半参数模型的统计推断及其在金融中的应用
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摘要
近二十年来,结合了参数模型和非参数模型的优点、把两种模型相结合的半参数模型,得到了学者们的广泛关注.本文首先针对部分线性模型进行了统计推断,在数据受污染情况下,使用Huber提出的M估计得到了模型参数部分的稳健估计,得到的估计是相合的并且是渐近正态的.模拟结果显示,当误差为正态分布时,M估计与最小二乘估计相比同样具有较高的精度；当误差为柯西分布时,M估计明显优于最小二乘估计,具有稳健性.随后我们提出一种新的半参数模型形式,称为部分非线性模型,不但囊括典型的部分线性模型,也将应用中常见的非线性模型关系涵盖进来.相比最一般的部分非线性模型,我们的模型可以在一个统一的框架下进行统计推断,非常方便.我们基于筛方法对这类部分非线性模型进行了参数估计,分别用分段线性函数与B-样条逼近模型的非参数部分,得到模型的最小二乘估计与极大似然估计.两类估计都具有强相合性与渐近正态性,模拟结果也支持了所得到的渐近理论.针对金融中的汇率与石油价格,我们首次将工程中的加速应力模型引入进来,建立了美元兑欧元汇率、石油价格以及美元兑人民币汇率三者之间的部分非线性模型关系,模拟效果很好.随后我们考察了全球股票市场之间的联系,针对日经指数、道琼斯指数、美元兑日元汇率三个变量进行建模,拟合效果较好.实证结果说明部分非线性模型可应用于金融中的这些建模问题.最后,针对金融时间序列的趋势项估计问题,我们基于copula函数构造了一个准则函数,能够衡量各种估计方法的优劣.
In recent 20 years, much research has been focused on the semiparametric models. The advantage of such semiparametric models is that they reduce the risk of misspecification relative to a fully parametric model and avoid some serious drawbacks of fully nonparametric methods. Firstly we discuss the statistical inference for partially linear models. Using Huber's M-estimator, we establish the robust estimator for the parametric part. The resulting estimator is shown to be consistent and asymptotically normally distributed. Simulation results show that the M-estimator performs as well as least square estimator under normally distributed errors. Especially, when the random errors are distributed as Cauchy distribution, the M-estimator is superior to the least square estimator. Secondly we propose a new kind of semiparametric models, called partially nonlinear models, which not only nest the partially linear models as a special case, but also contain many nonlinear functions widely used in applications. Compared to general partially nonlinear models, our models are convenient to make statistical inferences. We develop the sieve method to estimate the nonparametric part, which was approximated by the piecewise linear functions and B-spline functions, and then get the least square estimator and the maximum likelihood estimator, respectively. Both the two resulting estimators are consistent and asymptotically normally distributed. The simulation results support our theory procedures. By introducing the acceleration stress model, we establish the partially nonlinear model to simulate the relation among the exchange rate of USD/EUR, USD/RMB and oil price. And then we study the inner relations of the global stock market, the model is applied to simulate the relationship among the Dow Jones Industrial index, Nikkei index and USD/JPY exchange rate. The partially nonlinear models are found to be an appropriate setting in such cases in finance. Finally, we construct a criterion function using copula function, which can select the best one among the methods on trend estimation in financial time series.
In recent 20 years, much research has been focused on the semiparametric models. The advantage of such semiparametric models is that they reduce the risk of misspecification relative to a fully parametric model and avoid some serious drawbacks of fully nonparametric methods. Firstly we discuss the statistical inference for partially linear models. Using Huber's M-estimator, we establish the robust estimator for the parametric part. The resulting estimator is shown to be consistent and asymptotically normally distributed. Simulation results show that the M-estimator performs as well as least square estimator under normally distributed errors. Especially, when the random errors are distributed as Cauchy distribution, the M-estimator is superior to the least square estimator. Secondly we propose a new kind of semiparametric models, called partially nonlinear models, which not only nest the partially linear models as a special case, but also contain many nonlinear functions widely used in applications. Compared to general partially nonlinear models, our models are convenient to make statistical inferences. We develop the sieve method to estimate the nonparametric part, which was approximated by the piecewise linear functions and B-spline functions, and then get the least square estimator and the maximum likelihood estimator, respectively. Both the two resulting estimators are consistent and asymptotically normally distributed. The simulation results support our theory procedures. By introducing the acceleration stress model, we establish the partially nonlinear model to simulate the relation among the exchange rate of USD/EUR, USD/RMB and oil price. And then we study the inner relations of the global stock market, the model is applied to simulate the relationship among the Dow Jones Industrial index, Nikkei index and USD/JPY exchange rate. The partially nonlinear models are found to be an appropriate setting in such cases in finance. Finally, we construct a criterion function using copula function, which can select the best one among the methods on trend estimation in financial time series.
