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Bayesian Multivariate Regression Analysis with a New Class of Skewed Distributions


  • Jose T.A.S. Ferreira


  • Mark F.J. Steel



In this paper, we introduce a novel class of skewed multivariate distributions and, more generally, a method of building such a class on the basis of univariate skewed distributions. The method is based on a general linear transformation of a multidimensional random variable with independent components, each with a skewed distribution. Our proposed class of multivariate skewed distributions has a simple, intuitive form for the pdf, moment existence only depends on the existence of the moments of the underlying symmetric univariate distributions, and we avoid any conditioning on unobserved variables. In addition, we can freely allow for any mean and covariance structure in combination with any magnitude and direction of skewness. In order to deal with both skewness and fat tails, we introduce multivariate skewed regression models with fat tails, based on Student distributions. We present two main classes of such distributions, one of which is novel even under symmetry. Under standard non-informative priors on both regression and scale parameters, we derive conditions for propriety of the posterior and for existence of posterior moments. We describe MCMC samplers for conducting Bayesian inference and analyse two applications, one concerning the distribution of various measures of firm size and another on a set of biomedical data.

Suggested Citation

  • Jose T.A.S. Ferreira & Mark F.J. Steel, 2004. "Bayesian Multivariate Regression Analysis with a New Class of Skewed Distributions," Econometrics 0403001, EconWPA.
  • Handle: RePEc:wpa:wuwpem:0403001
    Note: Type of Document - pdf; prepared on WinXp; pages: 29

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    References listed on IDEAS

    1. John Sutton, 1997. "Gibrat's Legacy," Journal of Economic Literature, American Economic Association, vol. 35(1), pages 40-59, March.
    2. C. J. Hoggart & S. G. Walker & A. F. M. Smith, 2003. "Bivariate kurtotic distributions of garment fibre data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 52(3), pages 323-335.
    3. Fern ndez, Carmen & Steel, Mark F.J., 2000. "Bayesian Regression Analysis With Scale Mixtures Of Normals," Econometric Theory, Cambridge University Press, vol. 16(01), pages 80-101, February.
    4. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    5. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
    6. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew "t"-distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389.
    7. Barry Arnold & Robert Beaver & A. Azzalini & N. Balakrishnan & A. Bhaumik & D. Dey & C. Cuadras & J. Sarabia & Barry Arnold & Robert Beaver, 2002. "Skewed multivariate models related to hidden truncation and/or selective reporting," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(1), pages 7-54, June.
    8. Luc Bauwens & Sébastien Laurent, 2002. "A New Class of Multivariate skew Densities, with Application to GARCH Models," Computing in Economics and Finance 2002 5, Society for Computational Economics.
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    Cited by:

    1. Gianni Amisano & Roberto Casarin, 2008. "Particle Filters for Markov-Switching Stochastic-Correlation Models," Working Papers 0814, University of Brescia, Department of Economics.
    2. Roberto Casarin & Marco Tronzano & Domenico Sartore, 2013. "Bayesian Markov Switching Stochastic Correlation Models," Working Papers 2013:11, Department of Economics, University of Venice "Ca' Foscari".
    3. Ferreira, Jose T.A.S. & Steel, Mark F.J., 2007. "Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers," Journal of Econometrics, Elsevier, vol. 137(2), pages 641-673, April.
    4. M. Jones, 2004. "Families of distributions arising from distributions of order statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 1-43, June.
    5. José T. A. S. Ferreira & Mark F. J. Steel, 2005. "Modelling directional dispersion through hyperspherical log-splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 599-616.
    6. J. T. A. S. Ferreira & M. F. J. Steel, 2004. "On Describing Multivariate Skewness: A Directional Approach," Econometrics 0409010, EconWPA.
    7. Manabu Asai & Michael McAleer & Jun Yu, 2006. "Multivariate Stochastic Volatility," Microeconomics Working Papers 22058, East Asian Bureau of Economic Research.
    8. Siddhartha Chib & Yasuhiro Omori & Manabu Asai, 2007. "Multivariate stochastic volatility (Revised in May 2007, Handbook of Financial Time Series (Published in "Handbook of Financial Time Series" (eds T.G. Andersen, R.A. Davis, Jens-Peter Kreiss," CARF F-Series CARF-F-094, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. Balaev , Alexey, 2011. "Multivariate skewed t-distribution with degrees of freedom vector and its application to financial modeling," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 23(3), pages 79-97.
    10. Maximiano Pinheiro, 2010. "Marginal Distributions of Random Vectors Generated by Affine Transformations of Independent Two-Piece Normal Variables," Working Papers w201013, Banco de Portugal, Economics and Research Department.

    More about this item


    Asymmetric distributions; Heavy tails; Linear regression model; Mardia's measure of skewness; Orthogonal matrices; Posterior propriety.;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
    • C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs

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