无限方差的GARCH模型的WLAD估计及渐近性质
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摘要
在长期的实证研究中,人们发现诸如股票价格等金融时间序列通常表现出聚集性(volatility clustering)特征.所谓聚集性是指金融市场大幅波动往往伴随着另一次大的波动.因此如何准确刻画市场波动的这种异方差(Heteoskedastic)特征,并对未来市场的波动做出尽量准确的预测,对于金融学的理论研究以及金融监管政策的制定都具有极其重要的理论意义和现实意义.而通常我们为了解决问题的方便,往往假设序列服从同方差的正态分布,这恰好与实际情况相违背.在这种与实际情况出入较大的假设下,采用的参数估计方法、检验方法以及模型的选择上都将会出现较大的偏差,从而导致结论误差大甚至错误.同时,金融时间序列还表现出“重尾”现象,这种现象不可忽视.在波动性研究领域中,Engle和Tim Bollerslev先后提出的ARCH(自回归条件异方差)和GARCH(广义自回归条件异方差)模型,由于能够对金融时间序列的“尖峰厚尾”及。异方差。进行较为成功的刻画、计量,因此受到了许多金融学者的青睐.但是,Mikosch Starica发现残差服从正态分布的GARCH(1,1)模型的尾部比实际数据中的薄.因此,怎样在异方差、重尾的假设下建立一个恰当的模型及给出合理的估计方法成为当今理论界和实业界研究的热点.本文侧重于解决具有较厚尾部的异方差模型的参数估计问题.
本篇论文系统地阐述了重尾分布和GARCH模型的严平稳性,回顾了GARCH模型参数估计的历史进程.总结了GARCH模型在残差服从不同分布的假设下参数估计方法的选择.详细讨论了当时间序列呈现重尾性时,GARCH模型的准极大似然估计(QMLE)、最小一乘估计(LADE)方法,提出对LADE方法进行改进,称为带有权重的一乘估计(WLADE).用Monte-Carlo模拟一组具有重尾、聚集特征的数据,分别用QMLE、LADE、WLADE三种估计方法进行参数估计,并建立GARCH模型,比较它们的优良性.随后对上海和深圳两市股指数据进行了实证分析.计算结果表明,上海和深圳股指收益率具有重尾性、异方差性.与此同时,采用WLADE建立了GARCH模型,并对波动率给出了预测,从而给VAR值的计算提供了新的方法.实证结果表明,用本文提出的方法计算出的VAR值比常用的QMLE方法计算出的VAR值准确.
论文主要结论:
从整体上看,在具有重尾、聚集性的金融时间序列当中,应用WLADE方法建立GARCH模型,能得到令人满意的结果,WLADE方法优于我们常用的QMLE方法和LADE方法,WLADE方法在尾部数据较厚时估计精确性较高一些,而且受异常值的影响小.
In the long-term empirical study,it is found that as stock prices financial time series is usually gathered(volatility clustering) features.The so-called clustering of the financial markets is often accompanied by large fluctuations in another great fluctuatious. So how accurate characterization of this market volatility heteroskedasticity (Heteoskedastic) features,and future market fluctuations as much as possible to make accurate predictions,the theory of financial research and financial supervision policy formulation has an extremely important theoretical significance and the practical significance.Usually the hypothesis that the sequence is subject to normal distribution, breached the reality for the convenience of solution the problem.When it is greater difference with practice under this assumption,the parameter estimation methods, testing methods and the choice of models will have a larger deviations.Thus it will lead to the conclusion error even wrong.At the same time,financial time series also displayed the "heavy-tailed" phenomenon.This phenomenon should not be ignored.In the field of fluctuations research,Eagle and Tim Bollerslev has put the ARCH(autoregressive conditional heteroskedasticity) and GARCH(generalized autoregressive Conditional heteroskedasticity) model.Becanse of the financial time series to the "heavy-tailed" and "heteroscedasticity" more successful characterization,measurement, so were many financial scholars favour.However,Mikosch Starica found that when the error is subject to normal,the tail of GARCH(1,1) model of is thinner than the actual data.Therefore,how to construct appropriate model and reasonable estimation method under heteroscedasticity and heavy-tailed,for today it is center of theoretical circles and the business community.This paper focused on parameter estimation problem under heavy-tailed and Heteroscedasticity in the model.
