随机系数的随机前沿面模型
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摘要
生产函数是经济数学与数量经济学中的一个重要概念，它是一种技术关系，表明在一定的技术水平下，由每一组特定的生产要素组合构成的投入所能得到的最大产出。通常将严格符合上述定义的生产函数称为生产前沿或前沿生产函数，同时也与传统的回归统计方法确定的“平均生产函数”相区别。生产前沿与生产的技术无效性是联系在一起的，同时生产前沿又与生产可能集的边界很好的吻合起来。它是度量生产的有效性的重要工具，通过了解生产行为无效的根源及程度，进而提出相应的改进对策和目标，可以有效地节省能源、减少浪费，同时它又是与成本前沿函数对偶的，因而对前沿面的研究是很有价值的。
生产前沿的研究方法主要有，参数方法、非参数方法。本文的主要工作是：
(1) 本文首先在合理分析生产集与生产函数的理论的基础上，全面分析总结了研究生产前沿的非参数方法——数据包络分析(DEA)方法，指出了它的局限性。
(2) 研究生产前沿的参数方法包括确定性前沿面模型和随机前沿面模型方法。分析总结了确定性前沿模型的估计方法，在合理分析生产行为的随机性的基础上，针对随机前沿面模型因含有双误差而造成的估计上的困难，以C—D生产函数为例，采用极大似然估计法，全面讨论了管理偏差服从截断正态分布、半正态分布、指数分布及伽马分布时，随机前沿面模型的求解模型。在假定随机前沿面模型中的系数为多元正态分布的情形下，采用极大似然估计法，详细推导了管理偏差服从上述各分布时的随机系数的随机前沿面模型的求解模型。
(3) 采用将管理偏差先看作参数进行参数扩张的一般对随机前沿面的Bayes估计方法，基于MCMC方法中的Gibbs抽样，在假定管理偏差为不同分布时，详细推导了一般随机前沿面模型和随机系数的随机前沿面模型的Bayes估计方法，并给出了随机系数服从特殊的多元正态分布的情形下，随机系数的随机前沿面模型的Bayes估计的算法。推导证明了随机前沿面模型对生产技术无效性进行估计的方法。
(4) 实证中表明极大似然估计可以很好的解决随机前沿面模型中误差项的分解，并比较分析了随机系数的随机前沿面模型与一般随机前沿面模型的差异，指出了一般随机前沿面模型的局限性。
Production Function is an important idea in Economy Mathematics and Quantity Economics. It is a technology relationship which indicates the maximum production under certain technologies with a group of production requisites. Usually we call a production function a production frontier or frontier production function if it answers for the definition accurately, which different from the average production function based on conditional regression statistics. Production frontier contacts with the technical inefficiency, and it is also tallies with the boundary of possible production set. Production frontier is an important tool in measuring the efficiency of production. With the knowledge of the inefficient root and degree in production actions, one can bring forward advance countermeasures and aims, and with a result of saving energies and reducing waste, asd it is also the counterpart of cost frontier function, so it is meaningful to study the frontier.The main methods in studying the production frontier is parametric method and the non-parametric method. The main content of this dissertation is as follows:(1) Based on the rational analysis of production set and production function, the author analyses and summarizes the non-parametric method of studying the production frontier-Data Envelopment Analysis(DEA), and points out its finity.(2) The so called parametric method of studying the production frontier including positive frontier and stochastic frontier models. Analyze and summarize the methods of positive models. Based on the rational analysis of production action, for the difficulties of estimation with double error components, taking Cobb-Douglas production function as an example, the author adopts maximum likelihood estimation to construct the solving model of the stochastic frontier models with management components distributes according to truncated normal distribution, half normal distribution, exponential distribution and gamma distribution. Under the assumption of coefficients distributed according to multiply normal distributions, the solving models of stochastic frontier models with random coefficients are constructed by adopting maximum likelihood estimation.(3) Following standard practice in Bayesian analysis of stochastic frontiers, we treat the management components as parameters, based on one method of MCMC -Gibbs sampling, under the assumption of different distributions of management components, and infer the Bayesian estimation of common stochastic frontier
models and the stochastic frontiers with random coefficients. Also we provide the Bayesian estimation of stochastic frontier models with coefficients distributes according to special multiply normal. Measurement method of inefficiency also is inferred and proved.(4) Demonstrations show maximum likelihood estimation is a wonderful way to decompose the errors in stochastic frontier models. The difference between the common stochastic frontier models and the stochastic frontier models with random coefficients is compared and analyzed and the limitation of common stochastic frontier models is pointed out.
