函数型核加权估计法及其在经济学中的应用
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摘要
本文基于变系数模型提出了一个新的统计推断方法:函数型核函数加权最小二乘法.该方法将变系数模型中经典的核函数加权最小二乘法和参数模型中的函数型最小二乘法巧妙结合,通过条件特征函数构造损失函数进而定义了函数型核函数最小二乘估计量.该估计量既具有函数型最小二乘法的优势——在扰动项服从厚尾分布时也能够稳健估计参数,又具有非参数核估计的特点——估计量的相合性不依赖于参数模型的正确设定.同时,本文探讨了该估计量的大样本性质,证明了其相合性和渐近正态性.进一步,本文研究了该估计量的自适应估计,即基于估计量渐近方差的相合估计量来选择最优估计.最后,本文通过数值模拟来探究函数型核函数最小二乘法的有限样本性质,并将该方法应用到我国PM_(2.5)和经济增长关系的研究中.
In this article, we propose a new method called functional kernel-weighted least squares(FKLS)method to estimate the smooth coefficient function in semi-parametric smooth coefficient model. This novel proposal ingeniously combines the kernel-weighted least squares(KLS) and the functional least squares(FLS) methods. The corresponding FKLS estimator is defined based on the loss function constructed by the conditional characteristic function. It not only has the advantage of FLS method that can produce robust parameter estimation even if the disturbance is subject to heavy tailed distributions, but also has the characteristics of non-parametric kernel estimation that consistency can be achieved without the knowledge of the correct functional form. The consistency and asymptotic normality of the proposed estimator axe established. Furthermore, adaptive estimation is investigated based on the consistent estimator of the asymptotic variance. Finally, superiority of the FKLS estimator in finite samples, compared to the KLS estimator, is demonstrated through simulated numerical examples and the study of PM_(2.5) and economic growth in China.
In this article, we propose a new method called functional kernel-weighted least squares(FKLS)method to estimate the smooth coefficient function in semi-parametric smooth coefficient model. This novel proposal ingeniously combines the kernel-weighted least squares(KLS) and the functional least squares(FLS) methods. The corresponding FKLS estimator is defined based on the loss function constructed by the conditional characteristic function. It not only has the advantage of FLS method that can produce robust parameter estimation even if the disturbance is subject to heavy tailed distributions, but also has the characteristics of non-parametric kernel estimation that consistency can be achieved without the knowledge of the correct functional form. The consistency and asymptotic normality of the proposed estimator axe established. Furthermore, adaptive estimation is investigated based on the consistent estimator of the asymptotic variance. Finally, superiority of the FKLS estimator in finite samples, compared to the KLS estimator, is demonstrated through simulated numerical examples and the study of PM_(2.5) and economic growth in China.
引文
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[18]彭水军,包群.经济增长与环境污染——环境库兹涅茨曲线假说的中国检验[J].财经问题研究,2006(8):3-17.Peng S J, Bao Q. Economic growth and environmental pollution:An empirical test for the environmental Kuznets curve hypothesis in China[J]. Research on Financial and Economic Issues, 2006(8):3-17.
[19]虞义华,郑新业,张莉.经济发展水平、产业结构与碳排放强度——中国省级面板数据分析[J].经济理论与经济管理,2011(3):72-81.Yu Y H, Zheng X Y, Zhang L. Carbon dioxide emission and economic development:A panel data analysis[J].Economic Theory and Business Management, 2011(3):72-81.
[20]王志华,温宗国,闫芳,等.北京环境库兹涅茨曲线假设的验证[J].中国人口·资源与环境,2007(2):40-47.Wang Z H, Wen Z G, Yan F, et al. Verifying the environmental Kuznets curve hypothesis and its conditions in Beijing[J]. China Population, Resources and Environment, 2007(2):40-47.
[21]Bierens H J. Introduction to the mathematical and statistical foundations of econometrics[M]. Cambridge University Press, 2004.
