Multivariate Stochastic Volatility

  • Siddhartha ChibEmail author
  • Yasuhiro Omori
  • Manabu Asai


We provide a detailed summary of the large and vibrant emerging literature that deals with the multivariate modeling of conditional volatility of financial time series within the framework of stochastic volatility. The developments and achievements in this area represent one of the great success stories of financial econometrics. Three broad classes of multivariate stochastic volatility models have emerged: one that is a direct extension of the univariate class of stochastic volatility model, another that is related to the factor models of multivariate analysis and a third that is based on the direct modeling of time-varying correlation matrices via matrix exponential transformations, Wishart processes and other means. We discuss each of the various model formulations, provide connections and differences and show how the models are estimated. Given the interest in this area, further significant developments can be expected, perhaps fostered by the overview and details delineated in this paper, especially in the fitting of high-dimensional models.


Markov Chain Monte Carlo Stochastic Volatility Markov Chain Monte Carlo Algorithm Stochastic Volatility Model Full Conditional Distribution 
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.


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  1. Aas, K. and Haff, I. H. (2006): The generalized hyperbolic skew Student's t-distribution. Journal of Financial Econometrics 4, 275–309.CrossRefGoogle Scholar
  2. Aguilar, O. and West, M. (2000): Bayesian dynamic factor models and portfolio allocation. Journal of Business and Economic Statistics 18, 338–357.CrossRefGoogle Scholar
  3. Albert, J. H. and Chib, S. (1993): Bayesian inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts. Journal of Business and Economic Statistics 11, 1–15.CrossRefGoogle Scholar
  4. Andersen, T., Bollerslev, T., Diebold, F. X. and Labys, P. (2003): Modeling and forecasting realized volatility. Econometrica 71, 579–625.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Arellano-Valle, R. B. and Azzalini, A. (2006): On the unification of families of skew-normal distributions. Scandinavian Journal of Statistics 33, 561–574.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Asai, M. and McAleer, M. (2006): Asymmetric multivariate stochastic volatility. Econometric Reviews 25, 453–473.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Asai, M. and McAleer, M. (2007): The structure of dynamic correlations in multivariate stochastic volatility models. Unpublished paper: Faculty of Economics, Soka University.Google Scholar
  8. Asai, M., McAleer, M. and Yu, J. (2006). Multivariate stochastic volatility: A review. Econometric Reviews 25, 145–175.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Azzalini, A. (2005): The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics 32, 159–188.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Azzalini, A. and Capitanio, A. (2003). Distributions generated by perterbation of symmetry with emphasis on a multivariate skew t-distribution. Journal of the Royal Statistical Society Series B 65, 367–389.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Barndorff-Nielsen, O. E. (1977): Exponentially decreasing distributions for the logarithm of the particle size. Proceedings of the Royal Society London Series A Mathematical and Physical Sciences 353, 401–419.CrossRefGoogle Scholar
  12. Barndorff-Nielsen, O. E. and Shephard, N. (2001): Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society Series B 63, 167–241.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Barndorff-Nielsen, O. E. and Shephard, N. (2004): Econometric analysis of realised covariation: High frequency based covariance, regression and correlation in financial economics. Econometrica 72, 885–925.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Bauwens, L. and Laurent, S. (2005): A new class of multivariate skew densities, with application to generalized autoregressive conditional heteroscedasticity models. Journal of Business and Economic Statistics 23, 346–354.CrossRefMathSciNetGoogle Scholar