This paper systematically explain heavy-tailed distribution model and the strict stationary ergodic GARCG,and recalled the GARCH model parameters estimated historical process.It summed up parameter estimation method of the GARCH model with residual subject to different assumptions.When the time series show heavy tail, GARCH model discusses in detail quasi maximum likelihood estimation(QMLE),and least absolute deviations estimation(LADE),and improves the LADE,as weighed-least absolute deviations estimation(WLADE).Monte-Carlo simulates a group of heavytailed, heteroskedasticity of the data,and respectively QMLE,LADE,WLADE three estimation method estimate and compare construction of GARCH model.Subsequently the two cities of Shanghai and Shenzhen stock index data is empirical analysised. Calculated results show that,at the Shanghai and Shenzhen stock index benefits of a heavy tail,heteroscedasticity.At the same time,GARCH model is established by WLADE.volatility is forecasted and given to the calculation of VAR.The empirical results show that,the value of VAR is more precise by using of WLADE.
Main conclusion of this paper as follows:
Overall,in a heavy tail,aggregation of financial time series,to the application WLADE to establish GARCH model can be a satisfactory outcome,WLADE method is better than LADE,QMLE methods.When the tailed data is more thick,the WLADE method will be more accurate and the impact based on anomalies will be weak.
本篇论文系统地阐述了重尾分布和GARCH模型的严平稳性,回顾了GARCH模型参数估计的历史进程.总结了GARCH模型在残差服从不同分布的假设下参数估计方法的选择.详细讨论了当时间序列呈现重尾性时,GARCH模型的准极大似然估计(QMLE)、最小一乘估计(LADE)方法,提出对LADE方法进行改进,称为带有权重的一乘估计(WLADE).用Monte-Carlo模拟一组具有重尾、聚集特征的数据,分别用QMLE、LADE、WLADE三种估计方法进行参数估计,并建立GARCH模型,比较它们的优良性.随后对上海和深圳两市股指数据进行了实证分析.计算结果表明,上海和深圳股指收益率具有重尾性、异方差性.与此同时,采用WLADE建立了GARCH模型,并对波动率给出了预测,从而给VAR值的计算提供了新的方法.实证结果表明,用本文提出的方法计算出的VAR值比常用的QMLE方法计算出的VAR值准确.
论文主要结论:
从整体上看,在具有重尾、聚集性的金融时间序列当中,应用WLADE方法建立GARCH模型,能得到令人满意的结果,WLADE方法优于我们常用的QMLE方法和LADE方法,WLADE方法在尾部数据较厚时估计精确性较高一些,而且受异常值的影响小.
In the long-term empirical study,it is found that as stock prices financial time series is usually gathered(volatility clustering) features.The so-called clustering of the financial markets is often accompanied by large fluctuations in another great fluctuatious. So how accurate characterization of this market volatility heteroskedasticity (Heteoskedastic) features,and future market fluctuations as much as possible to make accurate predictions,the theory of financial research and financial supervision policy formulation has an extremely important theoretical significance and the practical significance.Usually the hypothesis that the sequence is subject to normal distribution, breached the reality for the convenience of solution the problem.When it is greater difference with practice under this assumption,the parameter estimation methods, testing methods and the choice of models will have a larger deviations.Thus it will lead to the conclusion error even wrong.At the same time,financial time series also displayed the "heavy-tailed" phenomenon.This phenomenon should not be ignored.In the field of fluctuations research,Eagle and Tim Bollerslev has put the ARCH(autoregressive conditional heteroskedasticity) and GARCH(generalized autoregressive Conditional heteroskedasticity) model.Becanse of the financial time series to the "heavy-tailed" and "heteroscedasticity" more successful characterization,measurement, so were many financial scholars favour.However,Mikosch Starica found that when the error is subject to normal,the tail of GARCH(1,1) model of is thinner than the actual data.Therefore,how to construct appropriate model and reasonable estimation method under heteroscedasticity and heavy-tailed,for today it is center of theoretical circles and the business community.This paper focused on parameter estimation problem under heavy-tailed and Heteroscedasticity in the model.
This paper systematically explain heavy-tailed distribution model and the strict stationary ergodic GARCG,and recalled the GARCH model parameters estimated historical process.It summed up parameter estimation method of the GARCH model with residual subject to different assumptions.When the time series show heavy tail, GARCH model discusses in detail quasi maximum likelihood estimation(QMLE),and least absolute deviations estimation(LADE),and improves the LADE,as weighed-least absolute deviations estimation(WLADE).Monte-Carlo simulates a group of heavytailed, heteroskedasticity of the data,and respectively QMLE,LADE,WLADE three estimation method estimate and compare construction of GARCH model.Subsequently the two cities of Shanghai and Shenzhen stock index data is empirical analysised. Calculated results show that,at the Shanghai and Shenzhen stock index benefits of a heavy tail,heteroscedasticity.At the same time,GARCH model is established by WLADE.volatility is forecasted and given to the calculation of VAR.The empirical results show that,the value of VAR is more precise by using of WLADE.
Main conclusion of this paper as follows:
Overall,in a heavy tail,aggregation of financial time series,to the application WLADE to establish GARCH model can be a satisfactory outcome,WLADE method is better than LADE,QMLE methods.When the tailed data is more thick,the WLADE method will be more accurate and the impact based on anomalies will be weak.
