CIR模型的统计诊断
|
摘要
关于金融模型的统计推断的研究已经有相当长的一段时间,并且产生了许多较为成熟的理论。然而将统计诊断的方法引入对金融模型的研究无论是在国内还是国外都是比较少见。目前运用到金融学领域的统计诊断方法仅限于时间序列诊断,而且关于时间序列诊断的理论尚不成熟。将回归诊断引入金融学的文章更为少见。本文以利率期限结构模型中的CIR模型为研究对象,采用回归诊断的思想,对CIR模型的统计诊断方法进行讨论。
本文首先介绍了一些关于扩散模型的估计方法和统计诊断的发展现状,然后针对CIR模型做了如下工作：
本文运用Girsanov定理和Ito定理,在等价概率测度下得到一个线性回归模型,然后采用回归诊断的思想给出了Cook距离,AP统计量和WK统计量。由于本文得到线性回归模型中误差项较为复杂,所以在数值模拟的情况下,本文从两方面考察了以上三个诊断统计量的诊断效果：1运用得到的诊断量诊断出异常点,并删除异常点后重新估计模型参数,估计值更接近预设值；2更改一些数据点使其明显偏离既定轨道,并重新估计模型参数,估计值与预设值距离增大。以上结果表明本文定义的Cook距离,AP统计量和WK统计量是合理的。本文同时也给出了均值漂移的诊断和CIR模型的似然距离。
在第四章,本文选取上海银行间同业拆借利率(SHIBOR)隔夜报价数据做为研究对象,对数据进行诊断。根据实际情况,将诊断出的异常点与货币政策出台时间进行比对,考察诊断效果。结果表明诊断方法有效。
It's been a long time for the study of statistical inference for financial model, and there have been a lot of maturity theories about the study. But there are few people to study the introduction of statistical diagnose to financial model in the world. Up to now, only the diagnosis for time series has been applied to finance, but the theory is not matured. And there are few paper which introduce regression diagnose to finance too. In order to extend regression diagnose to diffusion model, this paper applies the regression diagnose to CIR model, and proposes the diagnosis methods for CIR.
Firstly, this paper reviews the research status of the estimation of diffusion model and statistical diagnose, then studies the diagnosis for CIR model as followed:
This paper applies the Gisanov theorem and Ito theorem, under the equivalent probability defined in this paper, to obtain a linear regression model about CIR model, and gives the Cook distance, WK statistic and AP statistic. As the error term in the linear regression is more complex than the one in classical regression diagnose model, we use numerical simulation to examine the effect of the diagnostic statistics above in two ways. We first use the diagnose statistics above to indentify the outliers from the simulated data, then delete the outliers and reestimate the parameter of the diffusion term of CIR model, and the result is better. Secondly we change the values of some data to make them obviously deviate from the simulated track, and then reestimate the parameter, the result is not better. So we conclude that the diagnose statistics above is reasonable. In this paper, we also study about the diagnosis of mean-shift outlier model and likelihood distance for CIR model.
In the chapter 4, we take the ShangHai Interbank Offered Rate (SHIBOR) overnight quote as the study object, and detect the outliers. Then we compare the outliers to the date when the monetary policies released by government to examine the effect of the statistical diagnosis for CIR model. And the result show that the diagnosis for CIR model is effective.
本文首先介绍了一些关于扩散模型的估计方法和统计诊断的发展现状,然后针对CIR模型做了如下工作：
本文运用Girsanov定理和Ito定理,在等价概率测度下得到一个线性回归模型,然后采用回归诊断的思想给出了Cook距离,AP统计量和WK统计量。由于本文得到线性回归模型中误差项较为复杂,所以在数值模拟的情况下,本文从两方面考察了以上三个诊断统计量的诊断效果：1运用得到的诊断量诊断出异常点,并删除异常点后重新估计模型参数,估计值更接近预设值；2更改一些数据点使其明显偏离既定轨道,并重新估计模型参数,估计值与预设值距离增大。以上结果表明本文定义的Cook距离,AP统计量和WK统计量是合理的。本文同时也给出了均值漂移的诊断和CIR模型的似然距离。
在第四章,本文选取上海银行间同业拆借利率(SHIBOR)隔夜报价数据做为研究对象,对数据进行诊断。根据实际情况,将诊断出的异常点与货币政策出台时间进行比对,考察诊断效果。结果表明诊断方法有效。
It's been a long time for the study of statistical inference for financial model, and there have been a lot of maturity theories about the study. But there are few people to study the introduction of statistical diagnose to financial model in the world. Up to now, only the diagnosis for time series has been applied to finance, but the theory is not matured. And there are few paper which introduce regression diagnose to finance too. In order to extend regression diagnose to diffusion model, this paper applies the regression diagnose to CIR model, and proposes the diagnosis methods for CIR.