生产前沿的研究方法主要有，参数方法、非参数方法。本文的主要工作是：
(1) 本文首先在合理分析生产集与生产函数的理论的基础上，全面分析总结了研究生产前沿的非参数方法——数据包络分析(DEA)方法，指出了它的局限性。
(2) 研究生产前沿的参数方法包括确定性前沿面模型和随机前沿面模型方法。分析总结了确定性前沿模型的估计方法，在合理分析生产行为的随机性的基础上，针对随机前沿面模型因含有双误差而造成的估计上的困难，以C—D生产函数为例，采用极大似然估计法，全面讨论了管理偏差服从截断正态分布、半正态分布、指数分布及伽马分布时，随机前沿面模型的求解模型。在假定随机前沿面模型中的系数为多元正态分布的情形下，采用极大似然估计法，详细推导了管理偏差服从上述各分布时的随机系数的随机前沿面模型的求解模型。
(3) 采用将管理偏差先看作参数进行参数扩张的一般对随机前沿面的Bayes估计方法，基于MCMC方法中的Gibbs抽样，在假定管理偏差为不同分布时，详细推导了一般随机前沿面模型和随机系数的随机前沿面模型的Bayes估计方法，并给出了随机系数服从特殊的多元正态分布的情形下，随机系数的随机前沿面模型的Bayes估计的算法。推导证明了随机前沿面模型对生产技术无效性进行估计的方法。
(4) 实证中表明极大似然估计可以很好的解决随机前沿面模型中误差项的分解，并比较分析了随机系数的随机前沿面模型与一般随机前沿面模型的差异，指出了一般随机前沿面模型的局限性。
Production Function is an important idea in Economy Mathematics and Quantity Economics. It is a technology relationship which indicates the maximum production under certain technologies with a group of production requisites. Usually we call a production function a production frontier or frontier production function if it answers for the definition accurately, which different from the average production function based on conditional regression statistics. Production frontier contacts with the technical inefficiency, and it is also tallies with the boundary of possible production set. Production frontier is an important tool in measuring the efficiency of production. With the knowledge of the inefficient root and degree in production actions, one can bring forward advance countermeasures and aims, and with a result of saving energies and reducing waste, asd it is also the counterpart of cost frontier function, so it is meaningful to study the frontier.The main methods in studying the production frontier is parametric method and the non-parametric method. The main content of this dissertation is as follows:(1) Based on the rational analysis of production set and production function, the author analyses and summarizes the non-parametric method of studying the production frontier-Data Envelopment Analysis(DEA), and points out its finity.(2) The so called parametric method of studying the production frontier including positive frontier and stochastic frontier models. Analyze and summarize the methods of positive models. Based on the rational analysis of production action, for the difficulties of estimation with double error components, taking Cobb-Douglas production function as an example, the author adopts maximum likelihood estimation to construct the solving model of the stochastic frontier models with management components distributes according to truncated normal distribution, half normal distribution, exponential distribution and gamma distribution. Under the assumption of coefficients distributed according to multiply normal distributions, the solving models of stochastic frontier models with random coefficients are constructed by adopting maximum likelihood estimation.(3) Following standard practice in Bayesian analysis of stochastic frontiers, we treat the management components as parameters, based on one method of MCMC -Gibbs sampling, under the assumption of different distributions of management components, and infer the Bayesian estimation of common stochastic frontier
models and the stochastic frontiers with random coefficients. Also we provide the Bayesian estimation of stochastic frontier models with coefficients distributes according to special multiply normal. Measurement method of inefficiency also is inferred and proved.(4) Demonstrations show maximum likelihood estimation is a wonderful way to decompose the errors in stochastic frontier models. The difference between the common stochastic frontier models and the stochastic frontier models with random coefficients is compared and analyzed and the limitation of common stochastic frontier models is pointed out.