  15. Bauwens, L., Laurent, S. and Rombouts, J. V. K. (2006): Multivariate GARCH: A survey. Journal of Applied Econometrics 21, 79–109.CrossRefMathSciNetGoogle Scholar
  16. Bollerslev, T. (1990): Modelling the coherence in the short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72, 498–505.CrossRefGoogle Scholar
  17. Bollerslev, T., Engle, R. F. and Woodridge, J. (1988): A capital asset pricing model with time varying covariances. Journal of Political Economy 96, 116–131.CrossRefGoogle Scholar
  18. Bos, C. S. and Shephard, N. (2006): Inference for adaptive time series models: Stochastic volatility and conditionally Gaussian state space form. Econometric Reviews 25, 219–244.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Broto, C. and Ruiz, E. (2004): Estimation methods for stochastic volatility models: A survey. Journal of Economic Survey 18, 613–649.CrossRefGoogle Scholar
  20. Calvet, L. E. and Fisher, A. J. (2001): Forecasting multifractal volatility. Journal of Econometrics 105, 27–58.zbMATHCrossRefMathSciNetGoogle Scholar
  21. Calvet, L. E., Fisher, A. J. and Thompson, S. B. (2006): Volatility comovement: A multi-frequency approach. Journal of Econometrics 131, 179–215.CrossRefMathSciNetGoogle Scholar
  22. Carvalho, C. M. and West, M. (2006): Dynamic matrix-variate graphical models. Bayesian Analysis 1, 1–29.CrossRefMathSciNetGoogle Scholar
  23. Chan, D., Kohn, R. and Kirby, C. (2006): Multivariate stochastic volatility models with correlated errors. Econometric Reviews 25, 245–274.zbMATHCrossRefMathSciNetGoogle Scholar
  24. Chib, S. (2001): Markov chain Monte Carlo methods: Computation and inference. In: Heckman, J. J. and Leamer, E. (Eds.): Handbook of Econometrics 5, 3569–3649. North-Holland, Amsterdam.Google Scholar
  25. Chib, S. and Greenberg, E. (1994): Bayes inference for regression models with ARMA(p, q) errors. Journal of Econometrics 64, 183–206.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Chib, S. and Greenberg, E. (1995): Understanding the Metropolis-Hastings algorithm. The American Statistician 49, 327–335.CrossRefGoogle Scholar
  27. Chib, S. and Greenberg, E. (1996): Markov chain Monte Carlo simulation methods in econometrics. Econometric Theory 12, 409–431.CrossRefMathSciNetGoogle Scholar
  28. Chib, S. and Greenberg, E. (1998): Analysis of multivariate Probit models. Biometrika 85, 347–361.zbMATHCrossRefGoogle Scholar
  29. Chib, S., Nardari, F. and Shephard, N. (2002): Markov chain Monte Carlo methods for generalized stochastic volatility models. Journal of Econometrics 108, 281–316.zbMATHCrossRefMathSciNetGoogle Scholar
  30. Chib, S., Nardari, F. and Shephard, N. (2006): Analysis of high dimensional multivariate stochastic volatility models. Journal of Econometrics 134, 341–371.CrossRefMathSciNetGoogle Scholar
  31. Chiu, T., Leonard, T. and Tsui, K. (1996): The matrix-logarithmic covariance model. Journal of the American Statistical Association 91, 198–210.zbMATHCrossRefMathSciNetGoogle Scholar
  32. Cox, J., Ingersoll, J. and Ross, S. (1985): A theory of the term structure of interest rates. Econometrica 53, 385–407.CrossRefMathSciNetGoogle Scholar
  33. Dagpunar, J. S. (1989): An easily implemented generalized inverse Gaussian generator. Communications in Statistics Simulations 18, 703–710.CrossRefMathSciNetGoogle Scholar
  34. Daníelsson, J. (1994): Stochastic volatility in asset prices: Estimation with simulated maximum likelihood. Journal of Econometrics 64, 375–400.zbMATHCrossRefGoogle Scholar
  35. Daníelsson, J. (1998): Multivariate stochastic volatility models: Estimation and a comparison with VGARCH models. Journal of Empirical Finance 5, 155–173.CrossRefGoogle Scholar
  36. Dawid, A. P. and Lauritzen, S. L. (1993): Hyper-Markov laws in the statistical analysis. Annals of Statistics 3, 1272–1317.CrossRefMathSciNetGoogle Scholar
  37. de Jong, P. and Shephard, N. (1995): The simulation smoother for time series models. Biometrika 82, 339–350.zbMATHCrossRefMathSciNetGoogle Scholar
  38. Dey, D. and Liu, J. (2005): A new construction for skew multivariate distributions. Journal of Multivariate Analysis 95, 323–344.zbMATHCrossRefMathSciNetGoogle Scholar
  39. Diebold, F. X. and Nerlove, M. (1989): The dynamics of exchange rate volatility: A multivariate latent-factor ARCH model. Journal of Applied Econometrics 4, 1–22.CrossRefGoogle Scholar