Firstly, this paper reviews the research status of the estimation of diffusion model and statistical diagnose, then studies the diagnosis for CIR model as followed:
This paper applies the Gisanov theorem and Ito theorem, under the equivalent probability defined in this paper, to obtain a linear regression model about CIR model, and gives the Cook distance, WK statistic and AP statistic. As the error term in the linear regression is more complex than the one in classical regression diagnose model, we use numerical simulation to examine the effect of the diagnostic statistics above in two ways. We first use the diagnose statistics above to indentify the outliers from the simulated data, then delete the outliers and reestimate the parameter of the diffusion term of CIR model, and the result is better. Secondly we change the values of some data to make them obviously deviate from the simulated track, and then reestimate the parameter, the result is not better. So we conclude that the diagnose statistics above is reasonable. In this paper, we also study about the diagnosis of mean-shift outlier model and likelihood distance for CIR model.
In the chapter 4, we take the ShangHai Interbank Offered Rate (SHIBOR) overnight quote as the study object, and detect the outliers. Then we compare the outliers to the date when the monetary policies released by government to examine the effect of the statistical diagnosis for CIR model. And the result show that the diagnosis for CIR model is effective.
引文
[1]Sharpe, William F. Capital Asset Prices. A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance XIX (3)1964:425-42.
[2]Fama.Efficient Capital Market:A Review of Theory and Empirical Work.Journal of Finance, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York,N.Y.1970:383-417.
[3]Black F.,Scholes M. The pricing of options corporate liabilities[J].Journal of Plitical Economy,1973,81 (3):637-659.
[4]Merton,Robert C. An Intertemporal Capital Asset Pricing Model. Econometrica 1973 41:5,pp.867-87.
[5]Vasicek, O. An equilibrium characterization of the term structure. J. Finan. Econom.1977 5,177-188.
[6]Cox, J. C., Ingersoll, J. E. and Ross, S. A. A theory of the term structure of interest rates. Econometrica 1985 53,385-467.
[7]Cox, J. C., Ingersoll, J. E. and Ross, S. A. An analysis of variable rate loan contracts. J. Finance 1980 35,389-403.
[8]Chan, K. C., Karolyi, A. G., Longstaff, F. A. and Sanders, A. B. An empirical comparison of alternative models of the short-term interest rate. J.1992
[9]Ait-Sahalia Testing Continuous-Time Models of the Spot Interest Rate, Review of Financial Studies,1996,9,385-426
[10]Stanton A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk", Journal of Finance 1997,52,1973-2002,
[11]Hong, Y. and H. Li, Nonparametric specification testing for continuous-time models with applications to interest rate term structures, The Review of Financial Studies,2005,18, 37-84.
[12]Kim M.S.Consistent estimation in regression models for the drift function in some continuous time models, Computational Statistics& Data Analysis Volume 52, Issue 5,20 January 2008,2682-2691
[13]I.V. Basawa, B.L.S. Prakasa Rao, Statistical Inference for Stochastic Processes, AcademicPress, London.1980
[14]Dacunha-Castelle and Florens-Zmirou, Estimation of the coefficients of a diffusion from discrete observations, Stochastics1986,19, pp.263-284.
[15]Duffie, D. and K.J. Singleton, Simulated moments estimation of Markov models of asset prices, Econometrica,1993,61,929-952.
[16]Hansen, Lars Peter and Jose Alexandre Scheinkman.1995. Back to the Future: GeneratingMoment Implications for Continuous-Time Markov Processes. Econometrica 63 (4):767-804.
