金融资产动态相关性模型及其实证研究
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摘要
对金融资产间相关性的研究,是金融理论和实践中的一个基础性问题。在涉及到多个资产的许多场合,比如资产定价、资产选择、波动溢出以及风险管理中,都需要考虑资产间的相关性。而我们知道,金融市场是一个由复杂的动态系统,相关性结构也受到很多内外因素的影响,如果将相关性视为固定不变,会有失偏颇。因此在处理金融资产间的相关性时,考虑其动态结构是非常有意义的。
对相关性动态演化过程的描述有几种不同的方法,本文考虑了其中的两种描述方法。一种是时变相关性模型,其特点是相关性在时刻变化。另一种是状态转换模型,其特点是相关性指标在某一状态下是固定不变的,而在不同状态之间则有所区别,在不同状态间的转移过程服从一个马氏链。
在前一类模型中,具有代表性的是Engle提出的DCC模型,该模型简洁地刻画了相关性的时变演化过程,同时具有估计上的优势。我们通过引入Copula函数的概念,指出DCC模型是一种特殊的正态Copula模型,进而将其推广到广义形式。我们使用灵活的边缘分布来取代DCC模型中的正态假设,分别构建了正态Copula和t-Copula下的广义DCC模型。对我国股市的实证研究表明,使用厚尾分布如t分布、GED分布为边缘分布,可以更好地度量投资组合在99%置信水平下的VaR值。
在后一类模型中,我们通过在Copula函数中引入状态变量,构建了一类含有状态转换的Copula模型。我们讨论了该模型下的一般应用,并分析了该模型在相关性度量上的特点。对于我国股市的实证研究表明,使用状态转换Copula模型可以很好地解释相关性在不同行情走势下的区别。
The issue of dependence between financial assets is a foundational issue in modern financial theory and practice. In many applications related with assets of more than two, such as asset pricing, asset selection, volatility spillover and risk management, the measure of association between assets need to be taken into account. As we all know, financial markets are complicated dynamical systems. The dependence structure is impacted by lots of factors inside and outside the markets. It would be inappropriate to set the measure of association invariable. Therefore, when we deal with the measure of association between financial assets, the consideration of dynamic structure is meaningful.
There are several different ways to describe the dynamic structure of the dependence. In this article, we have considered two description methods. One is the time-varying model; the measure of association is different at each time. The other is the regime switching model; the measure of association is constant within a regime but different across regimes. The transitions between the regimes are governed by a Markov chain.
In the former kind of model, the Dynamic Conditional Correlation (DCC) model proposed by Engle is representative. The model describes the evolution path of the time-varying correlation with parsimonious parametric while providing a simple method of estimation. In this paper, we introduce the concept of Copula and point out that the DCC model is a particular case of multivariate Normal copula model, thus extend it to the general form. We use flexible marginal distribution to replace the normal distribution assumption, construct generalized DCC model under the multivariate Normal copula and the multivariate Student’t copula (t-Copula). Empirical study on Chinese stock markets have shown that using the heavy-tailed distribution such as student(t) distribution and Generalized error distribution for the marginal distribution can measure the Value at risk (VaR) of the portfolio in the 99% confidence level better .
In the latter kind of model, through introducing state variable into Copula, we build one regime switching Copula model. We discuss the general application under this model and analyze the features of the measure of association in this model. Empirical study on Chinese stock markets have shown that regime switching Copula model can be applied to explain the differences in the dependence between different market quotation .
