空间经济计量模型Bootstrap Moran检验有效性研究
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摘要
经过二十年的发展,空间经济计量分析已成为经济计量领域的一个重要分支,为处理经济管理活动中的空间相互作用和空间结构等问题提供了新的研究视角与分析工具。同时,由于处理空间问题的复杂性,空间经济计量领域中许多问题有待解决(Anselin,2007)。检验研究对象间的空间相关性存在与否是空间经济计量分析的必要环节。目前的空间经济计量分析中,当误差项服从正态独立同分布时,已有学者证明空间经济计量模型的常用检验统计量将渐近服从正态或χ~2等标准分布,可用于空间经济计量模型检验(Anselin,1988a)。然而,在大量实际经济管理研究工作中,通常可得样本有限、模型误差不满足正态独立同分布(比如,存在异方差或分布未知)的经典假设,常用的空间相关性Moran'sⅠ检验方法失效。在样本量较小或模型误差不服从正态独立同分布条件下,空间相关性Moran'sⅠ检验是目前国际学术界有待解决的难题。
本文将Bootstrap方法应用于构建空间相关性检验的Moran'sⅠ统计量,利用数理推导和Monte Carlo模拟实验,研究Bootstrap方法用于线性回归模型的OLS估计残差,以及空间滞后模型的2SLS或GMM估计残差构建的Moran'sⅠ统计量的有效性。
本文主要研究结论如下:
1、证明线性回归模型Bootstrap Moran检验有效。本研究通过数理推导及模拟实验证明,模型误差满足正态独立同分布经典假设时,线性回归模型Bootstrap Moran检验得到渐近改进;当模型误差不服从正态独立同分布,比如,存在异方差或分布未知时,Moran'sⅠ统计量的渐近理论失效,而线性回归模型Bootstrap Moran检验能够有效地进行空间相关性检验。其中,从水平扭曲角度看,无论模型误差满足正态独立同分布的假定条件与否,当Bootstrap模拟次数大于399时,线性回归模型Bootstrap Moran检验的水平扭曲都稳定地趋近理论值0;从功效角度看,尤其是空间负相关和样本量较小情况下,线性回归模型Bootstrap Moran检验的功效显著大于渐近检验。
2、证明空间滞后模型中Moran'sⅠ统计量,即OLL-Moran检验的渐近分布和精确分布。本研究利用空间滞后模型的2SLS或GMM估计残差,构建Moran'sⅠ统计量,检验空间滞后模型2SLS或GMM估计残差间是否仍存在空间相关性,提出OLL-Moran检验,并从数理角度证明OLL-Moran检验的渐近分布。进而,通过大量Monte Carlo实验发现,从水平扭曲和功效角度综合来看,当误差项服从正态独立同分布时,OLL-Moran检验具有良好的有限样本性质;与Kelejian & Prucha(2001)提出的KP-Moran检验相比,OLL-Moran检验更能有效地识别研究对象间的空间相关性。最后,从理论角度,推导出OLL-Moran检验的精确分布。
3、证明空间滞后模型Bootstrap Moran检验有效。本研究通过数理推导证明,当模型误差服从正态独立同分布时,空间滞后模型中Moran'sⅠ统计量的渐近Edgeworth展开的Bootstrap估计,近似Moran'sⅠ统计量真实分布的收敛速度为D(N~(-1)),空间滞后模型Bootstrap Moran检验获得渐近改进;当样本量较小或模型误差不服从正态独立同分布,存在异方差或分布未知时,无法利用OLL-Moran检验的渐近分布检验空间滞后模型2SLS或GMM估计残差间的空间相关性,而空间滞后模型Bootstrap Moran检验能够有效地进行空间相关性检验。同时,本研究Monte Carlo实验结果验证了空间滞后模型Bootstrap Moran检验有效性的数理推导结论。
基础理论研究与实际应用工具相结合是本文的突出特色。本文的创新之处主要表现在以下四个方面:
1、解决线性回归模型OLS估计残差间的空间相关性Moran'sⅠ检验问题。在样本量较小,以及模型误差不满足空间相关性检验Moran'sⅠ统计量渐近分布经典假设情况下,本文完成的线性回归模型Bootstrap Moran有效性的数理证明和模拟实验,为Bootstrap方法用于OLS估计残差间空间相关性Moran'sⅠ检验,提供数理与计算机模拟实验支撑,具有重要理论价值。本研究结论为有限样本情况下,线性回归模型OLS估计残差间的空间相关性检验提供了一种便捷、有效的分析工具,具有广泛的实际应用价值。
2、改进并拓展空间滞后模型估计残差间的空间相关Moran'sⅠ检验。本文基于空间滞后模型的2SLS或GMM估计残差,完成的OLL-Moran检验渐近分布的数理证明和模拟实验,将空间相关性Moran'sⅠ检验的构建基础从2SLS估计残差拓展至GMM估计残差,改进了空间滞后模型估计残差间空间相关性Moran'sⅠ检验的大样本性质,为空间滞后模型中大样本数据的空间相关性检验提供了便捷、有效的方法;研究中所完成的OLL-Moran检验精确分布的数理证明,为解决有限样本情况下,空间滞后模型2SLS或GMM估计残差间的空间相关性Moran'sⅠ检验问题提供了一种新的研究思路,具有理论研究价值。
3、解决空间滞后模型2SLS或GMM估计残差间的空间相关性Moran'sⅠ检验问题。在样本量较小,以及模型误差不满足OLL-Moran检验渐近分布情况下,本文完成的空间滞后模型Bootstrap Moran有效性的数理证明和模拟实验,为Bootstrap方法用于空间滞后模型2SLS或GMM估计残差间空间相关性Moran'sⅠ检验,提供数理与计算机模拟实验支撑,具有重要理论价值。本研究结论为有限样本情况下,空间滞后模型残差间的空间相关性Moran'sⅠ检验问题提供另一条研究思路,具有实际应用价值。
4、本文在Gauss软件中编写一系列Bootstrap Moran检验程序,研究线性回归模型和空间滞后模型的Bootstrap Moran检验的有效性,是在总结Bootstrap方法和空间相关性Moran'sⅠ检验的实际研究经验基础上的新拓展,大大丰富了Gauss软件工具箱,为空间经济计量研究者提供了便利的研究手段。
Research on spatial econometrics has been a very important branch of the econometric field in the past two decades.It will provide a new research perspective and analysis tool to deal with spatial interaction and structure in economy and management.Due to the complexity of spatial correlation,many problems need to be solved in spatial econometric analysis(Anselin,2007).It is very necessary to test spatial dependence among research objects in spatial econometric