[17]Gallant, A.R. and G. Tauchen, Which moments to match? Econometric Theory,1996,12, 657-681.
[18]Jacquier E., N.G. Polson and P.E. Rossi, Bayesian analysis of stochastic volatility models: reply, Journal of Business and Economic Statistics,1994,12,4,413-417.
[19]Eraker, B., Markov chain Monte Carlo analysis of diffusion models with application to finance, HAE Thesis, Norwegian School of Economics and Business Administration.1998
[20]Jones, C.S., Bayesian estimation of continuous-time finance models, Working Paper, Simon School of Business, University of Rochester.1998
[21]Ahn,D.H.and B.Gao,A parametric nonlinear model of term structure dynamics,Review of Financial Studies,1999,12,721-726.
[22]陈萍,随机波动率模型的统计推断及其衍生证券的定价,南京理工大学博士论文,2004.
[23]Hoffmann. M., Adaptive estimation in diffusion process, Stochastic processes and their applications,1999,79,135-163,
[24]Ait-Sahalia, Y. Nonparametric Pricing of Interest Rate Derivative Securities.Econometrica 1996a,64,527-560.
[25]Ait-Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusions:A closed-form approach, Econometrica,2002a,70,223-262.
[26]Jacod, J., Non-parametric Kernel Estimation of the Coefficient of a Diffusion, J. Scan.Stat.,2000,27(1),83-98
[27]Chen Ping, Wang Jinde, Wavelet estimation in time dependent diffusion models,science in China,2007.38 (6),1-16
[28]Pritsker,M., Nonparametric density estimation and tests of continuous time interest rate models, Review of Financial Studies,1998,11,449-487.
[29]Chapman, D. and N. Pearson, Is the short rate drift actually nonlinear? Journal of Finance, 2000,55,355-388.
[30]Fan, J. and C. Zhang, A re-examination of diffusion estimators with applications to financial model validation, Journal of the American Statistical Association,2003,98, 118-134.
[31]Cai, Z., J. Fan and Q. Yao, Functional-coefficient regression models for nonlinear time series, Journal of the American Statistical Association,2000,95,941-956.
[32]Fan, J., C. Zhang and J. Zhang, Generalized likelihood ratio statistics and Wilks phenomenon, The Annals of Statistics,2001,29,153-193.
[33]Chan, K.C., G.A. Karolyi, F.A. Longstaff and A.B. Sanders, An empirical comparison of alternative models of the short-term interest rate, Journal of Finance,1992,47,1209-1227.
[34]Conley, T.G., L.P. Hansen, E.G.J. Luttmer and J.A. Scheinkman, Short-term interest rates as subordinated diffusions, Review of Financial Studies,1997,10,525-577.
[35]Bliss, R.R. and D. Smith, The elasticity of interest rate volatility:Chan, Karolyi,Longstaff, and Sanders revisited, Journal of Risk,1998,1,21-46.
[36]Cai, Z. and Y. Hong, Nonparametric methods in continuous-time finance:A selectivereview. In Recent Advances and Trends in Nonparametric Statistics (M.G. Akritas and D.M. Politis, eds.) 2003,283-302.
[37]Cook, R. D. and Weisberg, S. Residuals and Influence in Regression. Chapman and Hall, New York.1982
[38]韦博成路过斌史建清统计诊断引论南京：东南大学出版社,1991.
[39]McCullagh, P. and Nelder, J. A.:Generalized Linear Models. Chapman and Hall, London.1983.
[40]Cook, R. D. Detection of influential observations in linear regression. Technometrics 19, 15-18.1977
[41]Cook, R. D. Assessment of local iofluence. J. R. Statist. Soc. B 48,133-169.1986.
[42]Belsley,D. A.,Kuh,E. and Welsch, R. E. Regression Diagnostics. John Wiley,New York,1980
[43]Simonoff, J. S. and Tsai, C. L.:Jackknife-based estimators and confidence regions in nonlinear regression. Tech nometrics 28,1986,103-112..
[44]Sber, G. A. F. and Wild, C. J.:Nonliear Regression. John Wiley, New York.1989.
