加权复合分位数回归方法在动态VaR风险度量中的应用
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摘要
风险价值(VaR)因为简单直观,成为了当今国际上最主流的风险度量方法之一,而基于时间序列自回归(AR)模型来计算无条件风险度量值在实业界有广泛应用。本文基于分位数回归理论对AR模型提出了一个估计方法——加权复合分位数回归(WCQR)估计,该方法可以充分利用多个分位数信息提高参数估计的效率,并且对于不同的分位数回归赋予不同的权重,使得估计更加有效,文中给出了该估计的渐近正态性质。有限样本的数值模拟表明,当残差服从非正态分布时,WCQR估计的的统计性质接近于极大似然估计,而该估计是不需要知道残差分布的,因此,所提出的WCQR估计更加具有竞争力。此方法在预测资产收益的VaR动态风险时有较好的应用,我们将所提出的理论分析了我国九只封闭式基金,实证分析发现,结合WCQR方法求得的VaR风险与用非参数方法求得的VaR风险非常接近,而结合WCQR方法可以计算动态的VaR风险值和预测资产收益的VaR风险值。
Value at Risk(VaR)is one of the most popular methods of measuring financial asset risk in the current finance industry.Calculating unconditional risk measure values based on autoregressive models(AR)is widely used in industries.Further investigation of this problem is valuable both in theory and application.In this paper,an estimating approach for autoregressive models based on weighted composite quantile regression(WCQR)is proposed.This new approach can make full use of information from several quantiles to improve the efficiency of parameter estimators,and given different weights for different quantiles regression,which result in making the estimation more efficient.The proposed estimator is shown to have asymptotic normality.Finite sample studies illustrate that when the error follows a non-normal distribution,the statistical properties of WCQR estimators are similar to those of maximum likelihood estimators.It implies that the proposed estimator is strongly competent to the existing estimators because the error is free of a specific distribution.The proposed method has a good application in the calculation of dynamic VaR.The empirical results via analyzing nine Chinese closed funds show that the VaR values based on WCQR method is similar to those based on the non-parametric method.In addition,one pronounced advantage based on the proposed WCQR estimation is that it can be used to calculate and forecast the values of dynamic VaR for asset returns.
Value at Risk(VaR)is one of the most popular methods of measuring financial asset risk in the current finance industry.Calculating unconditional risk measure values based on autoregressive models(AR)is widely used in industries.Further investigation of this problem is valuable both in theory and application.In this paper,an estimating approach for autoregressive models based on weighted composite quantile regression(WCQR)is proposed.This new approach can make full use of information from several quantiles to improve the efficiency of parameter estimators,and given different weights for different quantiles regression,which result in making the estimation more efficient.The proposed estimator is shown to have asymptotic normality.Finite sample studies illustrate that when the error follows a non-normal distribution,the statistical properties of WCQR estimators are similar to those of maximum likelihood estimators.It implies that the proposed estimator is strongly competent to the existing estimators because the error is free of a specific distribution.The proposed method has a good application in the calculation of dynamic VaR.The empirical results via analyzing nine Chinese closed funds show that the VaR values based on WCQR method is similar to those based on the non-parametric method.In addition,one pronounced advantage based on the proposed WCQR estimation is that it can be used to calculate and forecast the values of dynamic VaR for asset returns.
引文
[1]王顺.金融风险管理[M].北京:中国金融出版社,2007.
[2]Gouriéroux C,Scaillet O,Laurent J.Sensitivity analysis of values at risk[J].Journal of Empirical Finance,2000,7(3):225-245.
[3]Dowd K.Estimating VaR with order statistics[J].The Journal of Derivatives,2001,8(3):23-30.
[4]Cabedo D,Moya I.Estimationg oil price“value at risk”using the historical simulation approach[J].Energy Economics,2003,25(3):239-253.
[5]Kibzun A,Kuznetsov E.Analysis of criteria VaR and CVaR[J].Journal of Banking and Finance,2006,30(2):779-796.
[6]牛昂.银行风险管理的新方法[J].国际金融研究,1997,4(6):61-65.
[7]胡海鹏,方兆本.投资组合VaR及其分解[J].中国管理科学,2003,11(3):1-5.
[8]陈学华,杨辉耀.VaR—APARCH模型与证券投资风险量化分析[J].中国管理科学,2003,11(1):22-27.
[9]姚京,李仲飞.基于VaR的金融资产配置模型[J].中国管理科学,2004,12(1):8-14.
[10]刘晓倩,周勇.金融风险管理中VaR度量的光滑估计效率[J].应用数学学报,2011,34(4):752-768.
[11]王艺馨,周勇.极端情况下对我国股市风险的实证研究[J].中国管理科学,2012,20(3):20-27.