analysis.Under classical normality assumption of the model, several common test statistics are asymptotically normal orχ~2 distributed,and can be used to test spatial dependence(Anselin,1988a).However,the sample is limited,and the classical normality assumption of the model is violated in many economy and management research. Then Moran's I test is questionable.So far,under the conditions of small sample or the non-normal i.i.d errors,Moran's I test for spatial dependence is a difficult question in the international academia.
In this paper,bootstrap methods are applied to construct Moran's I test statistic for spatial dependence.Then we make use of mathematical derivation and simulation analysis, and research the efficiency of Bootstrap Moran's I statistic based on OLS residuals of the linear regression model,2SLS residuals of the spatial autoregressive model.
The main conclusions of this dissertation are as follows:
1.The efficiency of bootstrap Moran test in the linear regression model is proved. Specifically speaking,the bootstrap Moran test gains asymptotic refinements under the classical normality assumption of the model;the asymptotic theory of Moran's I statistic is invalidated under the non-normal i.i.d,errors(e.g.,the heteroscedastic or the unknown distributed error),and then spatial dependence can be effectively checked up by the bootstrap Moran test.When the number of bootstrap is more than 399,the size distortion of the bootstrap Moran test in the linear regression model is tending towards 0,and its' power is remarkably higher than asymptotic test under the negative spatial correlation and small samples.
2.The asymptotic and exact distribution of Moran's I statistic in the spatial autoregressive model are proved.In this paper,in order to test spatial dependence among 2SLS or GMM residuals of the spatial autoregressive model,we establish Moran's I statistic based the 2SLS or GMM residuals,obtain the OLL-Moran test,and prove its asymptotic distribution in theory.Then our Monte Carlo experiments indicate that OLL-Moran test has good finite sample properties,and OLL-Moran test is more effectively than KP-Moran test in Kelejian & Prucha(2001) under the normal error.Lastly,the exact distribution of Moran's I statistic is deduced in regards of theory.
3.The efficiency of bootstrap Moran test in the spatial autoregressive model is proved. In this paper,we deduce the following two conclusions.Firstly,bootstrap for the asymptotic Edgeworth expansion of Moran's I statistic in the spatial autoregressive model is approximately the true distribution of one at the rate of O(N~(-1)) under the normal i.i.d,error, and then the bootstrap Moran test gains asymptotic refinements.Secondly,when the error is not normal i.i.d,with heteroscedasticity or unknown distribution,OLL-Moran test is invalid, and the bootstrap Moran test can be effectively find spatial dependence among the 2SLS or GMM residuals of the spatial autoregressive model.Moreover,extensive Monte Carlo simulation results validate the theoretical conclusions about the efficiency of the bootstrap Moran test.