[45]Pregibon, D.:Logistic regression diagnostics. Ann. Statist.1981,9,705-724.
[46]Thomas, W. and Cook, R. D.:Assessing influence on regression coefficients in generalized linear models. Biometrika 1989,76,741-749.
[47]Thomas, W. and Cook, R. D.:Assessing influence on predictions from generalized linear models. Technometrics 1990,32,59-65.
[48]Bechman, R. J., Nachtscheim, C. J. and Cook, R. D.:Diagnostics for mixed-model analysis of variance.Technometrics 1987,29,413-426.
[49]Hodges J.S. Some algebra and geometry for hierarchical models, applied to diagnostics (with discussion). J. Roy. Stat. Soc., Series B 1998,60:497-536.
[50]宗序平,系统分析中非线性随机效应模型的统计分析,东南大学博士论文
[51]Fox,A.J. Ourliers in time series.J.Roy.Statist.Soc.B 1972,48,39-47.
[52]Abraham,B.and Box,G.E. Bayesian analysis of some outlier Problems in time serises.Biometrika 1979,66,229-236.
[53]Abraham, B.and Chuang, A. outlier detection and time series modeling. Technomerics 1989,31,241-248.
[54]Abraham, B. and Yatawara, N. A score test for detection of time series outliers. J.Time series analy.1988,9,109-119.
[55]Bruce, G and Martin, R. D.:Leave-k-out diagnostics for time series. J.Roy. Statist.Soc. B51,363-424.
[56]Tiao, G. C. Autoregressive moving average models, intervention problems and outlier detection in time series. Handbook of Statist.1985,5,85-118.
[57]Ferdousi&Maeda Unsupervised Fraud Detection in Time Series data.
[58]陈萍,杨孝平,Cox-Ingersoll-Ross模型的统计推断,应用概率统计,Vol.21,2005
[59]张玉桂,苏云鹏,杨宝成,基于Vasicek和CIR模型的SHIBOR期限结构实证分析,统计与信息论坛Vol.24,No.6,2009
[60]韦博成.林金官.解锋昌.统计诊断.第一版.北京：高等教育出版社.2009
[61]Egorov, A., H. Li and Y. Xu, Maximum likelihood estimation of time-inhomogeneous diffusions, Journal of Econometrics,2003,114,107-139.
[62]Ait-Sahalia, Y., Closed-form likelihood expansions for multivariate diffusion, Annals of Statistics,2008,36,906-937.
[63]Ait-Sahalia, Y. and R. Kimmel, Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics,2007,83,413-452.
[64]谢赤,吴雄伟.基于Vasicek和CIR模型的中国货币市场利率行为实证分析.中国管理科学.2002.
[65]Kytoyants,Yu.A. Parameter Estimation for Stochastic Process,Heldermann-Verlag,berlin,1984.
[66]Weisberg,S. Some principles for regression diagnostics and influence analysis.Technometrics 1983,25,240-244.
[67]Steven E.Shreve. Stochastic Calculus for Finance.Vol.2 2004.北京：世界图书出版社.
[2]Fama.Efficient Capital Market:A Review of Theory and Empirical Work.Journal of Finance, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York,N.Y.1970:383-417.
[3]Black F.,Scholes M. The pricing of options corporate liabilities[J].Journal of Plitical Economy,1973,81 (3):637-659.
[4]Merton,Robert C. An Intertemporal Capital Asset Pricing Model. Econometrica 1973 41:5,pp.867-87.
[5]Vasicek, O. An equilibrium characterization of the term structure. J. Finan. Econom.1977 5,177-188.
[6]Cox, J. C., Ingersoll, J. E. and Ross, S. A. A theory of the term structure of interest rates. Econometrica 1985 53,385-467.
[7]Cox, J. C., Ingersoll, J. E. and Ross, S. A. An analysis of variable rate loan contracts. J. Finance 1980 35,389-403.
[8]Chan, K. C., Karolyi, A. G., Longstaff, F. A. and Sanders, A. B. An empirical comparison of alternative models of the short-term interest rate. J.1992
[9]Ait-Sahalia Testing Continuous-Time Models of the Spot Interest Rate, Review of Financial Studies,1996,9,385-426
[10]Stanton A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk", Journal of Finance 1997,52,1973-2002,
[11]Hong, Y. and H. Li, Nonparametric specification testing for continuous-time models with applications to interest rate term structures, The Review of Financial Studies,2005,18, 37-84.
