广义部分线性违约概率模型
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摘要
在商业银行信用风险管理中,违约概率是指借款人在未来一定时期内不能按合同要求偿还银行贷款本息或履行相关义务的可能性。在现代商业银行的信用风险度量系统中,违约概率的测算已经成为商业银行计算预期损失和Var值、确定经济资本的核心工具之一。
违约概率模型的研究一直都是理论界和金融界的热点。基于多元线性统计理论的传统违约概率模型结构简单,解释能力强,计算强度小,因此被理论界和银行界广泛采用,但是其严格的统计假设条件、容易产生模型设定偏差、缺乏对违约风险的系统认识等缺陷,影响了模型预测的准确性。为此,本文使用一类基于广义部分线性模型理论的违约概率模型：P(Y=1|U,T)=E(Y|U,T)=G(UTβ+m(T))其中G(·)是给定连接函数,β是未知的参数向量,m(·)是未知的非参函数。这个模型同时含有参数向量和非参数向量,具有良好的解释性和适应性。
本文结合各种文献讨论了广义部分线性模型的估计理论和计算过程,并将二分类广义部分线性违约概率模型拓展到有序多分类的情形。最后通过处理加州大学欧文分校(UCI)机器学习数据库的德国信用卡数据,对广义部分线性模型的拟合精度和预测效果与传统的Logistic违约概率模型在二分类和有序多分类两种情形下进行了实证比较,并通过构造ROC曲线、CAP曲线、K-S检测曲线等对两类违约概率模型的功效进行了评价。
本文创新点：将广义部分线性模型应用到信用风险评估领域,研究了在二分类和有序多分类情形下的广义部分线性违约概率模型,为违约概率的测算提供了一种新的途径,并结合实例研究说明广义部分线性违约概率模型的可行性和有效性。
In the credit risk management of the commercial bank, probability of de-fault refers to the possibility that the borrowers who are not able to repay the principal and interest of the bank loan or fulfill the related obligations in a future certain time according to the contract requirement. In the credit risk management system of modern commercial bank, the evaluation and measure-ment of probability is an important asspect and has already been one of the core tools to compute the expected loss, Var and economic capital.
Probability of default model(PD) has always been a hotspot of finance and scientific research fields. Traditional PD models are easy to be used in practice for their good interpretation and computational simpleness. But it has some defects, such as, strict statistical hypothesis, being easy to produce model bias and loss of systemic comprehension about credit risks.and this affects models'forecasting accuracy. So this paper use a new PD model based on theories of generalized partial linear models: P(Y=1|U,T)= E(Y|U,T)= G(UTβ+m(T)) where G(-) is a given link function,βis the unknown parametric vector, m is the unknown nonparametric function. This model has parametric and non-parametric vectors simultaneously. So it has good interpretation and adaption.
This paper discusses estimate theories and computational process of GPLM model based on various literatures and put two-classify PD model to the or-dered multiple-classify situation. In the end, through dealing with German credit data from machine learning database of University of California at Irvine, I compare the fitness and accuracy between GPLM PD model and Logistic PD model in two-classify and ordered multiple-classify situations. I also appraise two models'effects through ROC curves, CAP curves and K-S testing curves. Research shows GPLM PD model has high accuracy and better manifestation.
The new idea in this paper is to make application in credit risks manage-ment fields with GPLM model and do some study about GPLM PD model in two-classify and ordered multiple-classify situations. This paper provides a new way to evaluate probability of default. Feasibility and effectiveness of GPLM PD model are illustrated through practical analysises.
违约概率模型的研究一直都是理论界和金融界的热点。基于多元线性统计理论的传统违约概率模型结构简单,解释能力强,计算强度小,因此被理论界和银行界广泛采用,但是其严格的统计假设条件、容易产生模型设定偏差、缺乏对违约风险的系统认识等缺陷,影响了模型预测的准确性。为此,本文使用一类基于广义部分线性模型理论的违约概率模型：P(Y=1|U,T)=E(Y|U,T)=G(UTβ+m(T))其中G(·)是给定连接函数,β是未知的参数向量,m(·)是未知的非参函数。这个模型同时含有参数向量和非参数向量,具有良好的解释性和适应性。
本文结合各种文献讨论了广义部分线性模型的估计理论和计算过程,并将二分类广义部分线性违约概率模型拓展到有序多分类的情形。最后通过处理加州大学欧文分校(UCI)机器学习数据库的德国信用卡数据,对广义部分线性模型的拟合精度和预测效果与传统的Logistic违约概率模型在二分类和有序多分类两种情形下进行了实证比较,并通过构造ROC曲线、CAP曲线、K-S检测曲线等对两类违约概率模型的功效进行了评价。
本文创新点：将广义部分线性模型应用到信用风险评估领域,研究了在二分类和有序多分类情形下的广义部分线性违约概率模型,为违约概率的测算提供了一种新的途径,并结合实例研究说明广义部分线性违约概率模型的可行性和有效性。
In the credit risk management of the commercial bank, probability of de-fault refers to the possibility that the borrowers who are not able to repay the principal and interest of the bank loan or fulfill the related obligations in a future certain time according to the contract requirement. In the credit risk management system of modern commercial bank, the evaluation and measure-ment of probability is an important asspect and has already been one of the core tools to compute the expected loss, Var and economic capital.