The distinguishing characteristic in this paper is that the basic theoretical research is combined with practical application.The theoretical innovation in this research separately lies in the followed four aspects:
1.The thesis solves the question of Moran's I test for spatial dependence based on OLS residuals of the linear regression model.The paper makes use of mathematical theories and simulation experiments,and then researches the efficiency of bootstrap Moran test in the linear regression model under the small sample or the non-normal i.i.d,error.It provides the sound foundation of mathematical theory and simulation experiments as well as the important theoretical value.These conclusions offer a convenient and effective analysis tool of spatial dependence test among OLS residuals of the linear regression model under the small sample. Furthermore,they have the extensive practical application value.
2.Moran's I test for spatial dependence in the spatial autoregressive model is improved and extended.The asymptotic distribution of OLL-Moran test is deduced by mathematical theories and simulation analysis,where the Moran's I statistic is based on 2SLS or GMM residuals of the spatial autoregressive model.The foundation is expanded from 2SLS residuals to GMM residuals,and then the large sample property of the Moran's I test for spatial dependence is improved.It provides a convenient and effective method in order to test spatial dependence of the spatial autoregressive model under the large sample.Moreover,we provide the exact distribution of OLL-Moran test in the spatial autoregressive model.Another research idea of the problem about Moran's I test for spatial dependence among 2SLS or GMM residuals in the spatial econometric model is proposed under small sample.It's very worthy of theoretical researches.
3.The thesis solves the question of Moran's I test for spatial dependence based on 2SLS or GMM residuals of the spatial autoregressive model.The paper makes use of mathematical theories and simulation experiments,and then researches the efficiency of bootstrap Moran test in the spatial autoregressive model when the sample is small,or the error assumption of the asymptotic distribution of OLL-Moran test isn't satisfied.It provides the sound foundation of mathematical theory and simulation experiments as well as the important theoretical value. These conclusions offer another research idea of spatial dependence test among 2SLS or GMM residuals of the spatial autoregressive model under the small sample.Moreover,there is the broad practical application value.
4.A series of programs about the bootstrap Moran test are designed and written in the Gauss software.The efficiencies of bootstrap Moran test in the linear regression model and the spatial autoregressive model are researched.It is an innovation based on the conclusion of bootstrap methods and the empirical researches of Moran's I test for spatial dependence. Furthermore,the toolbox of Gauss software is greatly enriched,and then the convenient method is provided for the spatial econometric researchers.
本文将Bootstrap方法应用于构建空间相关性检验的Moran'sⅠ统计量,利用数理推导和Monte Carlo模拟实验,研究Bootstrap方法用于线性回归模型的OLS估计残差,以及空间滞后模型的2SLS或GMM估计残差构建的Moran'sⅠ统计量的有效性。
本文主要研究结论如下:
1、证明线性回归模型Bootstrap Moran检验有效。本研究通过数理推导及模拟实验证明,模型误差满足正态独立同分布经典假设时,线性回归模型Bootstrap Moran检验得到渐近改进;当模型误差不服从正态独立同分布,比如,存在异方差或分布未知时,Moran'sⅠ统计量的渐近理论失效,而线性回归模型Bootstrap Moran检验能够有效地进行空间相关性检验。其中,从水平扭曲角度看,无论模型误差满足正态独立同分布的假定条件与否,当Bootstrap模拟次数大于399时,线性回归模型Bootstrap Moran检验的水平扭曲都稳定地趋近理论值0;从功效角度看,尤其是空间负相关和样本量较小情况下,线性回归模型Bootstrap Moran检验的功效显著大于渐近检验。
2、证明空间滞后模型中Moran'sⅠ统计量,即OLL-Moran检验的渐近分布和精确分布。本研究利用空间滞后模型的2SLS或GMM估计残差,构建Moran'sⅠ统计量,检验空间滞后模型2SLS或GMM估计残差间是否仍存在空间相关性,提出OLL-Moran检验,并从数理角度证明OLL-Moran检验的渐近分布。进而,通过大量Monte Carlo实验发现,从水平扭曲和功效角度综合来看,当误差项服从正态独立同分布时,OLL-Moran检验具有良好的有限样本性质;与Kelejian & Prucha(2001)提出的KP-Moran检验相比,OLL-Moran检验更能有效地识别研究对象间的空间相关性。最后,从理论角度,推导出OLL-Moran检验的精确分布。
3、证明空间滞后模型Bootstrap Moran检验有效。本研究通过数理推导证明,当模型误差服从正态独立同分布时,空间滞后模型中Moran'sⅠ统计量的渐近Edgeworth展开的Bootstrap估计,近似Moran'sⅠ统计量真实分布的收敛速度为D(N~(-1)),空间滞后模型Bootstrap Moran检验获得渐近改进;当样本量较小或模型误差不服从正态独立同分布,存在异方差或分布未知时,无法利用OLL-Moran检验的渐近分布检验空间滞后模型2SLS或GMM估计残差间的空间相关性,而空间滞后模型Bootstrap Moran检验能够有效地进行空间相关性检验。同时,本研究Monte Carlo实验结果验证了空间滞后模型Bootstrap Moran检验有效性的数理推导结论。
基础理论研究与实际应用工具相结合是本文的突出特色。本文的创新之处主要表现在以下四个方面:
1、解决线性回归模型OLS估计残差间的空间相关性Moran'sⅠ检验问题。在样本量较小,以及模型误差不满足空间相关性检验Moran'sⅠ统计量渐近分布经典假设情况下,本文完成的线性回归模型Bootstrap Moran有效性的数理证明和模拟实验,为Bootstrap方法用于OLS估计残差间空间相关性Moran'sⅠ检验,提供数理与计算机模拟实验支撑,具有重要理论价值。本研究结论为有限样本情况下,线性回归模型OLS估计残差间的空间相关性检验提供了一种便捷、有效的分析工具,具有广泛的实际应用价值。
2、改进并拓展空间滞后模型估计残差间的空间相关Moran'sⅠ检验。本文基于空间滞后模型的2SLS或GMM估计残差,完成的OLL-Moran检验渐近分布的数理证明和模拟实验,将空间相关性Moran'sⅠ检验的构建基础从2SLS估计残差拓展至GMM估计残差,改进了空间滞后模型估计残差间空间相关性Moran'sⅠ检验的大样本性质,为空间滞后模型中大样本数据的空间相关性检验提供了便捷、有效的方法;研究中所完成的OLL-Moran检验精确分布的数理证明,为解决有限样本情况下,空间滞后模型2SLS或GMM估计残差间的空间相关性Moran'sⅠ检验问题提供了一种新的研究思路,具有理论研究价值。
3、解决空间滞后模型2SLS或GMM估计残差间的空间相关性Moran'sⅠ检验问题。在样本量较小,以及模型误差不满足OLL-Moran检验渐近分布情况下,本文完成的空间滞后模型Bootstrap Moran有效性的数理证明和模拟实验,为Bootstrap方法用于空间滞后模型2SLS或GMM估计残差间空间相关性Moran'sⅠ检验,提供数理与计算机模拟实验支撑,具有重要理论价值。本研究结论为有限样本情况下,空间滞后模型残差间的空间相关性Moran'sⅠ检验问题提供另一条研究思路,具有实际应用价值。
4、本文在Gauss软件中编写一系列Bootstrap Moran检验程序,研究线性回归模型和空间滞后模型的Bootstrap Moran检验的有效性,是在总结Bootstrap方法和空间相关性Moran'sⅠ检验的实际研究经验基础上的新拓展,大大丰富了Gauss软件工具箱,为空间经济计量研究者提供了便利的研究手段。
Research on spatial econometrics has been a very important branch of the econometric field in the past two decades.It will provide a new research perspective and analysis tool to deal with spatial interaction and structure in economy and management.Due to the complexity of spatial correlation,many problems need to be solved in spatial econometric analysis(Anselin,2007).It is very necessary to test spatial dependence among research objects in spatial econometric analysis.Under classical normality assumption of the model, several common test statistics are asymptotically normal orχ~2 distributed,and can be used to test spatial dependence(Anselin,1988a).However,the sample is limited,and the classical normality assumption of the model is violated in many economy and management research. Then Moran's I test is questionable.So far,under the conditions of small sample or the non-normal i.i.d errors,Moran's I test for spatial dependence is a difficult question in the international academia.
In this paper,bootstrap methods are applied to construct Moran's I test statistic for spatial dependence.Then we make use of mathematical derivation and simulation analysis, and research the efficiency of Bootstrap Moran's I statistic based on OLS residuals of the linear regression model,2SLS residuals of the spatial autoregressive model.