对相关性动态演化过程的描述有几种不同的方法,本文考虑了其中的两种描述方法。一种是时变相关性模型,其特点是相关性在时刻变化。另一种是状态转换模型,其特点是相关性指标在某一状态下是固定不变的,而在不同状态之间则有所区别,在不同状态间的转移过程服从一个马氏链。
在前一类模型中,具有代表性的是Engle提出的DCC模型,该模型简洁地刻画了相关性的时变演化过程,同时具有估计上的优势。我们通过引入Copula函数的概念,指出DCC模型是一种特殊的正态Copula模型,进而将其推广到广义形式。我们使用灵活的边缘分布来取代DCC模型中的正态假设,分别构建了正态Copula和t-Copula下的广义DCC模型。对我国股市的实证研究表明,使用厚尾分布如t分布、GED分布为边缘分布,可以更好地度量投资组合在99%置信水平下的VaR值。
在后一类模型中,我们通过在Copula函数中引入状态变量,构建了一类含有状态转换的Copula模型。我们讨论了该模型下的一般应用,并分析了该模型在相关性度量上的特点。对于我国股市的实证研究表明,使用状态转换Copula模型可以很好地解释相关性在不同行情走势下的区别。
The issue of dependence between financial assets is a foundational issue in modern financial theory and practice. In many applications related with assets of more than two, such as asset pricing, asset selection, volatility spillover and risk management, the measure of association between assets need to be taken into account. As we all know, financial markets are complicated dynamical systems. The dependence structure is impacted by lots of factors inside and outside the markets. It would be inappropriate to set the measure of association invariable. Therefore, when we deal with the measure of association between financial assets, the consideration of dynamic structure is meaningful.
There are several different ways to describe the dynamic structure of the dependence. In this article, we have considered two description methods. One is the time-varying model; the measure of association is different at each time. The other is the regime switching model; the measure of association is constant within a regime but different across regimes. The transitions between the regimes are governed by a Markov chain.
In the former kind of model, the Dynamic Conditional Correlation (DCC) model proposed by Engle is representative. The model describes the evolution path of the time-varying correlation with parsimonious parametric while providing a simple method of estimation. In this paper, we introduce the concept of Copula and point out that the DCC model is a particular case of multivariate Normal copula model, thus extend it to the general form. We use flexible marginal distribution to replace the normal distribution assumption, construct generalized DCC model under the multivariate Normal copula and the multivariate Student’t copula (t-Copula). Empirical study on Chinese stock markets have shown that using the heavy-tailed distribution such as student(t) distribution and Generalized error distribution for the marginal distribution can measure the Value at risk (VaR) of the portfolio in the 99% confidence level better .
In the latter kind of model, through introducing state variable into Copula, we build one regime switching Copula model. We discuss the general application under this model and analyze the features of the measure of association in this model. Empirical study on Chinese stock markets have shown that regime switching Copula model can be applied to explain the differences in the dependence between different market quotation .
引文
罗付岩,邓光明.2007.基于时变Copula的VAR估计[J].系统工程, 25(8):28-33.
韦艳华.2004. Copula理论及其在多变量金融时间序列分析上的应用研究[D]:[博士].天津:天津大学,95-105.
韦艳华,张世英.2004.金融市场的相关性分析—Copula-GARCH模型及其应用[J].系统工程, 22(4):7-12.
韦艳华,张世英,郭焱.2004.金融市场相关程度与相关模式的研究[J].系统工程学报, 19(4):355-362.
吴振翔,陈敏,叶五一,缪柏其.2006.基于Copula-GARCH的投资组合风险分析[J].系统工程理论与实践, 26(3):45-52。
张蕾,郑振龙.2007.指数投资组合的VaR模型及检验[J].山西财经大学学报, 5:86-89.
张尧庭.2002.连接函数(copula)技术与金融风险分析[J].统计研究,4:48-51.
张尧庭.2002.我们应该选用什么样的相关性指标[J].统计研究, 9:41-44.
Ang A, Chen J.2002.Asymmetric Correlations of Equity Portfolios[J].Journal of Financial Economics, 63(3):443-494.
Billio M, Caporin M.2005.Multivariate Markov switching dynamic conditional correlation GARCH representations for contagion analysis[J].Statistical Methods & Application, 14: 145-461.
Bollerslev T.1986. A Conditionally Heteroskedastic Time series model for speculative prices and rates of return [J]. The Review of Economics and Statistics, 69(3):542-547.
Bollerslev T.1990.Modeling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model[J].Review of Economics and Statistics, 72:498-505.
Bollerslev T, Engle R, Wooldridge J.1988.A capital asset pricing model with time varying covariance[J]. Journal of Political Economy, 96: 116–131.
