend Choi, Saikkonen’s(2004,2005) test and estimating methods with fully modify OLS. The mathematics proofs and simulation show that the test statistics developed by this paper have same limiting distribution with Choi, Saikkonen(2005)and fully modify OLS is better than DNLLS under small-sample. (4) We extend threshold cointegration to panel data and use TR model to characterize panel cointegration vector. We base on Westerlund,Edgerton’s(2005) idea to develop panel threshold cointegration test statistic and derive the limiting distribution of our statistic, and found the limiting distribution is normal distribution and doesn’t depend on nuisance parameters. Furthermore, the Monte Carlo simulation result shows panel threshold cointegration statistic small-sample wonderful performances.
As the examples of empirical study, firstly three cases which are Taylor Rule in China’s monetary policy, the regime switch of Phillips Curve and the factors that affect on the farmer’s expending on health care are investigated respectively in the paper. The analysis of Taylor Rule is based on threshold cointegration defined on adjustment effect of ECM, while the study of Phillips Curve is based on threshold cointegration defined on cointegration vector, and the farmer’s expending on health care is based on panel data cointegration. Lastly, this paper studies the effect of the urban-rural income disparity to actual economic growth with regime switch panel data cointegration and threshold cointegration in China. We firstly define and calculate the Theil index ,which measure the urban rural income disparity , based on the dual structure in urban and rural economy, and then specify a panel cointegration model and threshold cointegration model with respect to the Chinese situation of rural-urban income disparity and economic restructure transforming. The estimated results indicate that there is nonlinear cointegartion relationship witch depending on the level of the rural-urban income disparity and the economic stage. Such result implies that the effect of rural-urban income disparity to economic growth is positive at the beginning of Chinese reform; however, the effect is negative at present. Estimated PVECM indicates that the long-run effects of the disparity depress the economic growth and stimulate the disparity in short run.
阈值协整的基本思想是在标准协整框架下,引入TR(STR)模型去刻画调节效应或长期协整关系。但是,阈值协整是最近才产生的计量经济学前沿热点领域,还有许多问题急待完善和发展。本文针对实际经济问题的需要,扩展和改进现有的非线性阈值协整方法,使之能够更好地应用于实际经济问题研究。本文的方法论贡献主要体现在以下四个方面:(1)Kapetanios, Shin(2004)在ECM中使用指数函数为转移函数刻画非线性调节效应,本文将转移函数扩展为未知,并设定检验阈值协整和转移函数形式的检验程序和构造相应的统计量。仿真结果显示,本文的改进有限样本性质很好。(2)Hansen,Seo(2002)在阈值协整向量未知条件下,在ECM中使用TR模型刻画非线性调节特征,因此,这种模型设定刻画的调节效应是急剧变化的,这与许多实际经济背景不一致。本文在协整向量和转移函数未知条件下,使用STR模型描述非线性调节效应,由此而刻画的调节效应是连续的。进一步,本文构造了相应的检验程序和估计方法对这一扩展模型进行估计和检验,仿真实验显示,本文的方法估计得到的参数具有一致性,构造的统计量具有较好的有限样本性质。(3)Choi, Saikkonen(2004)允许解释变量内生和随机误差项序列相关,研究在协整向量具有阈值效应时,基于动态最小二乘估计量检验阈值协整的方法。进一步,一旦检验结果表明协整向量具有非线性特征,Choi,Saikkonen(2005)提出使用动态非线性最小二乘估计量估计阈值协整向量,并进而对动态非线性最小二乘估计量的残差进行分块,使用分块残差构造统计量检验阈值协整。本文将上述基于动态非线性最小二乘估计量的一系列估计、检验方法扩展为完全修正的最小二乘法。我们的数学推导和仿真实验表明动态非线性最小二乘估计量和完全修正的最小二乘估计量具有相同的极限分布,但在有限样本下,完全修正的最小二乘估计量优于动态非线性最小二乘估计量。(4)本文将阈值协整扩展至非平稳的面板数据上,使用TR模型刻画异质面板协整向量,并基于Westerlund,Edgerton(2005)的思想构造检验面板阈值协整的统计量。本文的数学推导表明,面板阈值协整检验统计量的极限分布为正态分布,并不依赖未知参数,进一步,仿真实验发现,阈值面板协整统计量有较好的有限样本性质。
为对本文所介绍的方法提供较完整的应用案例,本文首先分别对泰勒规则在我国货币政策中的应用,我国菲利普斯曲线的机制转移以及我国农民医疗卫生支出的影响因素三个问题进行研究。其中,对泰勒规则的研究使用的是基于ECM调节效应的阈值协整,对菲利普斯曲线研究使用的是定义于协整向量的阈值协整方法,对农民医疗卫生支出的研究使用的是面板协整。最后,作为较完整的专题分析,本文使用阈值协整和面板数据机制转移协整对我国城乡收入差距与经济增长的关系进行研究。我们首先基于我国的城乡二元经济结构特征定义度量我国城乡收入差距的泰尔指数,然后根据我国城乡收入差距和经济增长的数据变化特征以及我国的实际经济结构转型背景,分别设定能够反映城乡收入差距对经济增长的效应因收入差距水平和经济发展阶段的不同而有机制变化特征的面板协整模型和阈值协整模型。模型估计结果表明:我国的城乡收入差距与经济增长之间存在非线性协整关系:城乡收入差距对经济增长的长期效应取决于城乡收入差距水平和经济发展阶段,并有地区差异。这一结论说明,改革初期的城乡收入差距促进了经济增长,而现阶段城乡收入差距的扩大对经济增长产生阻滞作用。并且,这种长期效应抑制了短期经济增长并对城乡收入差距的扩大产生刺激效应。
