具有AR（2）误差线性模型的研究
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摘要
本文在前人研究误差服从一阶自回归的线性模型的基础上,研究误差服从二阶自回归的线性模型的统计推断问题。误差服从自回归的线性模型,在许多领域,特别是在经济、管理、工程技术、林业和生态模型等领域具有广泛的应用。对于这种模型,如果忽略了相关性的存在,按照误差服从Gauss-Markov假设的情形,用标准的最小二乘法去处理,在许多情况下将会导致参数估计精度的下降和假设检验犯错误概率的升高,及其它一系列问题。因此,长期以来这种模型的研究受到统计学家的关注。
本文在考虑矩估计缺陷(只考虑相邻两个量的相关)的基础上,提出了新的方差参数的估计方法：距离偏相关系数迭代(Distance Partial correlation coefficient iteration)估计方法(DPC),对于平面数据不涉及到距离,所以简化为PC估计。以均方误差(mean square error (MSE))作为度量估计优劣的标准,通过模拟比较了矩估计(Moment estimation (MS))、Cochrane-Orcutt (CO)迭代估计与新估计的优劣。模拟显示当误差程度较高和误差具有更高阶相关性时,偏相关系数迭代估计优于其它两个估计。
关于误差服从二阶自回归的线性模型的参数估计问题,在误差服从一阶自回归的线性模型的参数估计的基础上,求出了误差的协方差阵及其逆矩阵,并进行了三角分解求出了差分加权变换矩阵。对变换后的模型采用了广义循环最小二乘估计(the cycle generalized least squares (CGLS))。通过模拟,以均方误差作为标准,比较误差服从二阶自回归的线性模型的广义循环最小二乘估计与两步广义最小二乘估计(two steps generalized least squares (TGLS))。模拟结果表明,广义循环最小二乘估计优于两步广义最小二乘估计。并研究当误差系数具有特殊关系时,用模拟比较了考虑特殊关系与不考虑特殊关系时的估计精度。表明考虑特殊关系优于不考虑特殊关系时的估计精度。
本文还对误差的相关性检验进行了研究,提出了误差高阶相关时的检验方法。以实际例子,阐述了误差存在高阶相关性时的处理方法。
Based on studying some statistical inference of linear models with AR(1) errors, the purpose of this statistical inference of linear models with AR(2) errors. This kind of models has extensive applications in many fields, in particular, such as economics, management, engineering, forestry and ecology. For these models, an approach for statistical inference is to ignore the existence of error correlation and then to apply standard least squares method under Gauss-Markov assumption test, this will cause some series problems in parameter estimation and hypothesis test in many cases. Thus study of these models has obtained a great attention of statisticians for a long period.
For parameter estimation of linear models with AR(2) errors, the paper obtains the covariance matrix of the error vector and its inverse matrix and obtains transformation matrix of weighted 2-order differences, based on studying linear models with AR(1) errors. Then estimates parameters of linear models with AR(2) errors with the cycle generalized least squares (CGLS). Under mean square error criterion, Simulation results show that efficiency of CGLS method is superior over the method of two steps generalized least squares (GLS). Furthermore the paper studies errors parameters with some special relationship in linear regression model with AR(2) errors. The result shows considering the special relationship is superior over not
In this paper we propose a new estimate, Distance Partial correlation coefficient iteration estimation of variance parameter, which takes defect of the moment estimation (only with adjacent variable information). Simulation results show that for high errors correlation case and high order errors correlation case, new estimate is superior over the moment estimate and the CO iteration estimate appeared in literature under mean square error criterion.
The paper also discusses autocorrelation test of the linear regression model and studies some methods of autocorrelation test of the linear regression model with high order autocorrelation errors, and explains treatment process of the linear regression model with high order autocorrelation errors with examples.
