Abstract
This paper presents an overview of the literature on applications of copulas in the modelling of financial time series. Copulas have been used both in multivariate time series analysis, where they are used to characterize the (conditional) cross-sectional dependence between individual time series, and in univariate time series analysis, where they are used to characterize the dependence between a sequence of observations of a scalar time series process. The paper includes a broad, brief, review of the many applications of copulas in finance and economics.
Keywords
- Marginal Distribution
- Asset Return
- Tail Dependence
- Multivariate Time Series
- Copula Model
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References
Alexander, C. 2008: Market Risk Analysis, Volume III. Wiley & Sons, London, forthcoming.
Alexander, C. and Chibumba, A. (1997): Multivariate Orthogonal Factor GARCH. Mimeo, University of Sussex.
Alsina, C., Nelsen, R.B. and Schweizer, B. (1993): On the characterization of a class of binary operations on distribution functions. Statistics and Probability Letters 17, 85–89.
Andersen, T.G., Bollerslev, T., Christoffersen, P.F. and Diebold, F.X. (2006): Volatility and Correlation Forecasting. In: Elliott, G., Granger, C.W.J. and Timmermann, A. (Eds.): The Handbook of Economic Forecasting. North Holland, Amsterdam.
Ang, A. and Bekaert, G. (2002): International Asset Allocation with Regime Shifts. Review of Financial Studies 15, 1137–1187.
Ang, A. and Chen, J. (2002): Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics 63, 443–494.
Arakelian, V. and Dellaportas, P. (2005): Contagion tests via copula threshold models. Mimeo, University of Crete.
Bartram, S.M., Taylor, S.J. and Wang, Y.-H. (2006): The euro and European financial market dependence. Journal of Banking and Finance forthcoming.
Bae, K.-H., Karolyi, G.A. and Stulz, R.M. (2003): A New Approach to Measuring Financial Contagion. Review of Financial Studies 16, 717–764.
Bauwens, L., Laurent, S. and Rombouts, J. (2006): Multivariate GARCH Models: A Survey. Journal of Applied Econometrics 21, 79–109.
Beare, B. (2007): Copula-based mixing conditions for Markov chains. Mimeo, University of Oxford.
Bennett, M.N. and Kennedy, J.E. (2004): Quanto Pricing with Copulas. Journal of Derivatives 12, 26–45.
Bollerslev, T. (1986): Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31, 307–327.
Bonhomme, S. and Robin, J.-M. (2004): Modeling Individual Earnings Trajectories using Copulas with an Application to the Study of Earnings Inequality: France, 1990–2002. Mimeo, Université de Paris 1.
Bouyé, E. and Salmon, M. (2002): Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets. Mimeo, University of Warwick.
Brendstrup, B. and Paarsch, H.J. (2007): Semiparametric Identification and Estimation in Multi-Object English Auctions. Journal of Econometrics 141, 84–108.
Breymann, W., Dias, A. and Embrechts, P. (2003): Dependence structures for multivariate high-frequency data in finance. Quantitative Finance 3, 1–16.
Capéraà, P., Fougères, A.-L. and Genest, C. (1997): A Nonparametric Estimation Procedure for Bivariate Extreme Value Copulas. Biometrika 84, 567–577.
Cappiello, L., Engle, R.F. and Sheppard, K. (2003): Evidence of Asymmetric Effects in the Dynamics of International Equity and Bond Return Covariance. Journal of Financial Econometrics forthcoming.
Carrasco M. and Chen X. (2002): Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.
Casella, G. and Berger, R.L. (1990): Statistical Inference Duxbury Press, U.S.A.
Chamberlain, G. (1983): A characterization of the distributions that imply mean-variance utility functions. Journal of Economic Theory 29, 185–201.
Chen, X. and Fan, Y. (2006a): Estimation of copula-based semiparametric time series models. Journal of Econometrics 130, 307–335.
Chen, X. and Fan, Y. (2006b): Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics 135, 125–154.
Chen, X., Fan, Y. and Tsyrennikov, V. (2006): Efficient estimation of semiparametric multivariate copula models. Journal of the American Statistical Association 101, 1228–1240.
Cherubini, U. and Luciano, E. (2001): Value at Risk trade-off and capital allocation with copulas. Economic Notes 30, 235–256.