[12]Carrasco M,Chen Xiaohong.Mixing and moment properties of various GARCH and stochastic volatility models[J].Econometrics Theory,2002,18(1):17-39.
[13]Genon-Catalot V,Jeantheau T,Larédo C.Stochastic volatility models as Hidden Markov models and statistical applications[J].Bernoulli,2000,6(6):1051-1079.
[14]刘晓倩.金融风险度量的VaR与ES光滑估计及比较[D].上海财经大学硕士学位论文,2010.
[15]Brockwell P,Davis A.Introduction to time series and forecasting,2nd Edition[M].New York:SpringerVerlag,2002.
[16]Cryer J,Chan K.Time Series Analysis with Applications in R,2nd Edition[M].New York:Springer-Verlag,2008.
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[19]Koenker R,Bassett G.Regression quantiles[J].Econometrica,1978,46(1):33-50.
[20]Koenker R,d'Orey V.Computing regression quantiles[J].Journal of the Royal Statistical Scociety.Series C(Applied Statistics),1987,36(3):383-393.
[21]Koenker R,Ng P,Portnoy S.Quantile smoothing splines[J].Biometrika,1994,81(4):673-680.
[22]Koenker R,Hallock K.Quantile regression:An introduction[J].Journal of Economic Perspectives,2001,15(4):43-56.
[23]Koenker R.Quantile Regression[M].London:Cambridge University Press,2005.
[24]Davis R,Knight K,Liu Jian.M-estimation for autoregressions with infinite variance[J].Stochastic Processes and Their Applications,1992,40(1):145-180.
[25]Koenker R,Zhao Quan Shui.Conditional quantile estimation and inference for ARCH models[J].Econometric Theory,1996,12(5):793-813.
[26]Davis R,Dunsmuir W.Least absolute deviation estimation for regression with ARMA errors[J].Journal of Theoretical Probability,1997,10(2):481-497.
[27]Breidt F,Davis R,Trindade A.Least absolute deviation estimation for all-pass time series models[J].The Annals of Statistics,2001,29(4):919-946.
[28]Koenker R,Xiao Zhijie.Quantile autoregression[J].Journal of the American Statistical Association,2006,101(475):980-990.
[29]Taylor J.A quantile regression approach to estimating the distribution of multiperiod returns[J].Journal of Derivatives,1999,7(1):64-78.
[30]Basset G,Chen H.Quantile style:Return-based attribution using regression quantiles[J].Empirical Economics,2001,26(1):293-305.
[31]关静,史道济.分位数回归与上证综指VaR研究[J].统计与信息论坛,2008,23(12):15-19.
[32]王新宇,赵绍娟.基于分位数回归模型的沪深股市风险测量研究[J].中国矿业大学,2008,37(3):416-421.
[33]陈星.期货市场量价关系:基于分位数回归模型的实证研究[J].南方经济,2009,(7):50-60.
[34]叶五一,陈杰成,缪柏其.基于虚拟变量分位点回归模型的条件VaR估计以及杠杆效应分析[J].中国管理科学,2010,18(4):1-7.
[35]罗玉波.房价影响因素分析:分位数回归方法[J].统计与决策,2011,(6):158-159.
[36]陈守东,王妍.我国金融机构的系统性金融风险评估——基于极端分位数回归技术的风险度量[J].中国管理科学,2014,22(7):10-17.
[37]Zou Hui,Yuan Ming.Composite quantile regression and the oracle model selection theory[J].The Annals of Statistics,2008,36(3):1108-1126.
[38]Jiang XueJun,Jiang Jiancheng,Song Xinyuan.Oracle model selection for nonlinear models based on weighted composite quantile regression[J].Statistica Sinica,2012,22(4):1479-1506.
[39]Jiang Xuejun,Jiang Jiancheng,Song Xinyuan.Weighted composite quantile regression of DTARCH models[J].The Econometrics Journal,2014,17(1):1-23.
[2]Gouriéroux C,Scaillet O,Laurent J.Sensitivity analysis of values at risk[J].Journal of Empirical Finance,2000,7(3):225-245.
[3]Dowd K.Estimating VaR with order statistics[J].The Journal of Derivatives,2001,8(3):23-30.
[4]Cabedo D,Moya I.Estimationg oil price“value at risk”using the historical simulation approach[J].Energy Economics,2003,25(3):239-253.
[5]Kibzun A,Kuznetsov E.Analysis of criteria VaR and CVaR[J].Journal of Banking and Finance,2006,30(2):779-796.
[6]牛昂.银行风险管理的新方法[J].国际金融研究,1997,4(6):61-65.
[7]胡海鹏,方兆本.投资组合VaR及其分解[J].中国管理科学,2003,11(3):1-5.
