时间序列在股指波动性建模中的应用
|
摘要
信息技术的快速发展和金融市场日趋全球化的倾向,导致新型金融产品大量涌现,极大地增加了金融投资和银行业务的复杂性,而区域性金融风暴和银行危机都表明,需要加强对金融产品稳定性的研究。对金融业进行谨慎的管制,除了需要发展新的更贴近现实的理论模型外,更迫切的任务是,需要以微观数据为基础的实证研究。金融资产价格的波动是金融体系风险积累的重要来源之一,几乎所有的金融危机都与金融资产波动有关。因而,对于波动性的定量计量建模是对金融资产波动性研究的核心内容之一,目前,各种统计建模的方法越来越多的应用于金融领域。大量的实证研究表明,金融数据中存在着波动集聚性和尖峰厚尾性,因此,用一般的时间序列模型来拟合金融数据的波动性显得不合适。GARCH模型是目前度量金融市场波动性的有力工具之一。参数GARCH模型是最常用的模型,但对于此模型的参数估计和相应的统计推断一直以来是个难题。传统的方法是基于极大似然估计(ML)的基础上运用最优化理论来对参数给出估计,然后导出参数的极限分布进行统计推断,但由于GARCH模型对其参数有一定的约束条件,使得此优化算法方法较为复杂且使得基于此方法而衍生出的进一步的统计检验,如:LM,Wald,LR,等方法的可靠性受到质疑。本文中,我们运用马尔科夫蒙特卡罗(MCMC)方法对残差基于正态分布的GARCH(1,1)模型进行估计,此方法通过构造收敛于待估计参数分布的随机过程而克服了运用最优化算法估计GARCH模型中参数以及对相应参数进行统计推断所产生的上述问题。实证分析结果表明:基于马尔科夫蒙特卡罗(MCMC)方法估计的GARCH模型比基于极大似然估计(ML)方法估计GARCH模型更灵活、统计推断的结论更可靠。最后,我们还给出了一个计算风险价值(VaR)的算例。
With the development of information technology and globalization of financial markets, the decision process of financial investment and banking operations are becoming more and more complex compared with previous ages. Meanwhile, regional financial crisis and bankruptcy of banks indicate that it is essential to strengthen the research on stability of financial product. The volatility of financial products is the core provoking risk factor of financial crisis for financial system, and almost all the crisis of finance are closely related with financial volatility. As a result, researches and studies on financial volatility have become one of important parts of financial econometrics. Nowadays, many statistical techniques have been used in these fields, and many researches show there are clustering effect and heavy-tail effect in financial data, which means that normal ARMA related models are not compatible with the financial data. GARCH model is a powerful tool for analyzing financial data, and the parametric GARCH models are the most commonly used models. However, the ways of estimation of parametric GARCH model’s coefficients are always hard problems. Traditional method is to use the ML-based method to estimate the parameters, then, do some appropriate statistical references and ML method is essentially an optimization method. But GARCH models typically have many constraints among parameters, which will result in the failure of trust of MLE results, and ML related statistical inferential methods, like: LM, Wald, LR, etc. This paper we use Markov Monte Carlo (MCMC) method to estimate the parameters of normal-based GARCH(1,1) model. The MCMC technique try to simulate several Markov chains to converge to the posterior distribution of parameters that we wish to estimate. Since MCMC method avoids the difficulties of constraints in optimization problems and statistical inferences about parameters avoid complicated asymptotic results , so the results based on MCMC are more reliable and we also show results based on MCMC are better than ones of ML based by using real financial data. Finally, we use our results to compute Value-at-risk (VaR) as an application of our research.
