Advertisement

Multivariate GARCH Models

  • Annastiina SilvennoinenEmail author
  • Timo Teräsvirta
Chapter

Abstract

This article contains a review of multivariate GARCH models. Most common GARCH models are presented and their properties considered. This also includes nonparametric and semiparametric models. Existing specification and misspecification tests are discussed. Finally, there is an empirical example in which several multivariate GARCH models are fitted to the same data set and the results compared.

Keywords

Conditional Variance GARCH Model Conditional Correlation Conditional Covariance Dynamic Conditional Correlation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alexander, C. O. and Chibumba, A. M. (1997): Multivariate orthogonal factor GARCH. University of Sussex Discussion Papers in Mathematics.Google Scholar
  2. Bae, K.-H., Karolyi, G. A. and Stulz, R. M. (2003): A new approach to measuring financial contagion. The Review of Financial Studies 16, 717–763.CrossRefGoogle Scholar
  3. Bauwens, L., Laurent, S. and Rombouts, J. V. K. (2006): Multivariate GARCH Models: A Survey. Journal of Applied Econometrics 21, 79–109.CrossRefMathSciNetGoogle Scholar
  4. Bera, A. K. and Kim, S. (2002): Testing constancy of correlation and other specifications of the BGARCH model with an application to international equity returns. Journal of Empirical Finance 9, 171–195.CrossRefGoogle Scholar
  5. Berben, R.-P. and Jansen, W. J. (2005): Comovement in international equity markets: A sectoral view. Journal of International Money and Finance 24, 832–857.CrossRefGoogle Scholar
  6. Billio, M. and Caporin, M. (2006): A generalized dynamic conditional correlation model for portfolio risk evaluation. Unpublished manuscript, Ca’Foscari University of Venice, Department of Economics.Google Scholar
  7. Bollerslev, T. (1990): Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72, 498–505.CrossRefGoogle Scholar
  8. Bollerslev, T., Engle, R. F. and Nelson, D. B. (1994): ARCH Models. In: Engle, R.F. and McFadden, D.L. (Eds.): Handbook of Econometrics 4, 2959–3038. North-Holland, Amsterdam.Google Scholar
  9. Bollerslev, T., Engle R. F. and Wooldridge, J. M. (1988): A capital asset pricing model with time-varying covariances. The Journal of Political Economy 96, 116–131.CrossRefGoogle Scholar
  10. Boussama, F. (1998): Ergodicité, mélange et estimation dans le modelès GARCH. PhD Thesis, Université 7 Paris.Google Scholar
  11. Brooks, C., Burke S. P. and Persand G. (2003): Multivariate GARCH models: software choice and estimation issues. Journal of Applied Econometrics 18, 725–734.CrossRefGoogle Scholar
  12. Cappiello, L., Engle, R. F. and Sheppard, K. (2006): Asymmetric dynamics in the correlations of global equity and bond returns. Journal of Financial Econometrics 4, 537–572.CrossRefGoogle Scholar
  13. Chib, S., Omori, Y. and Asai, M. (2008): Multivariate stochastic volatility. In: Andersen, T. G., Davis, R. A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 365–400. Springer, New York.Google Scholar
  14. Comte, F. and Lieberman, O. (2003): Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.zbMATHCrossRefMathSciNetGoogle Scholar
  15. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977): Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society 39, 1–38.zbMATHMathSciNetGoogle Scholar
  16. Diebold, F. X. and Nerlove, M. (1989): The dynamics of exchange rate volatility: a multivariate latent factor ARCH model. Journal of Applied Econometrics 4, 1–21.CrossRefGoogle Scholar
  17. Doornik, J. A. (2002): Object-Oriented Matrix Programming Using Ox 3rd edition. Timberlake Consultants Press.Google Scholar
  18. Duchesne, P. (2004): On matricial measures of dependence in vector ARCH models with applications to diagnostic checking. Statistics and Probability Letters 68, 149–160.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Engle, R. F. (1982): Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50, 987–1007.zbMATHCrossRefMathSciNetGoogle Scholar