本文在考虑矩估计缺陷(只考虑相邻两个量的相关)的基础上,提出了新的方差参数的估计方法：距离偏相关系数迭代(Distance Partial correlation coefficient iteration)估计方法(DPC),对于平面数据不涉及到距离,所以简化为PC估计。以均方误差(mean square error (MSE))作为度量估计优劣的标准,通过模拟比较了矩估计(Moment estimation (MS))、Cochrane-Orcutt (CO)迭代估计与新估计的优劣。模拟显示当误差程度较高和误差具有更高阶相关性时,偏相关系数迭代估计优于其它两个估计。
关于误差服从二阶自回归的线性模型的参数估计问题,在误差服从一阶自回归的线性模型的参数估计的基础上,求出了误差的协方差阵及其逆矩阵,并进行了三角分解求出了差分加权变换矩阵。对变换后的模型采用了广义循环最小二乘估计(the cycle generalized least squares (CGLS))。通过模拟,以均方误差作为标准,比较误差服从二阶自回归的线性模型的广义循环最小二乘估计与两步广义最小二乘估计(two steps generalized least squares (TGLS))。模拟结果表明,广义循环最小二乘估计优于两步广义最小二乘估计。并研究当误差系数具有特殊关系时,用模拟比较了考虑特殊关系与不考虑特殊关系时的估计精度。表明考虑特殊关系优于不考虑特殊关系时的估计精度。
本文还对误差的相关性检验进行了研究,提出了误差高阶相关时的检验方法。以实际例子,阐述了误差存在高阶相关性时的处理方法。
Based on studying some statistical inference of linear models with AR(1) errors, the purpose of this statistical inference of linear models with AR(2) errors. This kind of models has extensive applications in many fields, in particular, such as economics, management, engineering, forestry and ecology. For these models, an approach for statistical inference is to ignore the existence of error correlation and then to apply standard least squares method under Gauss-Markov assumption test, this will cause some series problems in parameter estimation and hypothesis test in many cases. Thus study of these models has obtained a great attention of statisticians for a long period.
For parameter estimation of linear models with AR(2) errors, the paper obtains the covariance matrix of the error vector and its inverse matrix and obtains transformation matrix of weighted 2-order differences, based on studying linear models with AR(1) errors. Then estimates parameters of linear models with AR(2) errors with the cycle generalized least squares (CGLS). Under mean square error criterion, Simulation results show that efficiency of CGLS method is superior over the method of two steps generalized least squares (GLS). Furthermore the paper studies errors parameters with some special relationship in linear regression model with AR(2) errors. The result shows considering the special relationship is superior over not
In this paper we propose a new estimate, Distance Partial correlation coefficient iteration estimation of variance parameter, which takes defect of the moment estimation (only with adjacent variable information). Simulation results show that for high errors correlation case and high order errors correlation case, new estimate is superior over the moment estimate and the CO iteration estimate appeared in literature under mean square error criterion.
The paper also discusses autocorrelation test of the linear regression model and studies some methods of autocorrelation test of the linear regression model with high order autocorrelation errors, and explains treatment process of the linear regression model with high order autocorrelation errors with examples.
引文
[1]王松桂、史建红等,线性模型引论,科学出版社,北京,2004
[2]陈希孺,王松桂著.线性模型中的最小二乘法.上海科学出版社,2003
[3]杨文礼.线性模型引论.北京师范大学出版社,1998
[4]何晓群、刘文卿,应用回归分析,中国人民大学出版社,2001
[5]王松桂、陈敏和陈立萍,线性统计模型,高等教育出版社,1999
[6]King M.L., Giles D.E.A.,Autocorrelation pre-testing in the linear model, Journal of Econometrics 25,35-48,1984
[7]王振龙,时间序列分析,中国统计出版社,2000
[8]周义仓、靳祯、秦军林,常微分方程及其应用科学出版社2003
[9]詹姆斯D.汉密尔顿著,刘志明译,时间序列分析,中国社会科学出版社1999
[10]韦博成、鲁国斌、史建清,统计诊断引论, 东南大学出版社,1991
[11]G.A.F.Seber and C.J.Wild. Nonlinear Regression. John Wilcy & Sons, New York,1989
[12]胡舒合,潘光明,高启兵,误差为线性过程时回归模型的估计问题。高校应用数学学报A辑,2003(18(1)：81-90
[13]林金官、韦博成,具有AR(1)误差的非线性随机效应模型中自相关系数的扰动诊断, 应用数学学报,2006(19(4))：818-822
[14]林金官、韦博成,具有ARMA (0,1,0)误差的非线性模型的异方差检验,数学杂志,2005(25)：71-76
[15]徐文科,基于微分方程的生态数学模型统计分析,东北林业大学博士论文,2009
[16]Cook, R.D. and Weisberg, S. Diagnostics for heteroscedasticity in regression, Biometrika, 1983(70):1-10