With the development of information technology and globalization of financial markets, the decision process of financial investment and banking operations are becoming more and more complex compared with previous ages. Meanwhile, regional financial crisis and bankruptcy of banks indicate that it is essential to strengthen the research on stability of financial product. The volatility of financial products is the core provoking risk factor of financial crisis for financial system, and almost all the crisis of finance are closely related with financial volatility. As a result, researches and studies on financial volatility have become one of important parts of financial econometrics. Nowadays, many statistical techniques have been used in these fields, and many researches show there are clustering effect and heavy-tail effect in financial data, which means that normal ARMA related models are not compatible with the financial data. GARCH model is a powerful tool for analyzing financial data, and the parametric GARCH models are the most commonly used models. However, the ways of estimation of parametric GARCH model’s coefficients are always hard problems. Traditional method is to use the ML-based method to estimate the parameters, then, do some appropriate statistical references and ML method is essentially an optimization method. But GARCH models typically have many constraints among parameters, which will result in the failure of trust of MLE results, and ML related statistical inferential methods, like: LM, Wald, LR, etc. This paper we use Markov Monte Carlo (MCMC) method to estimate the parameters of normal-based GARCH(1,1) model. The MCMC technique try to simulate several Markov chains to converge to the posterior distribution of parameters that we wish to estimate. Since MCMC method avoids the difficulties of constraints in optimization problems and statistical inferences about parameters avoid complicated asymptotic results , so the results based on MCMC are more reliable and we also show results based on MCMC are better than ones of ML based by using real financial data. Finally, we use our results to compute Value-at-risk (VaR) as an application of our research.
引文
[1] Baillie Richard T. and Tim Bollerslev, (1989), Common stochastic trends in a system of exchange rates, The Journal of Finance XLIV, 167-181.
[2] Black, F. ,1976, Studies of Stock Price Volatility Changes, Proceedings from the American Statistical Association[J], Business and Economic Statistics Section, 177-181.
[3] Bollerslev, T. , 1986, Generalized Autoregressive Conditional Heteroskedasticity[J], Journal of Econometrics, 31, 307-327.
[4] Bollerslev, T. , 1987, A conditionally Heteroskedasticity Time Series Model for Speculative Prices and Rates of Return[J], Review of Economics and Statistics, 69,542-547
[5] Bollerslev, T. ,R. Y. Chou and K.F.Kroner, 1992, ARCH modeling in Finance: A Review of the Theory and Empirical Evidence[J], Journal of Econometrics, 52, 5-59.
[6] Bollerslev,T., R.F.Engel, and D.N.Nelson, 1994, ARCH Models, in R.F.Engle and D.McFadden, eds., Handbook of Econometrics, Vol. IV., Amsterdam: North Holland, 2959-3038.
[7] Box, G.E.P, and Tiao, G.C.(1973), Bayesian Inference in Statistical Analysis. Addison-Wesley: Reading, MA.
[8] Campbell,J. and L.Hentschel,1992, No News is Good News: An Asymmetric Model of Changing Volatility in Stock Return[J], Journal of Financial Econometrics, 31, 281-318.
[9] Carlin,B.P, and Louis,T.A.(2000), Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed., Chapman and Hall:London.
[10] Christie, A.A. , 1982, The Stochastic Behavior of Common Stock Variances: Value, Leverage and Interest Rate Effects[J], Journal of Financial Economics,10, 407-432.
[11] Copeland, L, (1991), Cointegration test of daily foreign exchange data, Oxford Bulletin of Economics and Statistics 53, 185-198.
[12] Coleman, M, (1990), Cointegration-based test of daily foreign exchange marketefficiency, Economic Letters 32, 53-59.
[13] Corrrado,C. and T.Su, 1996, Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices[J], Journal of Financial Research, 19, 175-192.
[14] David Arida, 2008,Bayesain Estimation of a Markov-Switching Threshold Asymmetric GARCH Model with Student-t Innovations[J], Econometrics Journal,1-22.
[15] David Arida, 2006, Bayesain Estimation of the GARCH(1,1) Model with Normal Innovations[J], Student,5,283-298.
[16] Dickey, D.A. and Fuller, W.A, (1981), Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica 49, 1057-72.