  40. Doornik, J. A. (2002): Object-Oriented Matrix Programming Using Ox (3rd ed.). Timber-lake Consultants Press, London.
  41. Durbin, J. and Koopman, S. J. (2002): A simple and efficient simulation smoother for state space time series analysis. Biometrika 89, 603–616.zbMATHCrossRefMathSciNetGoogle Scholar
  42. 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
  43. Engle, R. F. and Kroner, K. F. (1995): Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150.CrossRefMathSciNetGoogle Scholar
  44. Ferreira, J. T. A. S. and Steel, M. F. J. (2004): Bayesian multivariate regression analysis with a new class of skewed distributions. Statistics Research Report 419, University of Warwick.Google Scholar
  45. Ghysels, E., Harvey, A. C. and Renault, E. (1996): Stochastic volatility. In: G. S. M. Rao, C. R. (Ed.): Statistical Models in Finance (Handbook of Statistics), 119–191. North-Holland, Amsterdam.Google Scholar
  46. Gilks, W. R., Best, N. G. and Tan, K. K. C. (1995): Adaptive rejection Metropolis sampling within Gibbs sampling. Applied Statistics 44, 455–472.zbMATHCrossRefGoogle Scholar
  47. Gourieroux, C. (2006): Continuous time Wishart process for stochastic risk. Econometric Reviews 25, 177–217.zbMATHCrossRefMathSciNetGoogle Scholar
  48. Gourieroux, C., Jasiak, J. and Sufana, R. (2004): The Wishart autoregressive process of multivariate stochastic volatility. Discussion paper: University of Toronto.Google Scholar
  49. Gupta, A. K., González-Farías, G. and Domínguez-Molina, J. A. (2004): A multivariate skew normal distribution. Journal of Multivariate Analysis 89, 181–190.zbMATHCrossRefGoogle Scholar
  50. Han, Y. (2006): The economics value of volatility modelling: Asset allocation with a high dimensional dynamic latent factor multivariate stochastic volatility model. Review of Financial Studies 19, 237–271.CrossRefGoogle Scholar
  51. Harvey, A. C., Ruiz, E. and Shephard, N. (1994): Multivariate stochastic variance models. Review of Economic Studies 61, 247–264.zbMATHCrossRefGoogle Scholar
  52. Harvey, A. C. and Shephard, N. (1996): Estimation of asymmetric stochastic volatility model for asset returns. Journal Journal of Business and Economic Statistics 14, 429–434.CrossRefGoogle Scholar
  53. Hörmann, W., Leydold, J. and Derflinger, G. (2004): Automatic Nonuniform Random Variate Generation. Springer, Berlin.zbMATHGoogle Scholar
  54. Jacquier, E., Polson, N. G. and Rossi, P. E. (1994): Bayesian analysis of stochastic volatility models (with discussion). Journal of Business and Economic Statistics 12, 371–389.CrossRefGoogle Scholar
  55. Jacquier, E., Polson, N. G. and Rossi, P. E. (1999): Stochastic volatility: Univariate and multivariate extensions. CIRANO Working paper 99s–26, Montreal.Google Scholar
  56. Jungbacker, B. and Koopman, S. J. (2006): Monte Carlo likelihood estimation for three multivariate stochastic volatility models. Econometric Reviews 25, 385–408.zbMATHCrossRefMathSciNetGoogle Scholar
  57. Kawakatsu, H. (2006): Matrix exponential GARCH. Journal of Econometrics 134, 95–128.CrossRefMathSciNetGoogle Scholar
  58. Kim, S., Shephard, N. and Chib, S. (1998): Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies 65, 361–393.zbMATHCrossRefGoogle Scholar
  59. King, M., Sentana, E. and Wadhwani, S. (1994): Volatility and links between national stock markets. Econometrica 62, 901–933.zbMATHCrossRefGoogle Scholar
  60. Liesenfeld, R. and Richard, J.-F. (2003): Univariate and multivariate stochastic volatility models: Estimation and diagnostics. Journal of Empirical Finance 10, 505–531.CrossRefGoogle Scholar
  61. Lopes, H. F. and Carvalho, C. M. (2007): Factor stochastic volatility with time varying loadings and Markov switching regimes. Journal of Statistical Planning and Inference 137, 3082–3091.zbMATHCrossRefMathSciNetGoogle Scholar