  20. Engle, R. F. (2002): Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350.CrossRefMathSciNetGoogle Scholar
  21. Engle, R. F. and Colacito, R. (2006): Testing and valuing dynamic correlations for asset allocation. Journal of Business and Economic Statistics 24, 238–253.CrossRefMathSciNetGoogle Scholar
  22. Engle, R. F. and Gonzalez-Rivera, G. (1991): Semiparametric ARCH models. Journal of Business and Economic Statistics 9, 345–359.CrossRefGoogle Scholar
  23. Engle, R. F., Granger, C. W. J. and Kraft, D. (1984): Combining competing forecasts of inflation using a bivariate ARCH model. Journal of Economic Dynamics and Control 8, 151–165.CrossRefMathSciNetGoogle Scholar
  24. Engle, R. F. and Kroner, K. F. (1995): Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150.CrossRefMathSciNetGoogle Scholar
  25. Engle, R. F. and Mezrich, J. (1996): GARCH for groups. Risk 9, 36–40.Google Scholar
  26. Engle, R. F. and Ng, V. K. (1993): Measuring and Testing the Impact of News on Volatility. Journal of Finance 48, 1749–1777.CrossRefGoogle Scholar
  27. Engle, R. F., Ng, V. K. and Rothschild, M. (1990): Asset pricing with a factor ARCH covariance structure: empirical estimates for treasury bills. Journal of Econometrics 45, 213–238.CrossRefGoogle Scholar
  28. Franke, J., Kreiss, J.-P. and Mammen, E. (2008): Nonparametric modeling in financial time series. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 926–952. Springer, New York.Google Scholar
  29. Gouriéroux, C. (1997): ARCH Models and Financial Applications. Springer, Berlin.zbMATHGoogle Scholar
  30. Hafner, C. M. (2003): Fourth moment structure of multivariate GARCH models. Journal of Financial Econometrics 1, 26–54.CrossRefGoogle Scholar
  31. Hafner, C. M. and Rombouts, J. V. K. (2007): Semiparametric multivariate volatility models. Econometric Theory 23, 251–280.CrossRefMathSciNetGoogle Scholar
  32. Hafner, C. M., van Dijk, D. and Franses, P. H. (2005): Semi-parametric modelling of correlation dynamics. In: Fomby, T., Hill, C. and Terrell, D. (Eds.): Advances in Econometrics 20/A, 59–103. Elsevier, Amsterdam.Google Scholar
  33. Hansen, B. E. (1996): Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.zbMATHCrossRefMathSciNetGoogle Scholar
  34. Hansson, B. and Hordahl, P. (1998): Testing the conditional CAPM using multivariate GARCH–M. Applied Financial Economics 8, 377–388.CrossRefGoogle Scholar
  35. He, C. and Teräsvirta, T. (2004): An extended constant conditional correlation GARCH model and its fourth-moment structure. Econometric Theory 20, 904–926.zbMATHCrossRefMathSciNetGoogle Scholar
  36. Jeantheau, T. (1998): Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86.CrossRefMathSciNetGoogle Scholar
  37. Kawakatsu, H. (2006): Matrix exponential GARCH. Journal of Econometrics 134, 95–128.CrossRefMathSciNetGoogle Scholar
  38. Kroner, K. F. and Ng, V. K. (1998): Modeling asymmetric comovements of asset returns. The Review of Financial Studies 11, 817–844.CrossRefGoogle Scholar
  39. Kwan, C. K., Li, W. K. and Ng, K. (inpress): A multivariate threshold GARCH model with time-varying correlations. Econometric Reviews to appear.Google Scholar
  40. Lanne, M. and Saikkonen, P. (2007): A multivariate generalized orthogonal factor GARCH model. Journal of Business and Economic Statistics 25, 61–75.CrossRefMathSciNetGoogle Scholar
  41. Li, W. K. and Mak, T. K. (1994): On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. Journal of Time Series Analysis 15, 627–636.zbMATHCrossRefMathSciNetGoogle Scholar
  42. Ling, S. and Li, W. K. (1997): Diagnostic checking of nonlinear multivariate time series with multivariate ARCH errors. Journal of Time Series Analysis 18, 447–464.zbMATHCrossRefMathSciNetGoogle Scholar
  43. Ling, S. and McAleer, M. (2003): Asymptotic theory for a vector ARMA–GARCH model. Econometric Theory 19, 280–310.MathSciNetGoogle Scholar
  44. Linton, O. B. (2008): Semiparametric and nonparametric ARCH modelling. In: Andersen, T. G., Davis, R. A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 156–167. Springer, New York.Google Scholar