[17] Engel,R.F., 1982, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation[J], Econometrica, 50, 987-1007.
[18] Engel,R.F. and V.K.Ng, 1993, Measuring and Testing the Impact of News on Volatility[J], Journal of Finance, 48, 1749-1778.
[19] Gelman,A.,Carlin,J.B.,Stern,H.S.,and Rubin,D.B.(1995), Bayesian Data Analysis, CRC Press:London.
[20] Hasting, W.K.(1970), Monte Carlo sampling methods using Markov chains and their applications, Biometrica, 57, 97-109
[21] Jonathan D.Cryer&Kung-Sik Chan, Time Series Analysis with Applications in R,Spriger,2008
[22] Johansen S. and K. Juselius, (1990), Maximum likelihood estimation and inference on cointegration-with application to Demand For Money, Oxford Bulletin of Economics and Statistics 52, 169-210.
[23] Karfakis, Costas I. and Ashok Parikh (1994), Exchange rate convergence and market efficiency, Applied Financial Economics 4, 93-98.
[24] Levich, R.M, (1989), Is the foreign exchange market efficient? Oxford Review of Economic Policy 5, 40-60.
[25] Ljung,L.G..M and G.E.P.Box, 1978, On a Measure of Lag Fit in Time Series Models[J], Biometrika, 67, 297-303.
[26] MacDonald, R. and M.P. Taylor (1989), Foreign exchange market efficiency and cointegration: some evidence from recent float, Economics Letters 29, 63-68.
[27] Masih A. M.M. and Masih R, (1994), On the robustness of cointegration tests of the market efficiency hypothesis: evidence from six European foreign exchange markets, Economia Internazionale XLVII, 160-180.
[28] McLeod,A.I.and W.K.Li(1983). Diagnostic checking ARMA time series models using squared residual sutocorrelations. Journal of Time Series Analysis,4,269-273
[29] Morelli. Simple Valuation Model and Growth Expections. Financial Analysis Journal, 1998(7):419-426
[30] Ruey S.Tsay.(2002),Analysis of Financial Time Series.John Wiely$Sons,Inc
[31] Sephton P.S. and Larsen H.K. (1991), Tests of exchange market efficiency: fragile evidence from cointegration tests, Journal of International Money and Finance, 10, 561-570.
[32] Sheather,S.J.and Jones,M.C.(1991). A reliable data-based bandwidth selection method for kernel estimation.Jouranl of the Royal Statistical Society,Series B,53,683-690
[33] Scott,D.W.(1979). On optimal and data-based histograms. Biometrika,66,605-610
[34] Teruo,Nakasuma,1998,A Markov-Chain Sampling Algorithm for GARCH models[J], Studies in Nonlinear Dynamics&Econometrics ,Berkeley Electronic Press.
[35] Teruo,Nakasuma,2000, Bayesian analysis of ARMA-GARCH models: A Markov chain sampling approach[J], Journal of Econometrics,95,57-69.
[36] Tierney,L.(1994), Markov chains for exploring posterior distributions, Annals of Statistics, 22, 1701-1762.
[37] Zakoian, J.M. (1994). Threshold heteroscedastic models, Journal of Economic Dynamics and Control 18, 931-955
[38] Fame. The Behavior of Stock-Market Price. Journal of Business,1965(7):38-41
[39] Osborne. Brownian Motion in the Stock Market. Operations Research, 1959(7): 69-74
[40]范龙振,胡畏.上海股票市场有效性实证检验.预测,2000(2):61-64
[41]高铁梅,计量经济分析方法与建模.2006
[42]胡朝霞.中国股市弱市有效性研究,1998(1):28-34
[43]胡海鹏,方兆本(2004),股市波动性预测模型改进研究[J],数理统计与管理,2004年9月,第23卷第5期,40-47.
[44]李雪松(2008),高级计量经济学,中国社会科学出版社
[45]李亚静等(2003),基于GARCH模型族的中国股市波动性预测[J],属性的实践与认识,2003年11月,第33卷第11期,65-71.