In macroeconomy, most of time series data are unit root process; therefore the theory of cointegration has undergone considerable development and various econometric and statistical methods have been developed for the analysis of cointegrated time series and many of them are now routinely applied in empirical studies. Under the standard cointegration frame, cointegration vector characterizes the long-run equilibrium relationship, and the ECM characterizes the short-run variety and adjustment effect, so the cointegration vector and ECM are supplement each other. Granger gained Nobel Prize of economics in 2003 for the contribution to cointegration. But standard cointegration theory imposes at least three strong restrictions on the underlying economic behavior:(1)the long-run equilibrium is unque;(2)the adjustment toward the equilibrium is symmetric; (3)the equilibrium correction is a constant proportion of the previous equilibrium error. Those indicate that the cointegration relationship and adjustment effect of standard cointegration are linear, which is according to economic theory and actual economic status quo, need not be the case. Once ignoring nonlinear of actual, your results maybe wrong when you instead of nonlinear model with linear model. Threshold cointegration expends standard cointegration with nonlinear aiming to the shortcomings of standard cointegration. The goal of threshold cointegration is to study the nonlinearity and nonstationarity in economics; therefore, threshold cointegration became a central part of modern time series econometrics.
The basic idea of threshold cointegration is using the TR (or STR) model to characterize the long-run equilibrium and adjustment effect in cointegration frame. But threshold cointegration is an advanced econometrics theory developed recently, there are much problems needing to solution. Base on the need of actual economic status quo, this paper expands and ameliorate the disadvantages of existing threshold cointegration. So the threshold cointegration developed in this paper can better describe actual economic. The contributions of this paper in theory are those: (1) Kapetanios, Shin (2004) use exponential function describe nonlinear adjustment effect in ECM. In this paper, we describe nonlinear adjustment effect in ECM with unknown transition function (exponential function or logistic function), and develop test procedures and test statistic to test transition function. The Monte Carlo simulation result shows its small-sample wonderful performances. (2) Under the condition of threshold cointegration vector is unknown, Hansen,Seo(2002) characterizes nonlinear adjustment effect with two-regime TR model in ECM, therefore the adjustment effect change sharply, which is too restrictive and not fit of many actual economic phenomena. This paper develops an ECM,in which the threshold cointegration vector and smooth transition function are unknown, to characterize nonlinear adjustment effect, therefore adjustment is continuous. Furthermore, we propose test procedures and estimation method to test and estimatethreshold cointegration vector and adjustment effect of the ECM. The simulation result shows that estimated parameters are consistent and test statistic have good small sample performances. (3) Choi, Saikkonen(2004) follow the common practice in threshold cointegration and permit both serial and contemporaneous correlations between the regressions and the error term of the model. In order to allow for this feature, they estimate the model with DNLS and propose LM statistic to test linearity. Once the test results show that the model is nonlinear (threshold cointegration), Choi,Saikkonen(2005) propose DNLS to estimate cointegration vector and employ KPSS statistic, calculating with subsamples of the DNLS residuals, to test threshold cointegration. This paper extend Choi, Saikkonen’s(2004,2005) test and estimating methods with fully modify OLS. The mathematics proofs and simulation show that the test statistics developed by this paper have same limiting distribution with Choi, Saikkonen(2005)and fully modify OLS is better than DNLLS under small-sample. (4) We extend threshold cointegration to panel data and use TR model to characterize panel cointegration vector. We base on Westerlund,Edgerton’s(2005) idea to develop panel threshold cointegration test statistic and derive the limiting distribution of our statistic, and found the limiting distribution is normal distribution and doesn’t depend on nuisance parameters. Furthermore, the Monte Carlo simulation result shows panel threshold cointegration statistic small-sample wonderful performances.