引文
[1]Benoit Mandelbort.The variation of certain speculative prices.Journal of Business,1963,ⅩⅩⅩⅥ,394-419.
[2]EugeneF.Fama.The behavior of stock-market prices.The Journal of Business,1965,Vol.38,No.1.,34-105.
[3]Engle,R.F..Autoregressive conditional heteroscedasticity with estimates of the variance of U.K.inflation.Econometrica,1982,50,987-1008.
[4]Bollerslev.T..Generalized autoregressive conditional heteroscedasticity.Economet,1986,31,307-327.
[5]岳朝龙.上海股市收益率GARCH模型族的实证研究.数量经济技术经济研究,2001,06,126-129.
[6]徐绪松,马莉莉,陈彦斌.Empirical study on GARCH effect in Shanghai stock market.武汉大学学报(理学版),2002,03,42-45.
[7]李亚静,朱宏泉,彭育威.基于GARCH模型族的中国股市波动性预测.数学的实践与认识,2003,11,65-71.
[8]陈敏,王国明,吴国富,蒋学雷.中国证券市场的ACD-GARCH模型及其应用.统计研究,2003,11,60-62.
[9]皮天雷.我国沪市波动聚集性GARCH效应的研究.管理科学,2003,06,31-35.
[10]王树娟,黄渝祥.基于GARCH-CVaR模型的我国股票市场风险分析.同济大学学报(自然科学版),2005,02,121-124.
[11]Mittnik,S.,Rachev,S.T.and Paolella,M.S..Stable paretian modeling in finance:some empirical and theoretical aspects.In A Practical Guide to Heavy Tail.(R.J.Adler,R.E.Feldman and M.S.Taqqu,ed.),Boston:Birkhauser,79-110.
[12]Shephed,N..Statistical aspects of ARCH and stochastic volatility.In Time Series Models in Econometrics,Finance and Other Fields.(D.R.Cox,D.V.Himnkley,and O.E.Barndorff-Nielsen,eds.).Chapman and Hall,London,1996,1-67.
[13]Rydberg,T..Realistic statistical modeling of financial data.International Statistical Review,2000,68,233-258.
[14]Straumann,D.and Mikosch,T..Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series:A stochastic recurrence equations approach.Ann.Statist,2006,Vol.34,No 5,2449-2495.
[15]Hall,P.and Yao,Q..Inference in ARCH and GARCH models with heavy-tailed error.Econometrica,2003,71,285-317.
[16]Giraitis,L.and Robinson,P.M..Whittle estimation of ARCH models.Econometric Theory,2001,17,608-623.
[17]Mikosch,T.and Straumann,D..Whittle estimation in a heavy-tailed GARCH(1,1)model.Technical Report,2000.
[18]Peng,L.and Yao,Q..Least absolute deviations estimation for ARCH and GAKCH model.Biometrika,2003,90,4,967-975.
[19]Davis,R.A.,Knight,K.and Liu,J..M-estimation for auto regressions with infinite variance.Stock Processes,1992,Appl.,40,145-180.
[20]Shiqing Ling.Self-weighted least absolute deviation estimation for infinite variance auto regressive models.J.R.Statist.Soc.B,2005,67,Part 3,381-393.
[21]Horvath,L.and F.Liese..L_1-estimators in ARCH models.Journal of Statistical Planning and Inference,2004,119,277-309.
[22]Jiazhu Pan,Hui Wang,Qiwei Yao.Weighted least absolute deviations estimation for ARMA models with infinite variance,http://stats.lse.ac.uk/q.yao/qyao.links/paper/pan.pdf.
[23]Ngai Hang Chan,Liang peng.Weighted least absolutions deviations estimation for an AR(1) process with ARCH(1) errors.Biometrika,2005,92,2,477-484.
[24]Jianqin Fan,Qiwei Yao.Nonlinear Time Series(Nonparametric and parametric methods).科技出版社.2006.
[25]Bougerol,P.and Picard,N..Strict stationarity of generalized autoressive processes.The Annals of Probability,1992,Vol.20,No.4,1714-1730.
[26]Mikosch,T.and Straumann,D..Stable limits of martingale thansforms with application to the estimation of GARCH parameters.The Annals of Statistics,2006,Vol.34,No.1,493-522:
[27]Basrak,B.,Dsvis,R.A.and Milosch,T.Regular variation of GARCH processes.Stochastic Processes and Their Applications,2002,Vol.99,No.1,95-115.
[28]Davis,R.A.and Dunsmuir,W.T.M..Least absolute deviation estimation for regression with ARMA errors.Theoretical Prob,1997,10,481-497.
[29]Knight,K..Limiting distributions for L_1 regression estimators under general conditions.Ann.Statist.,1998,26,755-770.