  45. Long, X. and Ullah, A. (2005): Nonparametric and semiparametric multivariate GARCH model. Unpublished manuscript.Google Scholar
  46. Luukkonen, R., Saikkonen, P. and Teräsvirta, T. (1988): Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499.zbMATHCrossRefMathSciNetGoogle Scholar
  47. McLeod, A. I. and Li, W. K. (1983): Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis 4, 269–273.zbMATHCrossRefMathSciNetGoogle Scholar
  48. Nelson, D. B. (1991): Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica 59, 347–370.zbMATHCrossRefMathSciNetGoogle Scholar
  49. Nelson, D. B. and Cao, C. Q. (1992): Inequality Constraints in the Univariate GARCH Model. Journal of Business and Economic Statistics 10, 229–235.CrossRefGoogle Scholar
  50. Ng, L. (1991): Tests of the CAPM with time-varying covariances: a multivariate GARCH approach. The Journal of Finance 46, 1507–1521.CrossRefGoogle Scholar
  51. Pagan, A. and Ullah, A. (1999): Nonparametric Econometrics. Cambridge University Press.Google Scholar
  52. Palm, F. C. (1996): GARCH Models of Volatility. In: Maddala, G. S. and Rao, C. R. (Eds.): Handbook of Statistics: Statistical Methods in Finance 14, 209–240. Elsevier, Amsterdam.Google Scholar
  53. Pelletier, D. (2006): Regime switching for dynamic correlations. Journal of Econometrics 131, 445–473.CrossRefMathSciNetGoogle Scholar
  54. Ross, S. A. (1976): The arbitrage theory of capital asset pricing. Journal of Economic Theory 13, 341–360.CrossRefMathSciNetGoogle Scholar
  55. Sentana, E. (1998): The relation between conditionally heteroskedastic factor models and factor GARCH models. Econometrics Journal 1, 1–9.zbMATHCrossRefGoogle Scholar
  56. Shephard, N. G. (1996): Statistical Aspects of ARCH and Stochastic Volatility. In: Cox, D. R., Hinkley, D. V. and Barndorff-Nielsen, O. E. (Eds.): Time Series Models in Econometrics, Finance and Other Fields, 1–67. Chapman and Hall, London.Google Scholar
  57. Silvennoinen, A. (2006): Numerical aspects of the estimation of multivariate GARCH models. Unpublished manuscript.Google Scholar
  58. Silvennoinen, A. (2008): Numerical aspects of the estimation of multivariate GARCH models. QFRC Research Paper, University of Technology, Sydney.Google Scholar
  59. Silvennoinen, A. and Teräsvirta, T. (2005): Multivariate autoregressive conditional heteroskedasticity with smooth transitions in conditional correlations. SSE/EFI Working Paper Series in Economics and Finance 577.Google Scholar
  60. Silvennoinen, A. and Teräsvirta, T. (2007): Modelling multivariate autoregressive conditional heteroskedasticity with the double smooth transition conditional correlation GARCH model. SSE/EFI Working Paper Series in Economics and Finance 625.Google Scholar
  61. Stone, C. (1980): Optimal rates of convergence for nonparametric estimators. Annals of Statistics 8, 1348–1360.zbMATHCrossRefMathSciNetGoogle Scholar
  62. Tse, Y. K. (2000): A test for constant correlations in a multivariate GARCH model. Journal of Econometrics 98, 107–127.zbMATHCrossRefGoogle Scholar
  63. Tse, Y. K. and Tsui, K. C. (1999): A note on diagnosing multivariate conditional heteroscedasticity models. Journal of Time Series Analysis. 20, 679–691.zbMATHCrossRefGoogle Scholar
  64. Tse, Y. K. and Tsui, K. C. (2002): A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. Journal of Business and Economic Statistics 20, 351–362.CrossRefMathSciNetGoogle Scholar
  65. van der Weide, R. (2002): GO–GARCH: A multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics 17, 549–564.CrossRefGoogle Scholar
  66. Vrontos, I. D., Dellaportas, P. and Politis, D. N. (2003): A full-factor multivariate GARCH model. Econometrics Journal 6, 312–334.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.School of Finance and EconomicsUniversity of Technology SydneyBroadway
  2. 2.CREATES Economics and Management,University of Aarhus, DK-8000 AarhusC, and Department of Economic Statistics, Stockholm School of EconomicsStockholm

Personalised recommendations