Probability of default model(PD) has always been a hotspot of finance and scientific research fields. Traditional PD models are easy to be used in practice for their good interpretation and computational simpleness. But it has some defects, such as, strict statistical hypothesis, being easy to produce model bias and loss of systemic comprehension about credit risks.and this affects models'forecasting accuracy. So this paper use a new PD model based on theories of generalized partial linear models: P(Y=1|U,T)= E(Y|U,T)= G(UTβ+m(T)) where G(-) is a given link function,βis the unknown parametric vector, m is the unknown nonparametric function. This model has parametric and non-parametric vectors simultaneously. So it has good interpretation and adaption.
This paper discusses estimate theories and computational process of GPLM model based on various literatures and put two-classify PD model to the or-dered multiple-classify situation. In the end, through dealing with German credit data from machine learning database of University of California at Irvine, I compare the fitness and accuracy between GPLM PD model and Logistic PD model in two-classify and ordered multiple-classify situations. I also appraise two models'effects through ROC curves, CAP curves and K-S testing curves. Research shows GPLM PD model has high accuracy and better manifestation.
The new idea in this paper is to make application in credit risks manage-ment fields with GPLM model and do some study about GPLM PD model in two-classify and ordered multiple-classify situations. This paper provides a new way to evaluate probability of default. Feasibility and effectiveness of GPLM PD model are illustrated through practical analysises.
引文
[1]中国银行业监督委员会(2006).统一资本计量和资本标准的国际协议：修订框架.中国金融出版社
[2]王松桂,史建红等(2004).线性模型引论.科学技术出版社.
[3]王济川,郭志刚(2001)Logistic回归模型—方法与应用.高等教育出版社.
[4]柴根象,洪圣岩(1995).半参数回归模型.安徽教育出版社.
[5]黎子良,邢海鹏(2009).金融市场中的统计模型和方法.高等教育出版社.
[6]张尧庭,方开泰(1982).多元统计分析.科学出版社.
[7]茆诗松,王静龙等(2006).高等数理统计.高等教育出版社.
[8]阿诺·德·瑟维吉尼,奥里维尔·雷劳特(2005),信用风险度量与管理.中国财政经济出版社
[9]毛振华,阎衍(2007).信用评级前沿理论与实践.中国金融出版社.
[10]郭敏华(2003).信用评级.中国人民大学出版社.
[11]庞素琳(2006)Logistic回归模型在信用风险分析中的应用.数学的实践与认识.第36卷第9期.
[12]管七海,冯宗宪(2004).信用违约概率测算研究：文献综述与比较.世界经济
[13]于力勇,詹捷辉(2004).基于Logistic回归分析的违约概率预测研究.财经研究.第30卷第9期
[14]周玮,杨兵兵,陈宏,徐晓肆(2005).商业银行违约概率测算相关问题研究.国际金融研究
[15]王小明(2008).关于一类广义可加违约概率模型的探讨.系统工程理论与实践第6期
[16]彭建刚,屠海波(2009).有序多分类logistic模型在违约概率测算中的应用.财经理论与实践第30卷第160期
[17]刘衡郁,连剑平.(2008).基于统计分析的主流信用风险模型评价.商业时代第16期
[18]孙月静(2007).违约概率测度研究方法与模型综述.东北财经大学学报2007年第2期
[19]白少布(2010).基于有序Logistic模型的企业供应链融资风险预警研究.经济经纬2010年第61期
[20]刘莉亚(2004).商业银行内部评级系统研究综述.外国经济与管理第26卷第8期
[21]张超(2010).公司违约概率模型及其在商业银行中的应用.华北金融2010年第4期
[22]Stone, C. J. (1977). Consistent nonparametric regression, Applied Statistics 5:595-635
[23]Engle, R. F. and Rice, J. (1986). Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association 81:310-320
[24]Wahba, G (1984). Partial spline models for semiparametric estimation of functions of several variables, Statistical Analysis of Time Series 319-329
[25]Hardle. W, Miiller. M, Sperlich. S and Werwatz. A. (2004). Nonparametric and Semiparametric Models. Springer.
[26]Severini, T. A. and Staniswalis, J. G. (1994). Quasi-likelihood Estimation in Semiparametric Models. Journal of the Royal Statistics Association,501-511.
[27]McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, Vol.37 Statistics and Applied Probablity,2edn, Chapman and Hall, London
[28]Severini, T. A. and Wong, W. H. (1992). Generalized profiled likelihood and conditionally parametric models, Annals of Statistics 20:1768-1802.
[29]Speckman, P. E. (1988). Regression analysis for partially linear models, Journal of the Royal Statistical Society, Series B 50:413-436.
[30]Robinson, P. (1988). Root-n-consistent semiparametric regression. Econometrica 56:931-954
[31]Chen, H. (1988). Convergent rates for parametric components in a partly linear model. Annals of Statistics 16:136-146.