[12]Kim M.S.Consistent estimation in regression models for the drift function in some continuous time models, Computational Statistics& Data Analysis Volume 52, Issue 5,20 January 2008,2682-2691
[13]I.V. Basawa, B.L.S. Prakasa Rao, Statistical Inference for Stochastic Processes, AcademicPress, London.1980
[14]Dacunha-Castelle and Florens-Zmirou, Estimation of the coefficients of a diffusion from discrete observations, Stochastics1986,19, pp.263-284.
[15]Duffie, D. and K.J. Singleton, Simulated moments estimation of Markov models of asset prices, Econometrica,1993,61,929-952.
[16]Hansen, Lars Peter and Jose Alexandre Scheinkman.1995. Back to the Future: GeneratingMoment Implications for Continuous-Time Markov Processes. Econometrica 63 (4):767-804.
[17]Gallant, A.R. and G. Tauchen, Which moments to match? Econometric Theory,1996,12, 657-681.
[18]Jacquier E., N.G. Polson and P.E. Rossi, Bayesian analysis of stochastic volatility models: reply, Journal of Business and Economic Statistics,1994,12,4,413-417.
[19]Eraker, B., Markov chain Monte Carlo analysis of diffusion models with application to finance, HAE Thesis, Norwegian School of Economics and Business Administration.1998
[20]Jones, C.S., Bayesian estimation of continuous-time finance models, Working Paper, Simon School of Business, University of Rochester.1998
[21]Ahn,D.H.and B.Gao,A parametric nonlinear model of term structure dynamics,Review of Financial Studies,1999,12,721-726.
[22]陈萍,随机波动率模型的统计推断及其衍生证券的定价,南京理工大学博士论文,2004.
[23]Hoffmann. M., Adaptive estimation in diffusion process, Stochastic processes and their applications,1999,79,135-163,
[24]Ait-Sahalia, Y. Nonparametric Pricing of Interest Rate Derivative Securities.Econometrica 1996a,64,527-560.
[25]Ait-Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusions:A closed-form approach, Econometrica,2002a,70,223-262.
[26]Jacod, J., Non-parametric Kernel Estimation of the Coefficient of a Diffusion, J. Scan.Stat.,2000,27(1),83-98
[27]Chen Ping, Wang Jinde, Wavelet estimation in time dependent diffusion models,science in China,2007.38 (6),1-16
[28]Pritsker,M., Nonparametric density estimation and tests of continuous time interest rate models, Review of Financial Studies,1998,11,449-487.
[29]Chapman, D. and N. Pearson, Is the short rate drift actually nonlinear? Journal of Finance, 2000,55,355-388.
[30]Fan, J. and C. Zhang, A re-examination of diffusion estimators with applications to financial model validation, Journal of the American Statistical Association,2003,98, 118-134.
[31]Cai, Z., J. Fan and Q. Yao, Functional-coefficient regression models for nonlinear time series, Journal of the American Statistical Association,2000,95,941-956.
[32]Fan, J., C. Zhang and J. Zhang, Generalized likelihood ratio statistics and Wilks phenomenon, The Annals of Statistics,2001,29,153-193.
[33]Chan, K.C., G.A. Karolyi, F.A. Longstaff and A.B. Sanders, An empirical comparison of alternative models of the short-term interest rate, Journal of Finance,1992,47,1209-1227.
[34]Conley, T.G., L.P. Hansen, E.G.J. Luttmer and J.A. Scheinkman, Short-term interest rates as subordinated diffusions, Review of Financial Studies,1997,10,525-577.
[35]Bliss, R.R. and D. Smith, The elasticity of interest rate volatility:Chan, Karolyi,Longstaff, and Sanders revisited, Journal of Risk,1998,1,21-46.
[36]Cai, Z. and Y. Hong, Nonparametric methods in continuous-time finance:A selectivereview. In Recent Advances and Trends in Nonparametric Statistics (M.G. Akritas and D.M. Politis, eds.) 2003,283-302.