The main conclusions of this dissertation are as follows:
1.The efficiency of bootstrap Moran test in the linear regression model is proved. Specifically speaking,the bootstrap Moran test gains asymptotic refinements under the classical normality assumption of the model;the asymptotic theory of Moran's I statistic is invalidated under the non-normal i.i.d,errors(e.g.,the heteroscedastic or the unknown distributed error),and then spatial dependence can be effectively checked up by the bootstrap Moran test.When the number of bootstrap is more than 399,the size distortion of the bootstrap Moran test in the linear regression model is tending towards 0,and its' power is remarkably higher than asymptotic test under the negative spatial correlation and small samples.
2.The asymptotic and exact distribution of Moran's I statistic in the spatial autoregressive model are proved.In this paper,in order to test spatial dependence among 2SLS or GMM residuals of the spatial autoregressive model,we establish Moran's I statistic based the 2SLS or GMM residuals,obtain the OLL-Moran test,and prove its asymptotic distribution in theory.Then our Monte Carlo experiments indicate that OLL-Moran test has good finite sample properties,and OLL-Moran test is more effectively than KP-Moran test in Kelejian & Prucha(2001) under the normal error.Lastly,the exact distribution of Moran's I statistic is deduced in regards of theory.
3.The efficiency of bootstrap Moran test in the spatial autoregressive model is proved. In this paper,we deduce the following two conclusions.Firstly,bootstrap for the asymptotic Edgeworth expansion of Moran's I statistic in the spatial autoregressive model is approximately the true distribution of one at the rate of O(N~(-1)) under the normal i.i.d,error, and then the bootstrap Moran test gains asymptotic refinements.Secondly,when the error is not normal i.i.d,with heteroscedasticity or unknown distribution,OLL-Moran test is invalid, and the bootstrap Moran test can be effectively find spatial dependence among the 2SLS or GMM residuals of the spatial autoregressive model.Moreover,extensive Monte Carlo simulation results validate the theoretical conclusions about the efficiency of the bootstrap Moran test.
The distinguishing characteristic in this paper is that the basic theoretical research is combined with practical application.The theoretical innovation in this research separately lies in the followed four aspects:
1.The thesis solves the question of Moran's I test for spatial dependence based on OLS residuals of the linear regression model.The paper makes use of mathematical theories and simulation experiments,and then researches the efficiency of bootstrap Moran test in the linear regression model under the small sample or the non-normal i.i.d,error.It provides the sound foundation of mathematical theory and simulation experiments as well as the important theoretical value.These conclusions offer a convenient and effective analysis tool of spatial dependence test among OLS residuals of the linear regression model under the small sample. Furthermore,they have the extensive practical application value.
2.Moran's I test for spatial dependence in the spatial autoregressive model is improved and extended.The asymptotic distribution of OLL-Moran test is deduced by mathematical theories and simulation analysis,where the Moran's I statistic is based on 2SLS or GMM residuals of the spatial autoregressive model.The foundation is expanded from 2SLS residuals to GMM residuals,and then the large sample property of the Moran's I test for spatial dependence is improved.It provides a convenient and effective method in order to test spatial dependence of the spatial autoregressive model under the large sample.Moreover,we provide the exact distribution of OLL-Moran test in the spatial autoregressive model.Another research idea of the problem about Moran's I test for spatial dependence among 2SLS or GMM residuals in the spatial econometric model is proposed under small sample.It's very worthy of theoretical researches.
3.The thesis solves the question of Moran's I test for spatial dependence based on 2SLS or GMM residuals of the spatial autoregressive model.The paper makes use of mathematical theories and simulation experiments,and then researches the efficiency of bootstrap Moran test in the spatial autoregressive model when the sample is small,or the error assumption of the asymptotic distribution of OLL-Moran test isn't satisfied.It provides the sound foundation of mathematical theory and simulation experiments as well as the important theoretical value. These conclusions offer another research idea of spatial dependence test among 2SLS or GMM residuals of the spatial autoregressive model under the small sample.Moreover,there is the broad practical application value.
4.A series of programs about the bootstrap Moran test are designed and written in the Gauss software.The efficiencies of bootstrap Moran test in the linear regression model and the spatial autoregressive model are researched.It is an innovation based on the conclusion of bootstrap methods and the empirical researches of Moran's I test for spatial dependence. Furthermore,the toolbox of Gauss software is greatly enriched,and then the convenient method is provided for the spatial econometric researchers.
引文
(?)由于处理区域经济计量模型中附属国数据的需要,20世纪70年代早期在欧洲出现了空间计量经济分析这一研究领域(Hordijk & Paelinck.1976)[100]。
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