[32]Cui Hengjian. and Li Rongcai. (1998). On parameter estimations for semi-linear errors-in-variables models. Journal of Multivariate Analysis 64:1-24.
[33]Wang qihua. (1999). Estimation of partial linear errors-in-variables models with validation data. Journal of Multivariate Analysis 69:30-64.
[34]Hamilton, S. A and Truong, Y. K(1997). Local linear estimation in partially linear models. Journal of Multivariate Analysis 60:1-19.
[35]Mammen. E and Van de Geer. (1997). Penalized estimation in partially linear models. Annals of Statistics 25:1014-1035.
[36]W. Hardle, H. Liang and J. T. Gao(2000). Partially Linear Models. Physica Verlag,Heidelberg.
[37]Staniswalis, J. G. and Thall, P. F. (2001). An explanation of generalized profile likelihoods. Statistics and Computing 11:293-298.
[2]王松桂,史建红等(2004).线性模型引论.科学技术出版社.
[3]王济川,郭志刚(2001)Logistic回归模型—方法与应用.高等教育出版社.
[4]柴根象,洪圣岩(1995).半参数回归模型.安徽教育出版社.
[5]黎子良,邢海鹏(2009).金融市场中的统计模型和方法.高等教育出版社.
[6]张尧庭,方开泰(1982).多元统计分析.科学出版社.
[7]茆诗松,王静龙等(2006).高等数理统计.高等教育出版社.
[8]阿诺·德·瑟维吉尼,奥里维尔·雷劳特(2005),信用风险度量与管理.中国财政经济出版社
[9]毛振华,阎衍(2007).信用评级前沿理论与实践.中国金融出版社.
[10]郭敏华(2003).信用评级.中国人民大学出版社.
[11]庞素琳(2006)Logistic回归模型在信用风险分析中的应用.数学的实践与认识.第36卷第9期.
[12]管七海,冯宗宪(2004).信用违约概率测算研究：文献综述与比较.世界经济
[13]于力勇,詹捷辉(2004).基于Logistic回归分析的违约概率预测研究.财经研究.第30卷第9期
[14]周玮,杨兵兵,陈宏,徐晓肆(2005).商业银行违约概率测算相关问题研究.国际金融研究
[15]王小明(2008).关于一类广义可加违约概率模型的探讨.系统工程理论与实践第6期
[16]彭建刚,屠海波(2009).有序多分类logistic模型在违约概率测算中的应用.财经理论与实践第30卷第160期
[17]刘衡郁,连剑平.(2008).基于统计分析的主流信用风险模型评价.商业时代第16期
[18]孙月静(2007).违约概率测度研究方法与模型综述.东北财经大学学报2007年第2期
[19]白少布(2010).基于有序Logistic模型的企业供应链融资风险预警研究.经济经纬2010年第61期
[20]刘莉亚(2004).商业银行内部评级系统研究综述.外国经济与管理第26卷第8期
[21]张超(2010).公司违约概率模型及其在商业银行中的应用.华北金融2010年第4期
[22]Stone, C. J. (1977). Consistent nonparametric regression, Applied Statistics 5:595-635
[23]Engle, R. F. and Rice, J. (1986). Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association 81:310-320
[24]Wahba, G (1984). Partial spline models for semiparametric estimation of functions of several variables, Statistical Analysis of Time Series 319-329
[25]Hardle. W, Miiller. M, Sperlich. S and Werwatz. A. (2004). Nonparametric and Semiparametric Models. Springer.
[26]Severini, T. A. and Staniswalis, J. G. (1994). Quasi-likelihood Estimation in Semiparametric Models. Journal of the Royal Statistics Association,501-511.
[27]McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, Vol.37 Statistics and Applied Probablity,2edn, Chapman and Hall, London
[28]Severini, T. A. and Wong, W. H. (1992). Generalized profiled likelihood and conditionally parametric models, Annals of Statistics 20:1768-1802.
[29]Speckman, P. E. (1988). Regression analysis for partially linear models, Journal of the Royal Statistical Society, Series B 50:413-436.
[30]Robinson, P. (1988). Root-n-consistent semiparametric regression. Econometrica 56:931-954
[31]Chen, H. (1988). Convergent rates for parametric components in a partly linear model. Annals of Statistics 16:136-146.
[32]Cui Hengjian. and Li Rongcai. (1998). On parameter estimations for semi-linear errors-in-variables models. Journal of Multivariate Analysis 64:1-24.
[33]Wang qihua. (1999). Estimation of partial linear errors-in-variables models with validation data. Journal of Multivariate Analysis 69:30-64.
[34]Hamilton, S. A and Truong, Y. K(1997). Local linear estimation in partially linear models. Journal of Multivariate Analysis 60:1-19.
[35]Mammen. E and Van de Geer. (1997). Penalized estimation in partially linear models. Annals of Statistics 25:1014-1035.
[36]W. Hardle, H. Liang and J. T. Gao(2000). Partially Linear Models. Physica Verlag,Heidelberg.
[37]Staniswalis, J. G. and Thall, P. F. (2001). An explanation of generalized profile likelihoods. Statistics and Computing 11:293-298.