Cappiello L,Engle R , Shepard K.2006.Asymmetric Dynamics in the correlations of Global Equity and Bond Returns[J]. Journal of Financial Econometrics, 4(4): 537-572
Edwards S, Susmel R.2001.Volatility dependence and contagion in emerging equity markets[J]. Journal of Development Economics, 66:505-532.
Embrechts P, McNeil A, Straum D. 2002. Correlation and Dependence in Risk Management: Properties and Pitfalls[C] //Dempster M. Risk Management: Value at Risk and Beyond. Cambridge University Press,176-223.
Engle R. 1982.Autoregressive Conditional Heteroscedasticity with estimates of the variance ofUnited Kingdom inflation[J].Econometrica,50(4):987-1007.
Engle R.2002.Dynamic conditional correlation-A simple class of multivariate GARCH models[J].Journal of Business and Economic Statistics, 20:339-350.
Engle R ,Kroner K.1995.Multivariate simultaneous GARCH[J]. Econometric Theory, 11: 122-150.
Engle R, Manganelli S.2004.CAViaR:Conditional autoregressive value at risk by regression quantiles[J].Journal of Business and Economic Statistics, 22:367-381.
Engle R, Sheppard K. 2003. Theoretical and Empirical properties of Dynamic Conditional Correlation Multivariate GARCH[R]. NBER Working Paper 8554.
Genest C, Rivest L. 1993. Statistical inference procedures for bivariate Archimedean Copulas[J].Journal of American Statistical Association, 88(243):1034-1043.
Greene W H. 2002.Econometric Analysis[M]. New Jersey: Prentice Hall.
Hamilton J D. 1989.A new approach to the economic analysis of nonstationary time series and the business cycle[J].Econometrica, 57(2):357-384.
Hamilton J D.1990.Analysis of time series subject to changes in regimes [J].Journal of Econometrics, 45:39-70.
Hamilton J D. 1994.Time Series Analysis[M]. Seattle: Princeton Press.677-703.
Hamilton J D, Susmel R.1994.Autoregressive conditional heteroskedasticity and change in regime[J]. Journal of Econometrics, 64:307-333.
Koenker R, Bassett G.1978.Regression quantiles[J].Econometrica. 46:33-50.
Nelson. 1991.An introduction to Copulas[M].New York: Springer.
Newey W K, McFadden D. 1994. Large sample estimation and hypothesis testing[M] //Engle R , McFadden D. Handbook of Econometrics: vol. IV. Amsterdam: North-Holland, 2111-2245.
Patton A. 2001.Modeling time-varying exchange rate dependence using the conditional copula.[R]Working paper of Department of Economics, University of California, San Diego.
Pelletier D. 2006.Regime switching for dynamic correlation[J].Journal of Econometrics, 131:445-473.
Rodriguez J C.2007.Measuring financial contagion: a Copula approach[J].Journal of Empirical Finance, 14(3):401-423.
Romano C.2002.Calibrating and simulating Copula functions: an Application to the Italian Stock Market[R].CIDEM, Working Paper.
Tse Y K, Tsui K C.2002. A multivariate generalized Autoregressive Conditional Heteroscedasticity model with time-varying correlations[J].Journal of Business and Economic Statistics, 20(3):351-362.
韦艳华.2004. Copula理论及其在多变量金融时间序列分析上的应用研究[D]:[博士].天津:天津大学,95-105.
韦艳华,张世英.2004.金融市场的相关性分析—Copula-GARCH模型及其应用[J].系统工程, 22(4):7-12.
韦艳华,张世英,郭焱.2004.金融市场相关程度与相关模式的研究[J].系统工程学报, 19(4):355-362.
吴振翔,陈敏,叶五一,缪柏其.2006.基于Copula-GARCH的投资组合风险分析[J].系统工程理论与实践, 26(3):45-52。
张蕾,郑振龙.2007.指数投资组合的VaR模型及检验[J].山西财经大学学报, 5:86-89.
张尧庭.2002.连接函数(copula)技术与金融风险分析[J].统计研究,4:48-51.