引文
[1]Engle R F, Granger C W J, Rice J, Weiss A. Semiparametric estimates of the re-lation between weather and electricity sales [J]. Journal of the American Statistical Association,1986,80:310-319.
[2]Heckman N. Spline smoothing in a partly linear model[J]. Journal of the Royal Statistical Society,1986, Ser. B 48:244-248.
[3]Rice J. Convergence rates for partially splined models [J]. Statistics and Probability Letters,1986,4:203-208.
[4]Robinson P M. Root-n-consistent semiparametric regression [J],1988,56:931-954.
[5]Speckman P. Kernel smoothing in partial linear models [J]. Journal of the Royal Statistical Society,1988, Ser. B,50:413-436.
[6]Liang H, Hardle W, Carroll R J. Estimation in a semiparametric partially linear errors-in-variables model[J]. The Annals of Statistics,1999,27(5):1519-1535.
[7]Hadle W, Liang H, Gao J. Partially Linear Models[M]. Heidelberg:Physica-Verlag, 2000.
[8]高集体,陈希孺,赵林城.部分线性模型中估计的渐近正态性[J].科学通报,1992,18：1726-1727.
[9]施沛德,李国英.部分线性模型参数分量的L1模估计的渐近正态性[J].数学年刊,1994,4(15A)：478-484.
[10]梁华.半参数部分线性模型在经济中的一些应用[J].统计研究,1996,3：57-61.
[11]薛宏旗,宋立新,李国英.Ⅰ型区间删失情形下部分线性模型Sieve极大似然估计渐近性质[J].数学年刊,2001,24(1)：139-151.
[12]薛留根.半参数回归模型中小波估计的随机加权逼近速度[J].应用数学学报,2003,26(1)：11-25.
[13]冉昊,朱仲义.半参数回归模型中的异方差统计分析[J].应用概率统计,2004,20(1)：83-90.
[14]金蛟,崔恒建.半参数回归模型中基于稳健散布阵的参数估计[J].北京师范大学学报(自然科学版),2006,42(3)：224-228.
[15]Wahba G. Spline Models for Observational Data. In:CBMS-NSF 59, Regional Conference Series in Applied Mathematics[M]. SIAM,1990.
[16]Gao J, H L. Statistical inference in single-index and partially nonlinear models[J]. Annals of the Institute of Statistical Mathematics,1997,49(3):493-517.
[17]Zhu Z, Wei B. Asymptotic efficient estimation in semiparametric nonlinear regression models [J]. Applied Mathematics-A Journal of Chinese Universities,1999,14(1):57-66.
[18]Liang H. Second-order asymptotic efficiency of pmle in generalized linear models[J]. Statistics and Probability Letters,1995,24:273-279.
[19]Li R, Nie L. Efficient statistical inference procedures for partially nonlinear models and their applications[J]. Biometrics,2008,64:904-911.
[20]Huang T, Chen H. Estimating the parametric component of nonlinear partial spline model[J]. Journal of Multivariate Analysis,2008,99:1665-1680.
[21]Shi P. On b-spline m-estimators in a semiparametric regression model[J]. ournal of Systems Science and Complexity,1994,7(3):270-281.
[22]Bianco A, Boente G. Robust estimators in semiparametric partly linear regression models[J]. Journal of Statistical Planning and Inference,2004,122(1-2):229-252.
[23]张日权,王静龙.部分线性回归模型的M-估计[J].应用数学学报,2005,28(1)：151-157.
[24]茆诗松,王玲玲.加速寿命试验[M].北京：科学出版社,1997.
[25]李子奈.计量经济学：方法和应用[M].北京：清华大学出版社,1992.
[26]Grenander U. Abstract Inference[M]. New York:John Wiley,1981.
[27]Beran R. Minimum hellinger distance estimates for parametric models[J]. The Annals of Statistics,1977,5(3):445-463.
[28]Huber P J. Robust Statistics[M]. New York:John Wiley,1981.
[29]Robinson P M. Root-n-consistent semiparametric regression[J],1988,56:931-954.
[30]Hamilton A, Truong K. Local linear estimation in partially linear models[J]. Journal of Multivariate Analysis,1997,60:1-19.
[31]Hampel F, Ronchetti E, Rousseeuw P, Stahel W. Robust Statistics, The Approach Based on Influence Functions[M]. New York:John Wiley,1986.