[46]李竹渝,鲁万波,龚金国(2005),经济、金融计量学中的非参数估计技术[R],社科基金解题报告[R],2005.
[47]陆懋祖(1999),高等时间序列计量学[R],上海人民出版社,1999.
[48]唐齐鸣等(2001),中国股市的ARCH效应分析[J],世界经济,2001年第3期,29-36
[49]吴长凤.利用回归GARCH模型对我国沪深股市的分析.预测,1999(4):46-47
[50]陈道富.当前股市走势分析及对策建议.金融与经济2008(5):41-44
[52]张世英、许启发、周红.金融时间序列分析.清华大学出版设:2008年1月
[2] Black, F. ,1976, Studies of Stock Price Volatility Changes, Proceedings from the American Statistical Association[J], Business and Economic Statistics Section, 177-181.
[3] Bollerslev, T. , 1986, Generalized Autoregressive Conditional Heteroskedasticity[J], Journal of Econometrics, 31, 307-327.
[4] Bollerslev, T. , 1987, A conditionally Heteroskedasticity Time Series Model for Speculative Prices and Rates of Return[J], Review of Economics and Statistics, 69,542-547
[5] Bollerslev, T. ,R. Y. Chou and K.F.Kroner, 1992, ARCH modeling in Finance: A Review of the Theory and Empirical Evidence[J], Journal of Econometrics, 52, 5-59.
[6] Bollerslev,T., R.F.Engel, and D.N.Nelson, 1994, ARCH Models, in R.F.Engle and D.McFadden, eds., Handbook of Econometrics, Vol. IV., Amsterdam: North Holland, 2959-3038.
[7] Box, G.E.P, and Tiao, G.C.(1973), Bayesian Inference in Statistical Analysis. Addison-Wesley: Reading, MA.
[8] Campbell,J. and L.Hentschel,1992, No News is Good News: An Asymmetric Model of Changing Volatility in Stock Return[J], Journal of Financial Econometrics, 31, 281-318.
[9] Carlin,B.P, and Louis,T.A.(2000), Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed., Chapman and Hall:London.
[10] Christie, A.A. , 1982, The Stochastic Behavior of Common Stock Variances: Value, Leverage and Interest Rate Effects[J], Journal of Financial Economics,10, 407-432.
[11] Copeland, L, (1991), Cointegration test of daily foreign exchange data, Oxford Bulletin of Economics and Statistics 53, 185-198.
[12] Coleman, M, (1990), Cointegration-based test of daily foreign exchange marketefficiency, Economic Letters 32, 53-59.
[13] Corrrado,C. and T.Su, 1996, Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices[J], Journal of Financial Research, 19, 175-192.
[14] David Arida, 2008,Bayesain Estimation of a Markov-Switching Threshold Asymmetric GARCH Model with Student-t Innovations[J], Econometrics Journal,1-22.
[15] David Arida, 2006, Bayesain Estimation of the GARCH(1,1) Model with Normal Innovations[J], Student,5,283-298.
[16] Dickey, D.A. and Fuller, W.A, (1981), Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica 49, 1057-72.
[17] Engel,R.F., 1982, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation[J], Econometrica, 50, 987-1007.
[18] Engel,R.F. and V.K.Ng, 1993, Measuring and Testing the Impact of News on Volatility[J], Journal of Finance, 48, 1749-1778.
[19] Gelman,A.,Carlin,J.B.,Stern,H.S.,and Rubin,D.B.(1995), Bayesian Data Analysis, CRC Press:London.
[20] Hasting, W.K.(1970), Monte Carlo sampling methods using Markov chains and their applications, Biometrica, 57, 97-109
[21] Jonathan D.Cryer&Kung-Sik Chan, Time Series Analysis with Applications in R,Spriger,2008
[22] Johansen S. and K. Juselius, (1990), Maximum likelihood estimation and inference on cointegration-with application to Demand For Money, Oxford Bulletin of Economics and Statistics 52, 169-210.