[8]陈学华,杨辉耀.VaR—APARCH模型与证券投资风险量化分析[J].中国管理科学,2003,11(1):22-27.
[9]姚京,李仲飞.基于VaR的金融资产配置模型[J].中国管理科学,2004,12(1):8-14.
[10]刘晓倩,周勇.金融风险管理中VaR度量的光滑估计效率[J].应用数学学报,2011,34(4):752-768.
[11]王艺馨,周勇.极端情况下对我国股市风险的实证研究[J].中国管理科学,2012,20(3):20-27.
[12]Carrasco M,Chen Xiaohong.Mixing and moment properties of various GARCH and stochastic volatility models[J].Econometrics Theory,2002,18(1):17-39.
[13]Genon-Catalot V,Jeantheau T,Larédo C.Stochastic volatility models as Hidden Markov models and statistical applications[J].Bernoulli,2000,6(6):1051-1079.
[14]刘晓倩.金融风险度量的VaR与ES光滑估计及比较[D].上海财经大学硕士学位论文,2010.
[15]Brockwell P,Davis A.Introduction to time series and forecasting,2nd Edition[M].New York:SpringerVerlag,2002.
[16]Cryer J,Chan K.Time Series Analysis with Applications in R,2nd Edition[M].New York:Springer-Verlag,2008.
[17]Burg J.A new analysis technique for time series data[R].Working Paper NATO Advanced Study Institute on Signal Processing with Emphasis on Underwater Acoustics,1968.
[18]Brockwell P,Dahlhaus R,Trindade A.Modified burg algorithms for multivariate subset autoregression[J].Statistica Sinica,2005,15(1):197-213.
[19]Koenker R,Bassett G.Regression quantiles[J].Econometrica,1978,46(1):33-50.
[20]Koenker R,d'Orey V.Computing regression quantiles[J].Journal of the Royal Statistical Scociety.Series C(Applied Statistics),1987,36(3):383-393.
[21]Koenker R,Ng P,Portnoy S.Quantile smoothing splines[J].Biometrika,1994,81(4):673-680.
[22]Koenker R,Hallock K.Quantile regression:An introduction[J].Journal of Economic Perspectives,2001,15(4):43-56.
[23]Koenker R.Quantile Regression[M].London:Cambridge University Press,2005.
[24]Davis R,Knight K,Liu Jian.M-estimation for autoregressions with infinite variance[J].Stochastic Processes and Their Applications,1992,40(1):145-180.
[25]Koenker R,Zhao Quan Shui.Conditional quantile estimation and inference for ARCH models[J].Econometric Theory,1996,12(5):793-813.
[26]Davis R,Dunsmuir W.Least absolute deviation estimation for regression with ARMA errors[J].Journal of Theoretical Probability,1997,10(2):481-497.
[27]Breidt F,Davis R,Trindade A.Least absolute deviation estimation for all-pass time series models[J].The Annals of Statistics,2001,29(4):919-946.
[28]Koenker R,Xiao Zhijie.Quantile autoregression[J].Journal of the American Statistical Association,2006,101(475):980-990.
[29]Taylor J.A quantile regression approach to estimating the distribution of multiperiod returns[J].Journal of Derivatives,1999,7(1):64-78.
[30]Basset G,Chen H.Quantile style:Return-based attribution using regression quantiles[J].Empirical Economics,2001,26(1):293-305.
[31]关静,史道济.分位数回归与上证综指VaR研究[J].统计与信息论坛,2008,23(12):15-19.
[32]王新宇,赵绍娟.基于分位数回归模型的沪深股市风险测量研究[J].中国矿业大学,2008,37(3):416-421.
[33]陈星.期货市场量价关系:基于分位数回归模型的实证研究[J].南方经济,2009,(7):50-60.
[34]叶五一,陈杰成,缪柏其.基于虚拟变量分位点回归模型的条件VaR估计以及杠杆效应分析[J].中国管理科学,2010,18(4):1-7.
[35]罗玉波.房价影响因素分析:分位数回归方法[J].统计与决策,2011,(6):158-159.
[36]陈守东,王妍.我国金融机构的系统性金融风险评估——基于极端分位数回归技术的风险度量[J].中国管理科学,2014,22(7):10-17.
[37]Zou Hui,Yuan Ming.Composite quantile regression and the oracle model selection theory[J].The Annals of Statistics,2008,36(3):1108-1126.
[38]Jiang XueJun,Jiang Jiancheng,Song Xinyuan.Oracle model selection for nonlinear models based on weighted composite quantile regression[J].Statistica Sinica,2012,22(4):1479-1506.
[39]Jiang Xuejun,Jiang Jiancheng,Song Xinyuan.Weighted composite quantile regression of DTARCH models[J].The Econometrics Journal,2014,17(1):1-23.