[37]Cook, R. D. and Weisberg, S. Residuals and Influence in Regression. Chapman and Hall, New York.1982
[38]韦博成路过斌史建清统计诊断引论南京：东南大学出版社,1991.
[39]McCullagh, P. and Nelder, J. A.:Generalized Linear Models. Chapman and Hall, London.1983.
[40]Cook, R. D. Detection of influential observations in linear regression. Technometrics 19, 15-18.1977
[41]Cook, R. D. Assessment of local iofluence. J. R. Statist. Soc. B 48,133-169.1986.
[42]Belsley,D. A.,Kuh,E. and Welsch, R. E. Regression Diagnostics. John Wiley,New York,1980
[43]Simonoff, J. S. and Tsai, C. L.:Jackknife-based estimators and confidence regions in nonlinear regression. Tech nometrics 28,1986,103-112..
[44]Sber, G. A. F. and Wild, C. J.:Nonliear Regression. John Wiley, New York.1989.
[45]Pregibon, D.:Logistic regression diagnostics. Ann. Statist.1981,9,705-724.
[46]Thomas, W. and Cook, R. D.:Assessing influence on regression coefficients in generalized linear models. Biometrika 1989,76,741-749.
[47]Thomas, W. and Cook, R. D.:Assessing influence on predictions from generalized linear models. Technometrics 1990,32,59-65.
[48]Bechman, R. J., Nachtscheim, C. J. and Cook, R. D.:Diagnostics for mixed-model analysis of variance.Technometrics 1987,29,413-426.
[49]Hodges J.S. Some algebra and geometry for hierarchical models, applied to diagnostics (with discussion). J. Roy. Stat. Soc., Series B 1998,60:497-536.
[50]宗序平,系统分析中非线性随机效应模型的统计分析,东南大学博士论文
[51]Fox,A.J. Ourliers in time series.J.Roy.Statist.Soc.B 1972,48,39-47.
[52]Abraham,B.and Box,G.E. Bayesian analysis of some outlier Problems in time serises.Biometrika 1979,66,229-236.
[53]Abraham, B.and Chuang, A. outlier detection and time series modeling. Technomerics 1989,31,241-248.
[54]Abraham, B. and Yatawara, N. A score test for detection of time series outliers. J.Time series analy.1988,9,109-119.
[55]Bruce, G and Martin, R. D.:Leave-k-out diagnostics for time series. J.Roy. Statist.Soc. B51,363-424.
[56]Tiao, G. C. Autoregressive moving average models, intervention problems and outlier detection in time series. Handbook of Statist.1985,5,85-118.
[57]Ferdousi&Maeda Unsupervised Fraud Detection in Time Series data.
[58]陈萍,杨孝平,Cox-Ingersoll-Ross模型的统计推断,应用概率统计,Vol.21,2005
[59]张玉桂,苏云鹏,杨宝成,基于Vasicek和CIR模型的SHIBOR期限结构实证分析,统计与信息论坛Vol.24,No.6,2009
[60]韦博成.林金官.解锋昌.统计诊断.第一版.北京：高等教育出版社.2009
[61]Egorov, A., H. Li and Y. Xu, Maximum likelihood estimation of time-inhomogeneous diffusions, Journal of Econometrics,2003,114,107-139.
[62]Ait-Sahalia, Y., Closed-form likelihood expansions for multivariate diffusion, Annals of Statistics,2008,36,906-937.
[63]Ait-Sahalia, Y. and R. Kimmel, Maximum likelihood estimation of stochastic volatility models, Journal of Financial Economics,2007,83,413-452.
[64]谢赤,吴雄伟.基于Vasicek和CIR模型的中国货币市场利率行为实证分析.中国管理科学.2002.
[65]Kytoyants,Yu.A. Parameter Estimation for Stochastic Process,Heldermann-Verlag,berlin,1984.
[66]Weisberg,S. Some principles for regression diagnostics and influence analysis.Technometrics 1983,25,240-244.
[67]Steven E.Shreve. Stochastic Calculus for Finance.Vol.2 2004.北京：世界图书出版社.