引文
[1] Pin-Huang Chou. A Gibbs sampling approach to the estimation of linear regression models under daily price limits. Pacific-Basin Finance Journal. 1997. 5: 39-62
[2] Mohamed T.madi. Bayesian inference for partially accelerated life tests using Gibbs sampling. Microelectron. Reliab, 1997, 37(8): 1165~1168
[3] Efthymios G, Tsionas. Combining DEA and stochastic frontier models: An empirical Bayes approach. European Journal of Operational Research, 2003, 147: 499-510
[4] John Ruggiero. Effciency estimation and error decomposition in the stochastic frontier model: A Monte Carlo analysis. European Journal of Operational Research,1999,115: 555-563
[5] Qi Li . Estimating a stochastic production frontier when the adjusted error is symmetric. Econamics Letters, 1996, 52: 221-228
[6] Robert Bifulco, Stuart Bretschneider. Estimating school efficiency a comparison of methods using simulated data. Economics of Education Review, 2001, 20 .- 417-429
[7] Philippe Vieu. A note on density mode estimation. Statistics & Probability Letters, 1996, 26 : 297-307
[8] By PJ.Bickel, Y.Ritov. Efficient estimation in the errors in variables model. The Annals of Statistics, 1987, 15(2): 513-540
[9] Marc Ivaldi, Sylvette Monier - Dilhan, Michel Simioni. Stochastic production frontier and panel data: A latent variable framework. European Journal of Operational Research, 1995, 80: 534-547
[10] B.B.M.Shao, W.T.Lin. Measuring the value of information technology in technical efficiency with stochastic production frontiers. Informatin and Software technology, 2001, 43: 447-456
[11] B.U.Park, R.C.Sickles, L.Simar . Stochastia panel frontier : A semiparametric approach . Journal of Econometrics, 1998, 84: 273-301
[12] Gilbert G. Walter, Gilbert G. Walter. Density estimation in the presence of noise. Statistics & Probability Letters, 1999, 41 : 237-246
[13] Hakan Orbay. Information Processing Hierarchies. Journal of Economic Theory, 2002,105: 370-407
[14] Uwe Jensen. Is it efficient to analyse efficiency rankings? . Empirical Economics, 2000, 25: 189-208
[15] Jirong Wang, Gail L.Cramer, Eric J.Wailes. Production efficiency fo Chinese agriculture: evidence from rural household survey data. Agricultural Economics, 1996, 15: 17-28
[16] Luis R. Murillo-Zamorano, Juan A. Vega-Cervera. The use of parametric and non-parametric frontier methods to measure the productive efficiency in the industrial sector: A comparative study. Production Economics, 2001, 69: 265～275
[17] Samuel T.Cooper, Elchanan Cohn. Estimation of a Frontier Production Function for the South Carolina Educational Process. Economics of Education, 1997, 16(3): 313-327
[18] Carmen Fernandez, Gary Koop, Mark Stee. A Bayesian analysis of ultiple-output production frontiers. Journal of Econometrics, 2000, 98: 47～79
[19] Martin A. Carree. Technological inefficiency and the skewness of the error component in stochastic frontier analysis. Economics Letters, 2002, 77: 101～107
[20] Geman, S. and Geman, D. Stochastical relaxation, Gibbs distributions and the Beyesian Restoration of images. IEEE Trans.Pattn.Anal.Mach. Intel, 1984, 6: 721-741
[21] Metropolis.N, Rosenbluth, A.W Rosenbluth etal. Equations of state calculations by fast computing machine. J.Chem.Phys, 1953, 21: 1089-10916.