[17]Puterman, M.L. Leverage and influence in autocorrelated regression models. Appl Statist, 1988(37):76-86
[18]Hossain A. Detection of influential observation in regression model with autocorrelated errors. Comm Statist Theory Methods 1990(19):1047-1060
[19]Richard F, Rob J.H, Alan V. Residual diagnostic plot for model mis-specification in time series regression, Aust N Z J Stat,2000(42):463-477
[20]杨爱军、林金官、韦博成,具有AR(1)误差的非线性随机效应模型中自相关系数的扰动诊断, 应用数学学报,2006(19(4))：818-822
[21]Chi, E.M Reinsel, GC Asymptotic properties of the score test for autocorrelation in effects with AR(1) errors models. Statistics & Probability Letter 1991(11):453-457
[22]Banerjee,A.N.,Magnus,J.R.(2000). On the sensitivity of the usual t-and F-tests to covariance misspecification. Journal of Econometrics 95,157-176
[23]Epps, T.M and Epps, M.I, The robustness of some standard test for autocorrelation and heteroscedasticity when both problems are present Econometrika,45(1977),745-735
[24]韦博成,胡跃清,非线性回归模型相关性和异方差性的检验,工程数学学报,1994(11(4))：1-12
[25]Tsai, C.L, Score test for the First-order autoregression model with heteroscedasticity, Biometrika,1986(73):455-460
[26]刘应安、韦博成、林金官,AR(1)误差的线性随机效应模型中方差齐性和自相关的检验,应用概率统计,2004(20(1)：27-36
[27]康新梅、王志忠,线性模型中自相关的经验似然比检验,数学的实践与认识,2006,36(6)
[28]Chi EM, Reinsel GC. Models for longitudinal data with random effects and AR(1) errors, t,. Am.Statist. Assoc.1989,84:452—459
[29]Owen A B. Empirical likelihood ratio for linear models[J]. Ann Statist,1991,19:1725— 1747
[30]Brockwell PJ, Davis RA. Time series:Theory and Methods. Second Edition [M]. Springer2 Verlag, New York,1991
[31]Owen A B. Empirical Iikelihood[M]. London, Chapman Hall,2001
[32]Owen AB. Empirical likelihood ratio confidence regions[J]. Ann Statist,1990,18:90— 120
[33]陈希孺、王松桂,近代回归分析, 安徽教育出版社,1987
[34]陈希孺,数理统计引论。科学出版社。1981
[35]李子奈,计量经济学,高等教育出版社,2000
[36]Walter Kramer, Christian Donninger*. Spatial Autocorrelation Among Errors and the Relative Efficiency of OLS in the Linear Regression Model [J].American Statistical Association,1987 82, p577-579
[37]Walter Kramer. High Correlation Among Errors and the Efficiency of Ordinary Least Squares in Linear Models[J].Statistische Hefte 1984 25,p135-142
[38]Ord K. Estimation Methods for Models of Spatial Interaction [J]. American Statistical Association,1975,70, p120-126
[39]Walter Kramer. Finite Sample Efficiency of Ordinary Least Squares in The Linear Regression Model with Autocorrelated Errors. American Statistical Association,1980, 75(372), p1005-1010
[40]Walter Kramer. When are Gauss-Markov and Least Squares Estimators Identical? A Coordinate-Free Approach. Annals of Mathematical Statistics,1968 39, p778-789
[41]Prais and Winsten, Trend Estimators and Serial Correlation, Discussion Paper No.383, Cowles Commission, Chicago
[42]Walter Kramer', Finite Sample Efficiency of Ordinary Least Squares in the Linear Regression Model with Autocorrelated Errors, Journal of American Statistical Association,1980,1005-1009
[43]王沫然,MATLAB6.0与科学计算,电子工业出版社,2001
[44]苏金明、阮沈永,MATLAB 6.1实用指南电子工业出版社,2002
[45]王维国,计量经济学,东北财经大学出版社,2001
[46]朱力行、许王莉,非参数蒙特卡罗检验及应用,科学出版社,2008
[47]余家林、肖枝洪,概率统计及SAS应用武汉大学出版社,2007
[48]张文彤,SPSS11统计分析教程(高级篇),北京希望电子出版社,2002
[49]Greg Kochanski, Monte Carlo Simulation, http://kochanski.org/gpk,2005
[50]Cochrane, D. and Orcutt,GH., Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms, Journal of American Statistical Association,1999 44,32-61
[51]Durbin, J.and Watson, G S., Testing for Serial Correlation in Least Squares Regression I, Biometrika,1950.37,409-28
[52]Prais, S. J. and Winsten, C. B., Trend Estimators and Serial Correlation, mimeographed, Chicago:Cowles Commission.1954.