张尧庭.2002.我们应该选用什么样的相关性指标[J].统计研究, 9:41-44.
Ang A, Chen J.2002.Asymmetric Correlations of Equity Portfolios[J].Journal of Financial Economics, 63(3):443-494.
Billio M, Caporin M.2005.Multivariate Markov switching dynamic conditional correlation GARCH representations for contagion analysis[J].Statistical Methods & Application, 14: 145-461.
Bollerslev T.1986. A Conditionally Heteroskedastic Time series model for speculative prices and rates of return [J]. The Review of Economics and Statistics, 69(3):542-547.
Bollerslev T.1990.Modeling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model[J].Review of Economics and Statistics, 72:498-505.
Bollerslev T, Engle R, Wooldridge J.1988.A capital asset pricing model with time varying covariance[J]. Journal of Political Economy, 96: 116–131.
Cappiello L,Engle R , Shepard K.2006.Asymmetric Dynamics in the correlations of Global Equity and Bond Returns[J]. Journal of Financial Econometrics, 4(4): 537-572
Edwards S, Susmel R.2001.Volatility dependence and contagion in emerging equity markets[J]. Journal of Development Economics, 66:505-532.
Embrechts P, McNeil A, Straum D. 2002. Correlation and Dependence in Risk Management: Properties and Pitfalls[C] //Dempster M. Risk Management: Value at Risk and Beyond. Cambridge University Press,176-223.
Engle R. 1982.Autoregressive Conditional Heteroscedasticity with estimates of the variance ofUnited Kingdom inflation[J].Econometrica,50(4):987-1007.
Engle R.2002.Dynamic conditional correlation-A simple class of multivariate GARCH models[J].Journal of Business and Economic Statistics, 20:339-350.
Engle R ,Kroner K.1995.Multivariate simultaneous GARCH[J]. Econometric Theory, 11: 122-150.
Engle R, Manganelli S.2004.CAViaR:Conditional autoregressive value at risk by regression quantiles[J].Journal of Business and Economic Statistics, 22:367-381.
Engle R, Sheppard K. 2003. Theoretical and Empirical properties of Dynamic Conditional Correlation Multivariate GARCH[R]. NBER Working Paper 8554.
Genest C, Rivest L. 1993. Statistical inference procedures for bivariate Archimedean Copulas[J].Journal of American Statistical Association, 88(243):1034-1043.
Greene W H. 2002.Econometric Analysis[M]. New Jersey: Prentice Hall.
Hamilton J D. 1989.A new approach to the economic analysis of nonstationary time series and the business cycle[J].Econometrica, 57(2):357-384.
Hamilton J D.1990.Analysis of time series subject to changes in regimes [J].Journal of Econometrics, 45:39-70.
Hamilton J D. 1994.Time Series Analysis[M]. Seattle: Princeton Press.677-703.
Hamilton J D, Susmel R.1994.Autoregressive conditional heteroskedasticity and change in regime[J]. Journal of Econometrics, 64:307-333.
Koenker R, Bassett G.1978.Regression quantiles[J].Econometrica. 46:33-50.
Nelson. 1991.An introduction to Copulas[M].New York: Springer.
Newey W K, McFadden D. 1994. Large sample estimation and hypothesis testing[M] //Engle R , McFadden D. Handbook of Econometrics: vol. IV. Amsterdam: North-Holland, 2111-2245.
Patton A. 2001.Modeling time-varying exchange rate dependence using the conditional copula.[R]Working paper of Department of Economics, University of California, San Diego.
Pelletier D. 2006.Regime switching for dynamic correlation[J].Journal of Econometrics, 131:445-473.
Rodriguez J C.2007.Measuring financial contagion: a Copula approach[J].Journal of Empirical Finance, 14(3):401-423.
Romano C.2002.Calibrating and simulating Copula functions: an Application to the Italian Stock Market[R].CIDEM, Working Paper.
Tse Y K, Tsui K C.2002. A multivariate generalized Autoregressive Conditional Heteroscedasticity model with time-varying correlations[J].Journal of Business and Economic Statistics, 20(3):351-362.
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