[22] Hasting, W.K. Monte Carlo sampling methods using Markov chain and their applicatins. Biometrika, 1970, 57: 97-109
[23] Gelfand, A., Smith, A.. Sampling based approaches to calculating marginal densities, JASA 85: 398-409
[24] Smith, A.F.M., Mao, S.S.. The problem of interpolation the grouped data, Reliability Theory and Applications. Proceedings of China-Japan Reliability Symposium, 1987
[25] Thanassou lis E. A comparison of regression analysis and data envelopment analysis alternative methods for performance assessment. JOpl Res Soc, 1993, 44(11): 1129～1144
[26] Efthymios G. Tsionas. Random Coefficients Stochastic Frontier Models. working paper.
[27] 孙文斌，孙巍．基于产出的生产资源配置效率的非参数测度与分解．吉林工业大学自然科学学报，2001，31(2)：45～50
[28] 郭京福、杨德礼．生产前沿参数方法与非参数方法的比较研究．系统工程理论与实践，1998，11：31～35
[29] 唐焱，吴群，刘友兆，金萍．基于C—D生产函数的农用地估价实证研究．南京农业大学学报，2003，26(3)：101—105．
[30] 田文霞，郭京福，张晓彤．生产前沿的参数与非参数方法的比较．决策借鉴，2000，13(2)：44～46
[31] 庄宁．随机前沿面分析在国外医院效率评价中的应用．卫生经济研究，2000．12：34～36
[32] 王丽，魏煜．企业效率研究方法比较．预测，1999，5：76～79
[33] 孙静养、李怀祖．测算行业企业技术进步的非参数前沿分析方法．西安交通大学学报(社会科学版)．2002，61(3)：30～35
[34] 王金祥，生产前沿面构造及应用研究：[博士学位论文]．天津：天津大学管理学院，2002
[35] 迟旭，杨德礼．生产过程状态分析的非参数方法．管理工程学报，1998，12(2)：1～6
[36] 王志江．生产函数中参数估计方法的改进．运筹与管理，1999，8(4)：80～84
[37] 童恒庆．经济回归模型及计算．武汉：湖北科学技术出版社，1999，3～7，362～368
[38] 陆璇．数理统计基础．北京：清华大学出版，2001
[39] 胡适耕．现代应用数学基础．科学出版社，2001
[40] 茆诗松．贝叶斯统计．中国统计出版社，1999
[41] 胡适耕．微观经济的数理分析．华中科技大学出版社，2003
[42] 茆诗松．高等数理统计．北京：高等教育出版社；德国：施普林格出版社，1998
[43] 孙毅，刘科伟，黄建国等．基于Gibbs采样的贝叶斯多目标高分辨方位估计．计算机工程与应用，2002，12：36～38
[44] 林秀光，程杞元．基于Gibbs抽样法的边际密度估计．北京理工大学学报，2002，22(1)：16～19
[45] 孙巍，杨庆芳，杨树绘．产出资源配置效率的参数测度与非参数测度及其比较分析．系统工程理论与实践，2000，6：118～122
[46] 尹慧英，解英男，李成红．科技进步贡献率概念的建立及测算方法．哈尔滨师范大学自然科学学报，1998，14(3)：37～40
[47] 叶阿忠．非参数计量经济学．南开大学出版社，2003
[48] 乔世君，张世英．用Gibbs抽样算法计算定数截尾时Weibull分布的贝叶斯估计．数理统计与管理，200019，(2)：35～40
[49] 王成军，何基报．压缩型正态——伽玛分布的Bayes统计分析研究．安徽师范大学学报(自然科学版)，2001，24(1)：5～8
[50] 李凌之．Poisson分布参数的渐近最优和可容许的经验Bayes估计．数学杂志，1998，18(4)：461～465
[51] 郭京福，杨德礼．数据包络分析方法综述．大连理工大学学报．1998，38(2)：236～241
[52] 魏权龄．崔宇刚．评价相对有效性的几个重要的DEA模型数据包络分析(二)．系统工程理论与实践，1989(5)：55～689
[53] 魏权龄，岳明．综合的DEA模型C2WY数据包络分析(四)．系统工程理论与实践，1989(4)：75～80
[54] [美]威廉 H 格林．经济计量分析，王明舰，王永宏等译．社会科学出版社，1999：488～505
[2] Mohamed T.madi. Bayesian inference for partially accelerated life tests using Gibbs sampling. Microelectron. Reliab, 1997, 37(8): 1165~1168
[3] Efthymios G, Tsionas. Combining DEA and stochastic frontier models: An empirical Bayes approach. European Journal of Operational Research, 2003, 147: 499-510
[4] John Ruggiero. Effciency estimation and error decomposition in the stochastic frontier model: A Monte Carlo analysis. European Journal of Operational Research,1999,115: 555-563