[53]Park, R.E., Mitchell, B.M.. Estimating the autocorrelated error model with trended data. Journal of Econometrics 1980.13,185-201
[54]Magnus, J.R.. The traditional pretest estimator. Theory of Probability and its Applications, to appear.1999
[55]Pitman, E.GJ.. The "closet" estimates of statistical parameters. Proceedings of the Cam bridge Philosophical Society 1937.33,212-222
[56]Rothenberg, T.J.. Approximate normality of generalized least squares estimates. Econometrica 1984.52,827-842
[57]Sarle, S.R.. Linear Models. Wiley, New York.1971
[58]C.Radhakrishna Rao and Helge Toutenburg, Linear Models, Spring,1999
[59]Wang, S.G., Ma, W.Q.. On exact test of Least hypothesis in linear models with nested error structure. Journal of Statistical Planning and Inference 2002.106,225-233
[60]King M.L., Giles D.E.. Specification Analysis in the Linear Model. New York.1987
[61]Henderson,C.R.. Estimation of Variance and Covariance Components. Biometrics,1953.9, 226-252
[62]Harvey A.c.. The Econometric Analysis of Times Series,2nd ed. Philip Allan.1990
[63]楼顺天、于卫、严华梁。MATLAB程序设计语言。西安电子科技大学出版,1999
[64]李志辉、罗平,SPSS for Windows统计分析教程,电子工业出版社,2003
[2]陈希孺,王松桂著.线性模型中的最小二乘法.上海科学出版社,2003
[3]杨文礼.线性模型引论.北京师范大学出版社,1998
[4]何晓群、刘文卿,应用回归分析,中国人民大学出版社,2001
[5]王松桂、陈敏和陈立萍,线性统计模型,高等教育出版社,1999
[6]King M.L., Giles D.E.A.,Autocorrelation pre-testing in the linear model, Journal of Econometrics 25,35-48,1984
[7]王振龙,时间序列分析,中国统计出版社,2000
[8]周义仓、靳祯、秦军林,常微分方程及其应用科学出版社2003
[9]詹姆斯D.汉密尔顿著,刘志明译,时间序列分析,中国社会科学出版社1999
[10]韦博成、鲁国斌、史建清,统计诊断引论, 东南大学出版社,1991
[11]G.A.F.Seber and C.J.Wild. Nonlinear Regression. John Wilcy & Sons, New York,1989
[12]胡舒合,潘光明,高启兵,误差为线性过程时回归模型的估计问题。高校应用数学学报A辑,2003(18(1)：81-90
[13]林金官、韦博成,具有AR(1)误差的非线性随机效应模型中自相关系数的扰动诊断, 应用数学学报,2006(19(4))：818-822
[14]林金官、韦博成,具有ARMA (0,1,0)误差的非线性模型的异方差检验,数学杂志,2005(25)：71-76
[15]徐文科,基于微分方程的生态数学模型统计分析,东北林业大学博士论文,2009
[16]Cook, R.D. and Weisberg, S. Diagnostics for heteroscedasticity in regression, Biometrika, 1983(70):1-10
[17]Puterman, M.L. Leverage and influence in autocorrelated regression models. Appl Statist, 1988(37):76-86
[18]Hossain A. Detection of influential observation in regression model with autocorrelated errors. Comm Statist Theory Methods 1990(19):1047-1060
[19]Richard F, Rob J.H, Alan V. Residual diagnostic plot for model mis-specification in time series regression, Aust N Z J Stat,2000(42):463-477
[20]杨爱军、林金官、韦博成,具有AR(1)误差的非线性随机效应模型中自相关系数的扰动诊断, 应用数学学报,2006(19(4))：818-822
[21]Chi, E.M Reinsel, GC Asymptotic properties of the score test for autocorrelation in effects with AR(1) errors models. Statistics & Probability Letter 1991(11):453-457
[22]Banerjee,A.N.,Magnus,J.R.(2000). On the sensitivity of the usual t-and F-tests to covariance misspecification. Journal of Econometrics 95,157-176
[23]Epps, T.M and Epps, M.I, The robustness of some standard test for autocorrelation and heteroscedasticity when both problems are present Econometrika,45(1977),745-735
[24]韦博成,胡跃清,非线性回归模型相关性和异方差性的检验,工程数学学报,1994(11(4))：1-12
[25]Tsai, C.L, Score test for the First-order autoregression model with heteroscedasticity, Biometrika,1986(73):455-460
[26]刘应安、韦博成、林金官,AR(1)误差的线性随机效应模型中方差齐性和自相关的检验,应用概率统计,2004(20(1)：27-36
[27]康新梅、王志忠,线性模型中自相关的经验似然比检验,数学的实践与认识,2006,36(6)
[28]Chi EM, Reinsel GC. Models for longitudinal data with random effects and AR(1) errors, t,. Am.Statist. Assoc.1989,84:452—459
[29]Owen A B. Empirical likelihood ratio for linear models[J]. Ann Statist,1991,19:1725— 1747