[32]Pollard D. Convergence of Stochastic Processes [M]. New York:Springer,1984.
[33]Liang H. Estimation in partially linear models and numerical comarisons[J]. Com-putational Statistics and Data Analysis,2006,50:675-687.
[34]van de Geer S A. Empirical Processes in M-estimation[M]. New York:Cambridge University Press,2000.
[35]Seber G, Wild C. Nonlinear Regression[M]. New York:John Wiley,1989.
[36]Huet S. Statistical Tools for Nonlinear Regression:A Practical Guide with S-PLUS and R Examples[M]. New York:Springer,2004.
[37]Robinson P M. Nearest-neighbour estimation of semiparametric regression models[J]. Journal of Nonparametric Statistics,1995,5(1):33-41.
[38]Chang X W, Qu L. Wavelet estimation of partially linear models[J],2004,47(1):31-48.
[39]Fan J, Huang T. Profile likelihood inferences on semiparametric varying-coefficient partially linear models[J]. The Annals of Statistics,2005, 11(6):1031-1057.
[40]Huang J, Rossini A. Sieve estimation for the proportional-odds failure-time regression model with interval censoring[J]. Journal of the American Statistical Association, 1997,92:960-967.
[41]Shen X. On methods of sieves and penalization[J]. Annals of Statistics,1997,25:2555-2591.
[42]Xue H, Lam K F, Li G. Sieve maximum likelihood estimator for semiparametric regression models with current status data[J]. Journal of the American Statistical Association,2004,99(466):346-356.
[43]Shen X, Wong W H. Convergence rate of sieve estimates[J]. Annals of Statistics, 1994,22:580-615.
[44]Pollard D. Empirical Processes:Theory and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics 2[M]. Hayward, California:Institute of Mathematical Statistics and American Statistical Association,1990.
Petrov V V. Sums of Independent Random Variables[M]. Berlin:Springer,1975.
[46]van der Vaart A, Wellner J A. Weak Convergence and Empirical Processes:With Applications to Statistics[M]. New York:Springer,1996.
[47]Huang J. Efficient estimation for the proportional hazareds molel with interval cenor-ing[J]. Annals of Statistics,1996,24:540-568.
[48]Bickel P, Klaassen C, Ritov Y, Wellner J. Efficient and Adaptive Estimation for Semiparametric Models[M]. Johns Hopkins University Press,1993.
[49]Amano R A, van Norden S. Oil prices and the rise and fall of the us real exchange rate[J]. Journal of International Money and Finance,1998,17:299-316.
[50]Benassy-Quere A, Mignon V, Penot A. China and the relationship between the oil price and the dollar [J]. Energy Policy,2007,35:5795-5805.
[51]Chen S S, Chen H C. Oil prices and real exchange rates[J]. Energy Economics,2007, 29:390-404.
[52]Huang Y, Guo F. The role of oil price shocks on china's real exchange rate[J]. China Economic Review,2007,18:244-265.
[53]Olomola P, Adejumo A. Oil price shocks and macroeconomic activities in nigeria[J]. International Research Journal of Finance and Economics,2006,3:28-34.
[54]Kutan A, Wyzan M. Explaining the real exchange rate in kazakhstan,1996-2003:is kazakhstan vulnerable to the dutch disease?[J]. Economic Systems,2005,29:242-255.
[55]Schumaker L L. Spline Functions[M]. New York:John Wiley,1981.
[56]Ossiander M. A central limit theorem under metric entropy with l2 bracketing [J]. The Annals of Probability,1987,15:897-919.
[57]匡继昌.常用不等式[M].济南：山东科学技术出版社,2004.
[58]Truong Y. Nonparametric curve estimation with time series errors [J]. Journal of Statistical Planning and Inference,1991,28:167-183.
[59]Beran J, Feng Y. Local polynomial fitting with long-memory, short-memory and antipersistent errors [J]. Annals of the Institute of Statistical Mathematics,2001, 54:291-311.
[60]Johnstone I, Silverman B W. Wavelet threshold estimators for data with correlated noise[J]. Journal of the Royal Statistical Society,1997, Series B 59:319-351.
[61]Nelsen R B. An Introduction to Copulas[M]. New York:Springer,1999.
[62]Neuhaus G. On weak convergence of stochastic processes with multidimensional time parameter[J]. The Annals of Mathematical Statistics,1971,42(4):1285-1295.
[63]Bickel P, Wichura M. Convergence criteria for multiparameter stochastic processes and some applications [J]. The Annals of Mathematical Statistics,1971,42(5):1656-1670.