[23] Karfakis, Costas I. and Ashok Parikh (1994), Exchange rate convergence and market efficiency, Applied Financial Economics 4, 93-98.
[24] Levich, R.M, (1989), Is the foreign exchange market efficient? Oxford Review of Economic Policy 5, 40-60.
[25] Ljung,L.G..M and G.E.P.Box, 1978, On a Measure of Lag Fit in Time Series Models[J], Biometrika, 67, 297-303.
[26] MacDonald, R. and M.P. Taylor (1989), Foreign exchange market efficiency and cointegration: some evidence from recent float, Economics Letters 29, 63-68.
[27] Masih A. M.M. and Masih R, (1994), On the robustness of cointegration tests of the market efficiency hypothesis: evidence from six European foreign exchange markets, Economia Internazionale XLVII, 160-180.
[28] McLeod,A.I.and W.K.Li(1983). Diagnostic checking ARMA time series models using squared residual sutocorrelations. Journal of Time Series Analysis,4,269-273
[29] Morelli. Simple Valuation Model and Growth Expections. Financial Analysis Journal, 1998(7):419-426
[30] Ruey S.Tsay.(2002),Analysis of Financial Time Series.John Wiely$Sons,Inc
[31] Sephton P.S. and Larsen H.K. (1991), Tests of exchange market efficiency: fragile evidence from cointegration tests, Journal of International Money and Finance, 10, 561-570.
[32] Sheather,S.J.and Jones,M.C.(1991). A reliable data-based bandwidth selection method for kernel estimation.Jouranl of the Royal Statistical Society,Series B,53,683-690
[33] Scott,D.W.(1979). On optimal and data-based histograms. Biometrika,66,605-610
[34] Teruo,Nakasuma,1998,A Markov-Chain Sampling Algorithm for GARCH models[J], Studies in Nonlinear Dynamics&Econometrics ,Berkeley Electronic Press.
[35] Teruo,Nakasuma,2000, Bayesian analysis of ARMA-GARCH models: A Markov chain sampling approach[J], Journal of Econometrics,95,57-69.
[36] Tierney,L.(1994), Markov chains for exploring posterior distributions, Annals of Statistics, 22, 1701-1762.
[37] Zakoian, J.M. (1994). Threshold heteroscedastic models, Journal of Economic Dynamics and Control 18, 931-955
[38] Fame. The Behavior of Stock-Market Price. Journal of Business,1965(7):38-41
[39] Osborne. Brownian Motion in the Stock Market. Operations Research, 1959(7): 69-74
[40]范龙振,胡畏.上海股票市场有效性实证检验.预测,2000(2):61-64
[41]高铁梅,计量经济分析方法与建模.2006
[42]胡朝霞.中国股市弱市有效性研究,1998(1):28-34
[43]胡海鹏,方兆本(2004),股市波动性预测模型改进研究[J],数理统计与管理,2004年9月,第23卷第5期,40-47.
[44]李雪松(2008),高级计量经济学,中国社会科学出版社
[45]李亚静等(2003),基于GARCH模型族的中国股市波动性预测[J],属性的实践与认识,2003年11月,第33卷第11期,65-71.
[46]李竹渝,鲁万波,龚金国(2005),经济、金融计量学中的非参数估计技术[R],社科基金解题报告[R],2005.
[47]陆懋祖(1999),高等时间序列计量学[R],上海人民出版社,1999.
[48]唐齐鸣等(2001),中国股市的ARCH效应分析[J],世界经济,2001年第3期,29-36
[49]吴长凤.利用回归GARCH模型对我国沪深股市的分析.预测,1999(4):46-47
[50]陈道富.当前股市走势分析及对策建议.金融与经济2008(5):41-44
[52]张世英、许启发、周红.金融时间序列分析.清华大学出版设:2008年1月