[22]van Der Vaart A W. Asymptotic statistics[M]. Cambridge University Press, 1998.
[2]Chen R, Tsay R S. Functional-coefficient autoregressive models[J]. Journal of the American Statistical Association, 1993, 88(421):298-308.
[3]Hastie T, Tibshirani R. Varying-coefficient models[J]. Journal of the Royal Statistical Society, 1993, 55(4):757-796.
[4]Li Q, Huang C J, Li D, et al. Semiparametric smooth coefficient models[J]. Journal of Business and Economic Statistics, 2002, 20(3):412-422.
[5]Cai Z, Fan J, Yao Q. Functional-coefficient regression models for nonlinear time series[J]. Journal of the American Statistical Association, 2000, 95:941-956.
[6]蔡宗武,陈琳娜,方颖.人民币汇率的半参数预测模型[J].系统工程理论与实践,2012, 32(4):685-692.Cai Z W,Chen L N, Fang Y. Semi-parametric forecasting model for USD/CNY exchange rate[J]. Systems Engineering—Theory&Practice, 2012, 32(4):685-692.
[7]Cai Z, Li Q, Park J Y. Functional-coefficient models for nonstationary time series data[J]. Journal of Econometrics, 2009, 148(2):101-113.
[8]Xiao Z. Functional-coefficient cointegration models[J]. Journal of Econometrics, 2009, 152(2):81-92.
[9]Tu Y, Wang Y. Adaptive estimation of functional-coefficient cointegration models with nonstationary volatility[R]. Technical Report, Peking University, 2017.
[10]Chambers R L, Heathcote C R. On the estimation of slope and the identification of outliers in linear regression[J].Biometrika, 1981, 68(1):21-33.
[11]Heathcote C R. Linear regression by functional least squares[J]. Journal of Applied Probability, 1982, 19(A):225-239.
[12]Csorgo S. The theory of functional least squares[J]. Journal of the Australian Mathematical Society, 1983, 34(3):336-355.
[13]Heathcote C R, Welsh A H. The robust estimation of autoregressive processes by functional least squares[J].Journal of Applied Probability, 1983, 20(4):737-753.
[14]Welsh A H, Nicholls D F. Robust estimation of regression models with dependent regressors:The functional least squares approach[J]. Econometric Theory, 1986, 2(1):132-150.
[15]Heathcote C R, Welsh A H. Multivariate functional least squares[J]. Journal of Multivariate Analysis, 1988,25(1):45-64.
[16]Silverman B W. Density estimation for statistics and data analysis[M]. Chapman and Hall, 1986.
[17]Grossman G M, Krueger A B. Environmental impacts of a North American free trade agreement[J]. Social Science Electronic Publishing, 1991, 8(2):223-250.
[18]彭水军,包群.经济增长与环境污染——环境库兹涅茨曲线假说的中国检验[J].财经问题研究,2006(8):3-17.Peng S J, Bao Q. Economic growth and environmental pollution:An empirical test for the environmental Kuznets curve hypothesis in China[J]. Research on Financial and Economic Issues, 2006(8):3-17.
[19]虞义华,郑新业,张莉.经济发展水平、产业结构与碳排放强度——中国省级面板数据分析[J].经济理论与经济管理,2011(3):72-81.Yu Y H, Zheng X Y, Zhang L. Carbon dioxide emission and economic development:A panel data analysis[J].Economic Theory and Business Management, 2011(3):72-81.
[20]王志华,温宗国,闫芳,等.北京环境库兹涅茨曲线假设的验证[J].中国人口·资源与环境,2007(2):40-47.Wang Z H, Wen Z G, Yan F, et al. Verifying the environmental Kuznets curve hypothesis and its conditions in Beijing[J]. China Population, Resources and Environment, 2007(2):40-47.
[21]Bierens H J. Introduction to the mathematical and statistical foundations of econometrics[M]. Cambridge University Press, 2004.
[22]van Der Vaart A W. Asymptotic statistics[M]. Cambridge University Press, 1998.