[64]回德俊.基于COPULA方法提取非线性时间序列的趋势项[D].大连：大连理工大学数学科学学院,2009.
[65]王仁宏,李崇君,朱春钢.计算几何教程[M].北京：科学出版社,2008.
[2]Heckman N. Spline smoothing in a partly linear model[J]. Journal of the Royal Statistical Society,1986, Ser. B 48:244-248.
[3]Rice J. Convergence rates for partially splined models [J]. Statistics and Probability Letters,1986,4:203-208.
[4]Robinson P M. Root-n-consistent semiparametric regression [J],1988,56:931-954.
[5]Speckman P. Kernel smoothing in partial linear models [J]. Journal of the Royal Statistical Society,1988, Ser. B,50:413-436.
[6]Liang H, Hardle W, Carroll R J. Estimation in a semiparametric partially linear errors-in-variables model[J]. The Annals of Statistics,1999,27(5):1519-1535.
[7]Hadle W, Liang H, Gao J. Partially Linear Models[M]. Heidelberg:Physica-Verlag, 2000.
[8]高集体,陈希孺,赵林城.部分线性模型中估计的渐近正态性[J].科学通报,1992,18：1726-1727.
[9]施沛德,李国英.部分线性模型参数分量的L1模估计的渐近正态性[J].数学年刊,1994,4(15A)：478-484.
[10]梁华.半参数部分线性模型在经济中的一些应用[J].统计研究,1996,3：57-61.
[11]薛宏旗,宋立新,李国英.Ⅰ型区间删失情形下部分线性模型Sieve极大似然估计渐近性质[J].数学年刊,2001,24(1)：139-151.
[12]薛留根.半参数回归模型中小波估计的随机加权逼近速度[J].应用数学学报,2003,26(1)：11-25.
[13]冉昊,朱仲义.半参数回归模型中的异方差统计分析[J].应用概率统计,2004,20(1)：83-90.
[14]金蛟,崔恒建.半参数回归模型中基于稳健散布阵的参数估计[J].北京师范大学学报(自然科学版),2006,42(3)：224-228.
[15]Wahba G. Spline Models for Observational Data. In:CBMS-NSF 59, Regional Conference Series in Applied Mathematics[M]. SIAM,1990.
[16]Gao J, H L. Statistical inference in single-index and partially nonlinear models[J]. Annals of the Institute of Statistical Mathematics,1997,49(3):493-517.
[17]Zhu Z, Wei B. Asymptotic efficient estimation in semiparametric nonlinear regression models [J]. Applied Mathematics-A Journal of Chinese Universities,1999,14(1):57-66.
[18]Liang H. Second-order asymptotic efficiency of pmle in generalized linear models[J]. Statistics and Probability Letters,1995,24:273-279.
[19]Li R, Nie L. Efficient statistical inference procedures for partially nonlinear models and their applications[J]. Biometrics,2008,64:904-911.
[20]Huang T, Chen H. Estimating the parametric component of nonlinear partial spline model[J]. Journal of Multivariate Analysis,2008,99:1665-1680.
[21]Shi P. On b-spline m-estimators in a semiparametric regression model[J]. ournal of Systems Science and Complexity,1994,7(3):270-281.
[22]Bianco A, Boente G. Robust estimators in semiparametric partly linear regression models[J]. Journal of Statistical Planning and Inference,2004,122(1-2):229-252.
[23]张日权,王静龙.部分线性回归模型的M-估计[J].应用数学学报,2005,28(1)：151-157.
[24]茆诗松,王玲玲.加速寿命试验[M].北京：科学出版社,1997.
[25]李子奈.计量经济学：方法和应用[M].北京：清华大学出版社,1992.
[26]Grenander U. Abstract Inference[M]. New York:John Wiley,1981.
[27]Beran R. Minimum hellinger distance estimates for parametric models[J]. The Annals of Statistics,1977,5(3):445-463.
[28]Huber P J. Robust Statistics[M]. New York:John Wiley,1981.
[29]Robinson P M. Root-n-consistent semiparametric regression[J],1988,56:931-954.
[30]Hamilton A, Truong K. Local linear estimation in partially linear models[J]. Journal of Multivariate Analysis,1997,60:1-19.
[31]Hampel F, Ronchetti E, Rousseeuw P, Stahel W. Robust Statistics, The Approach Based on Influence Functions[M]. New York:John Wiley,1986.
[32]Pollard D. Convergence of Stochastic Processes [M]. New York:Springer,1984.
[33]Liang H. Estimation in partially linear models and numerical comarisons[J]. Com-putational Statistics and Data Analysis,2006,50:675-687.