[30]Brockwell PJ, Davis RA. Time series:Theory and Methods. Second Edition [M]. Springer2 Verlag, New York,1991
[31]Owen A B. Empirical Iikelihood[M]. London, Chapman Hall,2001
[32]Owen AB. Empirical likelihood ratio confidence regions[J]. Ann Statist,1990,18:90— 120
[33]陈希孺、王松桂,近代回归分析, 安徽教育出版社,1987
[34]陈希孺,数理统计引论。科学出版社。1981
[35]李子奈,计量经济学,高等教育出版社,2000
[36]Walter Kramer, Christian Donninger*. Spatial Autocorrelation Among Errors and the Relative Efficiency of OLS in the Linear Regression Model [J].American Statistical Association,1987 82, p577-579
[37]Walter Kramer. High Correlation Among Errors and the Efficiency of Ordinary Least Squares in Linear Models[J].Statistische Hefte 1984 25,p135-142
[38]Ord K. Estimation Methods for Models of Spatial Interaction [J]. American Statistical Association,1975,70, p120-126
[39]Walter Kramer. Finite Sample Efficiency of Ordinary Least Squares in The Linear Regression Model with Autocorrelated Errors. American Statistical Association,1980, 75(372), p1005-1010
[40]Walter Kramer. When are Gauss-Markov and Least Squares Estimators Identical? A Coordinate-Free Approach. Annals of Mathematical Statistics,1968 39, p778-789
[41]Prais and Winsten, Trend Estimators and Serial Correlation, Discussion Paper No.383, Cowles Commission, Chicago
[42]Walter Kramer', Finite Sample Efficiency of Ordinary Least Squares in the Linear Regression Model with Autocorrelated Errors, Journal of American Statistical Association,1980,1005-1009
[43]王沫然,MATLAB6.0与科学计算,电子工业出版社,2001
[44]苏金明、阮沈永,MATLAB 6.1实用指南电子工业出版社,2002
[45]王维国,计量经济学,东北财经大学出版社,2001
[46]朱力行、许王莉,非参数蒙特卡罗检验及应用,科学出版社,2008
[47]余家林、肖枝洪,概率统计及SAS应用武汉大学出版社,2007
[48]张文彤,SPSS11统计分析教程(高级篇),北京希望电子出版社,2002
[49]Greg Kochanski, Monte Carlo Simulation, http://kochanski.org/gpk,2005
[50]Cochrane, D. and Orcutt,GH., Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms, Journal of American Statistical Association,1999 44,32-61
[51]Durbin, J.and Watson, G S., Testing for Serial Correlation in Least Squares Regression I, Biometrika,1950.37,409-28
[52]Prais, S. J. and Winsten, C. B., Trend Estimators and Serial Correlation, mimeographed, Chicago:Cowles Commission.1954.
[53]Park, R.E., Mitchell, B.M.. Estimating the autocorrelated error model with trended data. Journal of Econometrics 1980.13,185-201
[54]Magnus, J.R.. The traditional pretest estimator. Theory of Probability and its Applications, to appear.1999
[55]Pitman, E.GJ.. The "closet" estimates of statistical parameters. Proceedings of the Cam bridge Philosophical Society 1937.33,212-222
[56]Rothenberg, T.J.. Approximate normality of generalized least squares estimates. Econometrica 1984.52,827-842
[57]Sarle, S.R.. Linear Models. Wiley, New York.1971
[58]C.Radhakrishna Rao and Helge Toutenburg, Linear Models, Spring,1999
[59]Wang, S.G., Ma, W.Q.. On exact test of Least hypothesis in linear models with nested error structure. Journal of Statistical Planning and Inference 2002.106,225-233
[60]King M.L., Giles D.E.. Specification Analysis in the Linear Model. New York.1987
[61]Henderson,C.R.. Estimation of Variance and Covariance Components. Biometrics,1953.9, 226-252
[62]Harvey A.c.. The Econometric Analysis of Times Series,2nd ed. Philip Allan.1990
[63]楼顺天、于卫、严华梁。MATLAB程序设计语言。西安电子科技大学出版,1999
[64]李志辉、罗平,SPSS for Windows统计分析教程,电子工业出版社,2003
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