[5] Qi Li . Estimating a stochastic production frontier when the adjusted error is symmetric. Econamics Letters, 1996, 52: 221-228
[6] Robert Bifulco, Stuart Bretschneider. Estimating school efficiency a comparison of methods using simulated data. Economics of Education Review, 2001, 20 .- 417-429
[7] Philippe Vieu. A note on density mode estimation. Statistics & Probability Letters, 1996, 26 : 297-307
[8] By PJ.Bickel, Y.Ritov. Efficient estimation in the errors in variables model. The Annals of Statistics, 1987, 15(2): 513-540
[9] Marc Ivaldi, Sylvette Monier - Dilhan, Michel Simioni. Stochastic production frontier and panel data: A latent variable framework. European Journal of Operational Research, 1995, 80: 534-547
[10] B.B.M.Shao, W.T.Lin. Measuring the value of information technology in technical efficiency with stochastic production frontiers. Informatin and Software technology, 2001, 43: 447-456
[11] B.U.Park, R.C.Sickles, L.Simar . Stochastia panel frontier : A semiparametric approach . Journal of Econometrics, 1998, 84: 273-301
[12] Gilbert G. Walter, Gilbert G. Walter. Density estimation in the presence of noise. Statistics & Probability Letters, 1999, 41 : 237-246
[13] Hakan Orbay. Information Processing Hierarchies. Journal of Economic Theory, 2002,105: 370-407
[14] Uwe Jensen. Is it efficient to analyse efficiency rankings? . Empirical Economics, 2000, 25: 189-208
[15] Jirong Wang, Gail L.Cramer, Eric J.Wailes. Production efficiency fo Chinese agriculture: evidence from rural household survey data. Agricultural Economics, 1996, 15: 17-28
[16] Luis R. Murillo-Zamorano, Juan A. Vega-Cervera. The use of parametric and non-parametric frontier methods to measure the productive efficiency in the industrial sector: A comparative study. Production Economics, 2001, 69: 265～275
[17] Samuel T.Cooper, Elchanan Cohn. Estimation of a Frontier Production Function for the South Carolina Educational Process. Economics of Education, 1997, 16(3): 313-327
[18] Carmen Fernandez, Gary Koop, Mark Stee. A Bayesian analysis of ultiple-output production frontiers. Journal of Econometrics, 2000, 98: 47～79
[19] Martin A. Carree. Technological inefficiency and the skewness of the error component in stochastic frontier analysis. Economics Letters, 2002, 77: 101～107
[20] Geman, S. and Geman, D. Stochastical relaxation, Gibbs distributions and the Beyesian Restoration of images. IEEE Trans.Pattn.Anal.Mach. Intel, 1984, 6: 721-741
[21] Metropolis.N, Rosenbluth, A.W Rosenbluth etal. Equations of state calculations by fast computing machine. J.Chem.Phys, 1953, 21: 1089-10916.