[34]van de Geer S A. Empirical Processes in M-estimation[M]. New York:Cambridge University Press,2000.
[35]Seber G, Wild C. Nonlinear Regression[M]. New York:John Wiley,1989.
[36]Huet S. Statistical Tools for Nonlinear Regression:A Practical Guide with S-PLUS and R Examples[M]. New York:Springer,2004.
[37]Robinson P M. Nearest-neighbour estimation of semiparametric regression models[J]. Journal of Nonparametric Statistics,1995,5(1):33-41.
[38]Chang X W, Qu L. Wavelet estimation of partially linear models[J],2004,47(1):31-48.
[39]Fan J, Huang T. Profile likelihood inferences on semiparametric varying-coefficient partially linear models[J]. The Annals of Statistics,2005, 11(6):1031-1057.
[40]Huang J, Rossini A. Sieve estimation for the proportional-odds failure-time regression model with interval censoring[J]. Journal of the American Statistical Association, 1997,92:960-967.
[41]Shen X. On methods of sieves and penalization[J]. Annals of Statistics,1997,25:2555-2591.
[42]Xue H, Lam K F, Li G. Sieve maximum likelihood estimator for semiparametric regression models with current status data[J]. Journal of the American Statistical Association,2004,99(466):346-356.
[43]Shen X, Wong W H. Convergence rate of sieve estimates[J]. Annals of Statistics, 1994,22:580-615.
[44]Pollard D. Empirical Processes:Theory and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics 2[M]. Hayward, California:Institute of Mathematical Statistics and American Statistical Association,1990.
Petrov V V. Sums of Independent Random Variables[M]. Berlin:Springer,1975.
[46]van der Vaart A, Wellner J A. Weak Convergence and Empirical Processes:With Applications to Statistics[M]. New York:Springer,1996.
[47]Huang J. Efficient estimation for the proportional hazareds molel with interval cenor-ing[J]. Annals of Statistics,1996,24:540-568.
[48]Bickel P, Klaassen C, Ritov Y, Wellner J. Efficient and Adaptive Estimation for Semiparametric Models[M]. Johns Hopkins University Press,1993.
[49]Amano R A, van Norden S. Oil prices and the rise and fall of the us real exchange rate[J]. Journal of International Money and Finance,1998,17:299-316.
[50]Benassy-Quere A, Mignon V, Penot A. China and the relationship between the oil price and the dollar [J]. Energy Policy,2007,35:5795-5805.
[51]Chen S S, Chen H C. Oil prices and real exchange rates[J]. Energy Economics,2007, 29:390-404.
[52]Huang Y, Guo F. The role of oil price shocks on china's real exchange rate[J]. China Economic Review,2007,18:244-265.
[53]Olomola P, Adejumo A. Oil price shocks and macroeconomic activities in nigeria[J]. International Research Journal of Finance and Economics,2006,3:28-34.
[54]Kutan A, Wyzan M. Explaining the real exchange rate in kazakhstan,1996-2003:is kazakhstan vulnerable to the dutch disease?[J]. Economic Systems,2005,29:242-255.
[55]Schumaker L L. Spline Functions[M]. New York:John Wiley,1981.
[56]Ossiander M. A central limit theorem under metric entropy with l2 bracketing [J]. The Annals of Probability,1987,15:897-919.
[57]匡继昌.常用不等式[M].济南：山东科学技术出版社,2004.
[58]Truong Y. Nonparametric curve estimation with time series errors [J]. Journal of Statistical Planning and Inference,1991,28:167-183.
[59]Beran J, Feng Y. Local polynomial fitting with long-memory, short-memory and antipersistent errors [J]. Annals of the Institute of Statistical Mathematics,2001, 54:291-311.
[60]Johnstone I, Silverman B W. Wavelet threshold estimators for data with correlated noise[J]. Journal of the Royal Statistical Society,1997, Series B 59:319-351.
[61]Nelsen R B. An Introduction to Copulas[M]. New York:Springer,1999.
[62]Neuhaus G. On weak convergence of stochastic processes with multidimensional time parameter[J]. The Annals of Mathematical Statistics,1971,42(4):1285-1295.
[63]Bickel P, Wichura M. Convergence criteria for multiparameter stochastic processes and some applications [J]. The Annals of Mathematical Statistics,1971,42(5):1656-1670.
[64]回德俊.基于COPULA方法提取非线性时间序列的趋势项[D].大连：大连理工大学数学科学学院,2009.
[65]王仁宏,李崇君,朱春钢.计算几何教程[M].北京：科学出版社,2008.