Cherubini, U., Luciano, E. and Vecchiato, W. (2004): Copula Methods in Finance John Wiley & Sons, England.
Chollete, L. (2005): Frequent extreme events? A dynamic copula approach. Mimeo, Norwegian School of Economics and Business.
Chollete, L., de la Peña, V. and Lu, C.-C. (2005): Comovement of international financial markets. Mimeo, Norwegian School of Economics and Business.
Christoffersen, P. (2008): Value–at–Risk models. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 752–766. Springer Verlag, New York.
Clayton, D.G. (1978): A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151.
Coles, S., Heffernan, J. and Tawn, J. (1999): Dependence measures for extreme value analyses. Extremes 2, 339–365.
Cook, R.D. and Johnson, M.E. (1981): A family of distributions for modelling non-elliptically symmetric multivariate data. Journal of the Royal Statistical Society 43, 210–218.
Corradi V. and Swanson, N.R. (2005): Predictive Density Evaluation. In: Elliott, G., Granger, C.W.J. and Timmermann, A. (Eds.): Handbook of Economic Forecasting. North Holland, Amsterdam.
Darsow, W.F., Nguyen, B. and Olsen, E.T. (1992): Copulas and Markov processes. Illinois Journal of Mathematics 36, 600–642.
Daul, S., De Giorgi, E., Lindskog, F. and McNeil, A. (2003): The grouped t-copula with an application to credit risk. RISK 16, 73–76.
Demarta, S. and McNeil, A.J. (2005): The t copula and related copulas. International Statistical Review 73, 111–129.
Denuit, M. and Lambert, P. (2005): Constraints on concordance measures in bivariate discrete data. Journal of Multivariate Analysis 93, 40–57.
Diebold, F.X., Gunther, T. and Tay, A.S. (1998): Evaluating Density Forecasts with Applications to Financial Risk Management. International Economic Review 39, 863–883.
Diebold, F.X., Hahn, J. and Tay, A.S. (1999): Multivariate Density Forecast Evaluation and Calibration in Financial Risk Management: High Frequency Returns on Foreign Exchange. Review of Economics and Statistics 81, 661–673.
Duffie, D. (2004): Clarendon Lecture in Finance, mimeo Stanford University. http://www.finance.ox.ac.uk/NR/rdonlyres/9A26FC79-980F-4114-8033-B73899EADE88/0/slides_duffie_clarendon_3.pdf
Embrechts, P. and Höing, A. (2006): Extreme VaR scenarios in higher dimensions. Mimeo ETH Zürich.
Embrechts, P., McNeil, A. and Straumann, D. (2002): Correlation and Dependence Properties in Risk Management: Properties and Pitfalls. In: Dempster, M. (Ed.): Risk Management: Value at Risk and Beyond. Cambridge University Press.
Embrechts, P., Höing, A. and Juri, A. (2003): Using Copulae to bound the Value-at-Risk for functions of dependent risks. Finance & Stochastics 7, 145–167.
Embrechts, P., Furrer, H. and Kaufmann, R. (2008): Different Kinds of Risk. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 729–751. Springer Verlag, New York.
Engle, R.F. (1982): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation. Econometrica 50, 987–1007.
Engle, R.F. (2002): Dynamic Conditional Correlation - A Simple Class of Multivariate GARCH Models. Journal of Business and Economic Statistics 20, 339–350.
Engle, R.F. and Russell, J.R. (1998): Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data. Econometrica 66, 1127–1162.
Erb, C.B., Harvey, C.R. and Viskanta, T.E. (1994): Forecasting International Equity Correlations. Financial Analysts Journal 50, 32–45.
Fermanian, J.-D. (2005): Goodness of fit tests for copulas. Journal of Multivariate Analysis 95, 119–152.
Fermanian, J.-D. and Scaillet, O. (2003): Nonparametric estimation of copulas for time series. Journal of Risk 5, 25–54.
Fermanian, J.-D. and Scaillet, O. (2005): Some statistical pitfalls in copula modeling for financial applications. In: Klein, E. (Ed.): Capital Formation, Governance and Banking. Nova Science Publishing.
Fermanian, J.-D. and Wegkamp, M. (2004): Time dependent copulas. Mimeo, CREST.
Fisher, R.A. (1932): Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh.