[22] Hasting, W.K. Monte Carlo sampling methods using Markov chain and their applicatins. Biometrika, 1970, 57: 97-109
[23] Gelfand, A., Smith, A.. Sampling based approaches to calculating marginal densities, JASA 85: 398-409
[24] Smith, A.F.M., Mao, S.S.. The problem of interpolation the grouped data, Reliability Theory and Applications. Proceedings of China-Japan Reliability Symposium, 1987
[25] Thanassou lis E. A comparison of regression analysis and data envelopment analysis alternative methods for performance assessment. JOpl Res Soc, 1993, 44(11): 1129～1144
[26] Efthymios G. Tsionas. Random Coefficients Stochastic Frontier Models. working paper.
[27] 孙文斌，孙巍．基于产出的生产资源配置效率的非参数测度与分解．吉林工业大学自然科学学报，2001，31(2)：45～50
[28] 郭京福、杨德礼．生产前沿参数方法与非参数方法的比较研究．系统工程理论与实践，1998，11：31～35
[29] 唐焱，吴群，刘友兆，金萍．基于C—D生产函数的农用地估价实证研究．南京农业大学学报，2003，26(3)：101—105．
[30] 田文霞，郭京福，张晓彤．生产前沿的参数与非参数方法的比较．决策借鉴，2000，13(2)：44～46
[31] 庄宁．随机前沿面分析在国外医院效率评价中的应用．卫生经济研究，2000．12：34～36
[32] 王丽，魏煜．企业效率研究方法比较．预测，1999，5：76～79
[33] 孙静养、李怀祖．测算行业企业技术进步的非参数前沿分析方法．西安交通大学学报(社会科学版)．2002，61(3)：30～35
[34] 王金祥，生产前沿面构造及应用研究：[博士学位论文]．天津：天津大学管理学院，2002
[35] 迟旭，杨德礼．生产过程状态分析的非参数方法．管理工程学报，1998，12(2)：1～6
[36] 王志江．生产函数中参数估计方法的改进．运筹与管理，1999，8(4)：80～84
[37] 童恒庆．经济回归模型及计算．武汉：湖北科学技术出版社，1999，3～7，362～368
[38] 陆璇．数理统计基础．北京：清华大学出版，2001
[39] 胡适耕．现代应用数学基础．科学出版社，2001
[40] 茆诗松．贝叶斯统计．中国统计出版社，1999
[41] 胡适耕．微观经济的数理分析．华中科技大学出版社，2003
[42] 茆诗松．高等数理统计．北京：高等教育出版社；德国：施普林格出版社，1998
[43] 孙毅，刘科伟，黄建国等．基于Gibbs采样的贝叶斯多目标高分辨方位估计．计算机工程与应用，2002，12：36～38
[44] 林秀光，程杞元．基于Gibbs抽样法的边际密度估计．北京理工大学学报，2002，22(1)：16～19
[45] 孙巍，杨庆芳，杨树绘．产出资源配置效率的参数测度与非参数测度及其比较分析．系统工程理论与实践，2000，6：118～122
[46] 尹慧英，解英男，李成红．科技进步贡献率概念的建立及测算方法．哈尔滨师范大学自然科学学报，1998，14(3)：37～40
[47] 叶阿忠．非参数计量经济学．南开大学出版社，2003
[48] 乔世君，张世英．用Gibbs抽样算法计算定数截尾时Weibull分布的贝叶斯估计．数理统计与管理，200019，(2)：35～40
[49] 王成军，何基报．压缩型正态——伽玛分布的Bayes统计分析研究．安徽师范大学学报(自然科学版)，2001，24(1)：5～8
[50] 李凌之．Poisson分布参数的渐近最优和可容许的经验Bayes估计．数学杂志，1998，18(4)：461～465
[51] 郭京福，杨德礼．数据包络分析方法综述．大连理工大学学报．1998，38(2)：236～241
[52] 魏权龄．崔宇刚．评价相对有效性的几个重要的DEA模型数据包络分析(二)．系统工程理论与实践，1989(5)：55～689
[53] 魏权龄，岳明．综合的DEA模型C2WY数据包络分析(四)．系统工程理论与实践，1989(4)：75～80
[54] [美]威廉 H 格林．经济计量分析，王明舰，王永宏等译．社会科学出版社，1999：488～505