As the examples of empirical study, firstly three cases which are Taylor Rule in China’s monetary policy, the regime switch of Phillips Curve and the factors that affect on the farmer’s expending on health care are investigated respectively in the paper. The analysis of Taylor Rule is based on threshold cointegration defined on adjustment effect of ECM, while the study of Phillips Curve is based on threshold cointegration defined on cointegration vector, and the farmer’s expending on health care is based on panel data cointegration. Lastly, this paper studies the effect of the urban-rural income disparity to actual economic growth with regime switch panel data cointegration and threshold cointegration in China. We firstly define and calculate the Theil index ,which measure the urban rural income disparity , based on the dual structure in urban and rural economy, and then specify a panel cointegration model and threshold cointegration model with respect to the Chinese situation of rural-urban income disparity and economic restructure transforming. The estimated results indicate that there is nonlinear cointegartion relationship witch depending on the level of the rural-urban income disparity and the economic stage. Such result implies that the effect of rural-urban income disparity to economic growth is positive at the beginning of Chinese reform; however, the effect is negative at present. Estimated PVECM indicates that the long-run effects of the disparity depress the economic growth and stimulate the disparity in short run.
引文
①王少平.宏观计量的若干前沿理论与应用.天津:南开大学出版社,2003.168-178.
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[53] Galor,O., and Moav,O., From Physical to Human Capital Accumulation: Inequalityin the Process of Development, Review of Economic Studies, 2004(4): 1001-1026.
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[66] Hansen, B.E., and Seo,B., Testing for Two-regime Threshold Cointegration inVector Error Correction Models, Journal of Econometrics, 2002(110): 293–318.
[67] Hongyi,L., and Maddala,G.S., Bootstrapping Cointegrating Regression, Journal of Econometrics, 1997(80):297-318.
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[73] Kapetanios,G.,Shin,Y.and Snell.A., Testing for Cointegration in Nonlinear Smooth Transition Error Correction Models, Econometric Theory, 2006(22):279-303.
[74] Kapetanios,G..and Shin, Y.,Unit Root Tests in Three-regime SETAR Models, Econometrics Journal, 2006(9):252–278.
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[80] Lundbergh,S., Ter a svirta, T. and Dijk,D.V.,Time-varying Smooth Transition Autoregressive Models, Journal of Business and Economic Statistics,2003(21):104- 121
[81] Luukkonen,R., Saikkonen, P.and Tera svirta,T., Testing Linearity Against Smooth Transition Autoregressive Models, Biometrika, 1988(75): 491-499.
[82] Maddala,G.S.,and Kim,I., Unit Roots Cointegration and Structural Change, Cambridge University Press,1998.
[83] Maki,D., The Performance of Variance Ratio Unit Root Tests Under Nonlinear Stationary TAR and STAR Processes: Evidence from Monte Carlo Simulations and Applications, Computational Economics,2007(1):77-94.
[84] McCoskey,S.and Kao, C., A Residual-Based Test of the Null Cointegration in Panel Data, Econometric Reviews ,1998(17):57-84.
[85] Michael,P., Nobay, R.A.and Peel,D.A.,Transactions Costs and Nonlinear Adjustment in Real Exchange Rates:An Empirical Investigation, Journal of Political Economy,1997(105):862-879.
[86] Ng,S.and Perron, P.,Unit Root Tests in ARMA Models with Data-dependent Methods for the Selection of the Truncation Lag,Journal of the American Statistical Association,1995(90): 268-281.
[87] Panizza, U., Income and Economic Growth: Evidence from America Data, Journal of Economic Growth,2002(7): 25-41.
[88] Pedroni, P., Critical Values For Cointegration Tests in Heterogeneous Panels With multiple Regressors, Oxford Bulletin of Economics and Statistics ,1999(61): 653-670.
[89] Pedroni, P., Panel Cointegration: Asymptotic and Finite Sample Properties ofPooled Time Series Tests, with an Application to the PPP Hypothesis, Econometric Theory, 2004(8):597-625.
[90] Perotti,R., Growth, Income Distribution and Democracy: What the Data Say, Journal of Economic Growth, 1996(2):149-187.
[91] Perron, P., Test Consistency with Varying Sampling Frequency. Econometric Theory. 1991(7):341-368.
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