Frey, R. and McNeil, A.J. (2001): Modelling dependent defaults. ETH, Zürich, E-Collection, http://e-collection.ethbib.ethz.ch/show?type=bericht&nr=273
Gagliardini, P. and Gouriéroux, C. (2007a): An Efficient Nonparametric Estimator for Models with Non-linear Dependence. Journal of Econometrics 137, 187–229.
Gagliardini, P. and Gouriéroux, C. (2007b): Duration Time Series Models with Proportional Hazard. Journal of Time Series Analysis forthcoming.
Galambos, J. (1978): The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.
Garcia, R. and Tsafack, G. (2007): Dependence Structure and Extreme Comovements in International Equity and Bond Markets. Working paper, Université de Montreal.
Genest, C. and Rivest, L.-P. (1993): Statistical Inference Procedures for Bivariate Archimedean Copulas. Journal of the American Statistical Association 88, 1034–1043.
Genest, C., Ghoudi, K. and Rivest, L.-P. (1995): A Semiparametric Estimation Procedure of Dependence Parameters in Multivariate Families of Distributions. Biometrika 82, 543–552.
Genest, C., Quasada Molina, J.J., Rodríguez Lallena, J.A. and Sempi, C. (1999): A characterization of quasi-copulas. Journal of Multivariate Analysis 69, 193–205.
Genest, C., Rémillard, B. and Beaudoin, D. (2007): Goodness-of-Fit Tests for Copulas: A Review and Power Study. Insurance: Mathematics and Economics forthcoming.
Giesecke, K. (2004): Correlated Default with Incomplete Information. Journal of Banking and Finance 28, 1521–1545.
Grammig, J., Heinen, A. and Rengifo, E. (2004): An analysis of the submission of orders on Xetra, using multivariate count data. CORE Discussion Paper 2004/58.
Granger, C.W.J., Teräsvirta, T. and Patton, A.J. (2006): Common factors in conditional distributions for bivariate time series. Journal of Econometrics 132, 43–57.
Hamilton, J.D. (1989): A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle. Econometrica 57, 357–384.
Heinen, A. and Rengifo, E. (2003): Modelling Multivariate Time Series of Count Data Using Copulas. CORE Discussion Paper 2003/25.
Hu, L. (2006): Dependence Patterns across Financial Markets: A Mixed Copula Approach. Applied Financial Economics 16, 717–729.
Hull, J. and White, A. (1998): Value–at–Risk when daily changes in market variables are not normally distributed. Journal of Derivatives 5, 9–19.
Hurd, M., Salmon, M. and Schleicher, C. (2005): Using copulas to construct bivariate foreign exchange distributions with an application to the Sterling exchange rate index. Mimeo, Bank of England.
Ibragimov, R. (2005): Copula-based dependence characterizations and modeling for time series. Harvard Institute of Economic Research Discussion Paper 2094.
Ibragimov, R. (2006): Copula-based characterizations and higher-order Markov processes. Mimeo, Department of Economics, Harvard University.
Joe, H. (1997): Multivariate Models and Dependence Concepts. Monographs in Statistics and Probability 73. Chapman and Hall, London.
Joe, H. and Xu, J.J. (1996): The Estimation Method of Inference Functions for Margins for Multivariate Models. Working paper, Department of Statistics, University of British Columbia.
Jondeau, E. and Rockinger, M. (2006): The copula-GARCH model of conditional dependencies: an international stock market application. Journal of International Money and Finance 25, 827–853.
Lee, L.-F. (1983): Generalized econometric models with selectivity Econometrica 51, 507–512.
Lee, T.-H. and Long, X. (2005): Copula-based multivariate GARCH model with uncorrelated dependent standardized returns. Journal of Econometrics forthcoming.
Li, D.X. (2000): On default correlation: a copula function approach. Journal of Fixed Income 9, 43–54.
Longin, F. and Solnik, B. (2001): Extreme Correlation of International Equity Markets. Journal of Finance 56, 649–676.
Malevergne, Y. and Sornette, D. (2003): Testing the Gaussian Copula Hypothesis for Financial Assets Dependences. Quantitative Finance 3, 231–250.
McNeil, A.J., Frey, R. and Embrechts, P. (2005): Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, New Jersey.
Meitz M. and Saikkonen P. (2004): Ergodicity, mixing, and the existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory forthcoming.
