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Estimation of Copula Models for Time Series of Possibly Different Length

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  • Patton, Andrew J

Abstract

The theory of conditional copulas provides a means of constructing flexible multivariate density models, allowing for time varying conditional densities of each individual variable, and for time-varying conditional dependence between the variables. Further, the use of copulas in constructing these models often allows for the partitioning of the parameter vector into elements relating only to a marginal distribution, and elements relating to the copula. This paper presents a two-stage (or multi-stage) maximum likelihood estimator for the case that such a partition is possible. We extend the existing statistics literature on the estimation of copula models to consider data that exhibit temporal dependence and heterogeneity. The estimator is flexible enough that the case that unequal amounts of data are available on each variable is easily handled. We investigate the small sample properties of the estimator in a Monte Carlo study, and find that it performs well in comparisons with the standard (one-stage) maximum likelihood estimator. Finally, we present an application of the estimator to a model of the joint distribution of daily Japanese yen - U.S. dollar and euro - U.S. dollar exchange rates. We find some evidence that a copula that captures asymmetric dependence performs better than those that assume symmetric dependence.

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Bibliographic Info

Paper provided by Department of Economics, UC San Diego in its series University of California at San Diego, Economics Working Paper Series with number qt3fc1c8hw.

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Date of creation: 12 Nov 2001
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Handle: RePEc:cdl:ucsdec:qt3fc1c8hw

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Related research

Keywords: copulas; maximum likelihood; two-stage estimation; exchange rates; missing data;

References

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  1. William N. Goetzmann & Philippe Jorion, 1997. "Re-emerging Markets," NBER Working Papers 5906, National Bureau of Economic Research, Inc.
  2. Joshua Rosenberg, 1999. "Semiparametric Pricing of Multivariate Contingent Claims," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-028, New York University, Leonard N. Stern School of Business-.
  3. Kee-Hong Bae & G. Andrew Karolyi & Rene M. Stulz, 2000. "A New Approach to Measuring Financial Contagion," NBER Working Papers 7913, National Bureau of Economic Research, Inc.
  4. Asger Lunde & Peter R. Hansen, 2005. "A forecast comparison of volatility models: does anything beat a GARCH(1,1)?," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 20(7), pages 873-889.
  5. Diebold, Francis X & Gunther, Todd A & Tay, Anthony S, 1998. "Evaluating Density Forecasts with Applications to Financial Risk Management," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 863-83, November.
  6. Bouye, Eric & Durlleman, Valdo & Nikeghbali, Ashkan & Riboulet, Gaël & Roncalli, Thierry, 2000. "Copulas for finance," MPRA Paper 37359, University Library of Munich, Germany.
  7. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  8. Jean-David FERMANIAN & Olivier SCAILLET, 2003. "Nonparametric Estimation of Copulas for Time Series," FAME Research Paper Series rp57, International Center for Financial Asset Management and Engineering.
  9. ROCKINGER, Michael & JONDEAU, Eric, 2001. "Conditional dependency of financial series : an application of copulas," Les Cahiers de Recherche 723, HEC Paris.
  10. Newey, Whitney K., 1988. "Asymptotic Properties of One-step Estimator Obtained from an Optimal Step-size," Econometric Theory, Cambridge University Press, vol. 4(02), pages 359-361, August.
  11. François Longin, 2001. "Extreme Correlation of International Equity Markets," Journal of Finance, American Finance Association, vol. 56(2), pages 649-676, 04.
  12. Rockinger, M. & Jondeau, E., 2001. "Conditional Dependency of Financial Series: An Application of Copulas," Working papers 82, Banque de France.
  13. Paul Kofman & Ian Sharpe, 2000. "Imputation Methods for Incomplete Dependent Variables in Finance," Research Paper Series 33, Quantitative Finance Research Centre, University of Technology, Sydney.
  14. Gourieroux Christian & Monfort Alain & Trognon A, 1981. "Pseudo maximum likelihood methods : theory," CEPREMAP Working Papers (Couverture Orange) 8129, CEPREMAP.
  15. Joshua V. Rosenberg, 2003. "Nonparametric pricing of multivariate contingent claims," Staff Reports 162, Federal Reserve Bank of New York.
  16. Paul Kofman & Ian Sharpe, 2000. "Imputation Methods for Incomplete Dependent Variables in Finance," Econometric Society World Congress 2000 Contributed Papers 0409, Econometric Society.
  17. Andrews, Donald W.K., 1988. "Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables," Econometric Theory, Cambridge University Press, vol. 4(03), pages 458-467, December.
  18. Farebrother, R.W., 1993. "A Class of Bivariate Density Functions with Common Marginals," Econometric Theory, Cambridge University Press, vol. 9(01), pages 148-149, January.
  19. Farebrother, R. W., 1992. "A Class of Bivariate Density Functions with Common Marginals," Econometric Theory, Cambridge University Press, vol. 8(01), pages 145-146, March.
  20. Patton, Andrew J, 2001. "Modelling Time-Varying Exchange Rate Dependence Using the Conditional Copula," University of California at San Diego, Economics Working Paper Series qt01q7j1s2, Department of Economics, UC San Diego.
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Cited by:
  1. Granger, Clive W.J. & Teräsvirta, Timo & Patton, Andrew J., 2002. "Common factors in conditional distributions," Working Paper Series in Economics and Finance 515, Stockholm School of Economics.
  2. Timo Terasvirta & Clive W.J Granger & Andrew Patton, 2003. "Common factors in conditional distributions for Bivariate time series," FMG Discussion Papers dp455, Financial Markets Group.
  3. Y. Malevergne & D. Sornette, 2002. "Tail Dependence of Factor Models," Papers cond-mat/0202356, arXiv.org.
  4. Söehnke Bartram & Stephen Taylor & Yaw-Huei Wang, 2004. "The Euro and European Financial Market Integration," Money Macro and Finance (MMF) Research Group Conference 2004 49, Money Macro and Finance Research Group, revised 13 Oct 2004.
  5. Y. Malevergne & D. Sornette, 2002. "Investigating Extreme Dependences: Concepts and Tools," Papers cond-mat/0203166, arXiv.org.

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