[30]朱世武.基于SAS系统的金融计算[M].清华大学出版社.2004.
[2]EugeneF.Fama.The behavior of stock-market prices.The Journal of Business,1965,Vol.38,No.1.,34-105.
[3]Engle,R.F..Autoregressive conditional heteroscedasticity with estimates of the variance of U.K.inflation.Econometrica,1982,50,987-1008.
[4]Bollerslev.T..Generalized autoregressive conditional heteroscedasticity.Economet,1986,31,307-327.
[5]岳朝龙.上海股市收益率GARCH模型族的实证研究.数量经济技术经济研究,2001,06,126-129.
[6]徐绪松,马莉莉,陈彦斌.Empirical study on GARCH effect in Shanghai stock market.武汉大学学报(理学版),2002,03,42-45.
[7]李亚静,朱宏泉,彭育威.基于GARCH模型族的中国股市波动性预测.数学的实践与认识,2003,11,65-71.
[8]陈敏,王国明,吴国富,蒋学雷.中国证券市场的ACD-GARCH模型及其应用.统计研究,2003,11,60-62.
[9]皮天雷.我国沪市波动聚集性GARCH效应的研究.管理科学,2003,06,31-35.
[10]王树娟,黄渝祥.基于GARCH-CVaR模型的我国股票市场风险分析.同济大学学报(自然科学版),2005,02,121-124.
[11]Mittnik,S.,Rachev,S.T.and Paolella,M.S..Stable paretian modeling in finance:some empirical and theoretical aspects.In A Practical Guide to Heavy Tail.(R.J.Adler,R.E.Feldman and M.S.Taqqu,ed.),Boston:Birkhauser,79-110.
[12]Shephed,N..Statistical aspects of ARCH and stochastic volatility.In Time Series Models in Econometrics,Finance and Other Fields.(D.R.Cox,D.V.Himnkley,and O.E.Barndorff-Nielsen,eds.).Chapman and Hall,London,1996,1-67.
[13]Rydberg,T..Realistic statistical modeling of financial data.International Statistical Review,2000,68,233-258.
[14]Straumann,D.and Mikosch,T..Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series:A stochastic recurrence equations approach.Ann.Statist,2006,Vol.34,No 5,2449-2495.
[15]Hall,P.and Yao,Q..Inference in ARCH and GARCH models with heavy-tailed error.Econometrica,2003,71,285-317.
[16]Giraitis,L.and Robinson,P.M..Whittle estimation of ARCH models.Econometric Theory,2001,17,608-623.
[17]Mikosch,T.and Straumann,D..Whittle estimation in a heavy-tailed GARCH(1,1)model.Technical Report,2000.
[18]Peng,L.and Yao,Q..Least absolute deviations estimation for ARCH and GAKCH model.Biometrika,2003,90,4,967-975.
[19]Davis,R.A.,Knight,K.and Liu,J..M-estimation for auto regressions with infinite variance.Stock Processes,1992,Appl.,40,145-180.
[20]Shiqing Ling.Self-weighted least absolute deviation estimation for infinite variance auto regressive models.J.R.Statist.Soc.B,2005,67,Part 3,381-393.
[21]Horvath,L.and F.Liese..L_1-estimators in ARCH models.Journal of Statistical Planning and Inference,2004,119,277-309.
[22]Jiazhu Pan,Hui Wang,Qiwei Yao.Weighted least absolute deviations estimation for ARMA models with infinite variance,http://stats.lse.ac.uk/q.yao/qyao.links/paper/pan.pdf.
[23]Ngai Hang Chan,Liang peng.Weighted least absolutions deviations estimation for an AR(1) process with ARCH(1) errors.Biometrika,2005,92,2,477-484.
[24]Jianqin Fan,Qiwei Yao.Nonlinear Time Series(Nonparametric and parametric methods).科技出版社.2006.
[25]Bougerol,P.and Picard,N..Strict stationarity of generalized autoressive processes.The Annals of Probability,1992,Vol.20,No.4,1714-1730.
[26]Mikosch,T.and Straumann,D..Stable limits of martingale thansforms with application to the estimation of GARCH parameters.The Annals of Statistics,2006,Vol.34,No.1,493-522:
[27]Basrak,B.,Dsvis,R.A.and Milosch,T.Regular variation of GARCH processes.Stochastic Processes and Their Applications,2002,Vol.99,No.1,95-115.
[28]Davis,R.A.and Dunsmuir,W.T.M..Least absolute deviation estimation for regression with ARMA errors.Theoretical Prob,1997,10,481-497.
[29]Knight,K..Limiting distributions for L_1 regression estimators under general conditions.Ann.Statist.,1998,26,755-770.
[30]朱世武.基于SAS系统的金融计算[M].清华大学出版社.2004.