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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.

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Paper provided by EconWPA in its series Econometrics with number 0403001.

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Length: 29 pages
Date of creation: 04 Mar 2004
Date of revision:
Handle: RePEc:wpa:wuwpem:0403001
Note: Type of Document - pdf; prepared on WinXp; pages: 29
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  1. 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, vol. 11(1), pages 7-54, June.
  2. BAUWENS, Luc & LAURENT, Sébastien, 2002. "A new class of multivariate skew densities, with application to GARCH models," CORE Discussion Papers 2002020, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. 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.
  4. Carmen Fernandez & Mark F J Steel, 1999. "Bayesian Regression Analysis with scale mixtures of normals," ESE Discussion Papers 27, Edinburgh School of Economics, University of Edinburgh.
  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. 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.
  8. John Sutton, 1997. "Gibrat's Legacy," Journal of Economic Literature, American Economic Association, vol. 35(1), pages 40-59, March.
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