Mikosch, T. (2006): Copulas: Tales and Facts, with discussion and rejoinder. Extremes 9, 3–62.
Miller, D.J. and Liu, W.-H. (2002): On the recovery of joint distributions from limited information. Journal of Econometrics 107, 259–274.
Mills, F.C. (1927): The Behavior of Prices. National Bureau of Economic Research, New York.
Nelsen, R.B. (2006): An Introduction to Copulas, second Edition. Springer, New York.
Newey, W.K. and McFadden, D. (1994): Large Sample Estimation and Hypothesis Testing. In: Engle, R.F. and McFadden, D. (Eds.): Handbook of Econometrics 4. North-Holland, Amsterdam.
Okimoto, T. (2006): New evidence of asymmetric dependence structure in international equity markets: further asymmetry in bear markets: Journal of Financial and Quantitative Analysis forthcoming.
Panchenko, V. (2005a): Goodness-of-fit Tests for Copulas. Physica A 355, 176–182.
Panchenko, V. (2005b): Estimating and evaluating the predictive abilities of semiparametric multivariate models with application to risk management. Mimeo, University of Amsterdam.
Patton, A.J. (2002): Applications of Copula Theory in Financial Econometrics. Unpublished Ph.D. dissertation, University of California, San Diego.
Patton, A.J. (2004): On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation. Journal of Financial Econometrics 2, 130–168.
Patton, A.J. (2006a): Modelling Asymmetric Exchange Rate Dependence. International Economic Review 47, 527–556.
Patton, A.J. (2006b): Estimation of Multivariate Models for Time Series of Possibly Different Lengths. Journal of Applied Econometrics 21, 147–173.
Rivers, D. and Vuong, Q. (2002): Model Selection Tests for Nonlinear Dynamic Models. The Econometrics Journal 5, 1–39.
Rodriguez, J.C. (2007): Measuring financial contagion: a copula approach. Journal of Empirical Finance 14, 401–423.
Rosenberg, J.V. (2003): Nonparametric pricing of multivariate contingent claims. Journal of Derivatives 10, 9–26.
Rosenberg, J.V. and Schuermann, T. (2006): A general approach to integrated risk management with skewed, fat-tailed risks. Journal of Financial Economics 79, 569–614.
Salmon, M. and Schleicher, C. (2006): Pricing Multivariate Currency Options with Copulas. In: Rank, J. (Ed.): Copulas: From Theory to Application in Finance. Risk Books, London.
Sancetta, A. and Satchell, S. (2004): The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20, 535–562.
Scaillet, O. (2007): Kernel based goodness-of-fit tests for copulas with fixed smoothing parameters. Journal of Multivariate Analysis 98, 533–543.
Schönbucher, P. and Schubert, D. (2001): Copula Dependent Default Risk in Intensity Models. Mimeo, Bonn University.
Shih, J.H. and Louis, T.A. (1995): Inferences on the Association Parameter in Copula Models for Bivariate Survival Data. Biometrics 51, 1384–1399.
Silvennoinen, A. and Teräsvirta, T. (2008): Multivariate GARCH Models. In: Andersen, T.G., Davis, R. A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 201–229. Springer, New York.
Sklar, A. (1959): Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut Statistique de l’Universite de Paris 8, 229–231.
Smith, M.D. (2003): Modelling sample selection using Archimedean copulas. Econometrics Journal 6, 99–123.
Taylor, S.J. and Wang, Y.-H. (2004): Option prices and risk-neutral densities for currency cross-rates. Mimeo, Department of Accounting and Finance, Lancaster University.
van den Goorbergh, R.W.J., C. Genest and Werker, B.J.M. (2005): Multivariate Option Pricing Using Dynamic Copula Models. Insurance: Mathematics and Economics 37, 101–114.
van der Weide, R. (2002): GO-GARCH: A Multivariate Generalized Orthogonal GARCH Model. Journal of Applied Econometrics 17, 549–564.
Vuong, Q. (1989): Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses. Econometrica 57, 307–333.
White, H. (1994): Estimation, Inference and Specification Analysis. Econometric Society Monographs 22, Cambridge University Press, Cambridge, U.K.
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Patton, A.J. (2009). Copula–Based Models for Financial Time Series. In: Mikosch, T., Kreiß, JP., Davis, R., Andersen, T. (eds) Handbook of Financial Time Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71297-8_34
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