  62. Omori, Y., Chib, S., Shephard, N. and Nakajima, J. (2007): Stochastic volatility with leverage: Fast and efficient likelihood inference. Journal of Econometrics 140, 425–449.CrossRefMathSciNetGoogle Scholar
  63. Philipov, A. and Glickman, M. E. (2006a): Factor multivariate stochastic volatility via Wishart processes. Econometric Reviews 25, 311–334.zbMATHCrossRefMathSciNetGoogle Scholar
  64. Philipov, A. and Glickman, M. E. (2006b): Multivariate stochastic volatility via Wishart processes. Journal of Business and Economic Statistics 24, 313–328.CrossRefMathSciNetGoogle Scholar
  65. Pitt, M. K., Chan, D. and Kohn, R. (2006): Efficient Bayesian inference for Gaussian copula regression models. Biometrika 93, 537–554.zbMATHCrossRefMathSciNetGoogle Scholar
  66. Pitt, M. K. and Shephard, N. (1999): Time varying covariances: a factor stochastic volatility approach. In: Bernardo, J. M., Berger, J. O., Dawid, A. P. and Smith, A. F. M. (Eds.): Bayesian Statistics 6, 547–570. Oxford University Press, Oxford.Google Scholar
  67. Protassov, R. S. (2004): EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ. Statistics and Computing 14, 67–77.CrossRefMathSciNetGoogle Scholar
  68. Quintana, J. M. and West, M. (1987): An analysis of international exchange rates using multivariate DLMs. The Statistician 36, 275–281.CrossRefGoogle Scholar
  69. Ray, B. K. and Tsay, R. S. (2000): Long-range dependence in daily stock volatilities. Journal of Business and Economic Statistics 18, 254–262.CrossRefGoogle Scholar
  70. Schmidt, R., Hrycej, T. and Stützle, E. (2006): Multivariate distribution models with generalized hyperbolic margins. Computational Statistics and Data Analysis 50, 2065–20096.zbMATHCrossRefMathSciNetGoogle Scholar
  71. Shephard, N. (2004): Stochastic Volatility: Selected Readings. Oxford University Press, Oxford.Google Scholar
  72. Shephard, N. and Pitt, M. K. (1997): Likelihood analysis of non-Gaussian measurement time series. Biometrika 84, 653–667.zbMATHCrossRefMathSciNetGoogle Scholar
  73. Smith, M. and Pitts, A. (2006): Foreign exchange intervention by the Bank of Japan: Bayesian analysis using a bivariate stochastic volatility model. Econometric Reviews 25, 425–451.zbMATHCrossRefMathSciNetGoogle Scholar
  74. So, M. K. P. and Kwok, W. Y. (2006): A multivariate long memory stochastic volatility model. Physica A 362, 450–464.CrossRefGoogle Scholar
  75. So, M. K. P., Li, W. K. and Lam, K. (1997): Multivariate modelling of the autoregressive random variance process. Journal of Time Series Analysis 18, 429–446.zbMATHCrossRefMathSciNetGoogle Scholar
  76. So, M. K. P., Lam, K. and Li, W. K. (1998): A stochastic volatility model with Markov switching. Journal of Business and Economic Statistics 16, 244–253.CrossRefMathSciNetGoogle Scholar
  77. Tims, B. and Mahieu, R. (2006): A range-based multivariate stochastic volatility model for exchange rates. Econometric Reviews 25, 409–424.zbMATHCrossRefMathSciNetGoogle Scholar
  78. Tsay, R. S. (2005). Analysis of Financial Time Series: Financial Econometrics (2nd ed.). Wiley, New York.Google Scholar
  79. Watanabe, T. and Omori, Y. (2004): A multi-move sampler for estimating non-Gaussian times series models: Comments on Shephard and Pitt (1997). Biometrika 91, 246–248.zbMATHCrossRefMathSciNetGoogle Scholar
  80. Wong, F., Carter, C. and Kohn, R. (2003): Efficient estimation of covariance matrix selection models. Biometrika 90, 809–830.CrossRefMathSciNetGoogle Scholar
  81. Yu, J. (2005): On leverage in a stochastic volatility model. Journal of Econometrics 127, 165–178.CrossRefMathSciNetGoogle Scholar
  82. Yu, J. and Meyer, R. (2006): Multivariate stochastic volatility models: Bayesian estimation and model comparison. Econometric Reviews 25, 361–384.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Washington University in St. LouisUSA
  2. 2.Faculty of EconomicsUniversity of TokyoBunkyo-KuJapan
  3. 3.Faculty of EconomicsSoka UniversityTangi-choJapan

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