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A Generalized Asymmetric Student-t Distribution with Application to Financial Econometrics

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  • Dongming Zhu
  • John Galbraith

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Abstract

This paper proposes a new class of asymmetric Student-t (AST) distributions, and investigates its properties, gives procedures for estimation, and indicates applications in financial econometrics. We derive analytical expressions for the cdf, quantile function, moments, and quantities useful in financial econometric applications such as the expected shortfall. A stochastic representation of the distribution is also given. Although the AST density does not satisfy the usual regularity conditions for maximum likelihood estimation, we establish consistency, asymptotic normality and efficiency of ML estimators and derive an explicit analytical expression for the asymptotic covariance matrix. A Monte Carlo study indicates generally good finite-sample conformity with these asymptotic properties. Le présent document propose une nouvelle catégorie de distributions asymétriques suivant la loi t de Student (Asymmetric Student-t Distribution - AST). Il en examine les propriétés, suggère des procédures d’estimation et propose des applications dans le domaine de l’économétrie financière. Nous établissons des expressions analytiques pour la fonction de distribution cumulative, la fonction quantile, les moments et les quantités, ces aspects étant utiles dans certaines applications liées à l’économétrie financière, par exemple l’estimation du manque à gagner prévu. Nous mettons aussi de l’avant une représentation stochastique de la distribution. Même si la densité suivant la loi t de Student ne répond pas aux conditions habituelles de régularité pour l’estimation du maximum de vraisemblance, nous établissons néanmoins la consistance, la normalité asymptotique et l’efficacité des estimateurs du maximum de vraisemblance et arrivons à une expression analytique explicite en ce qui concerne la matrice de covariance asymptotique. Une étude selon la méthode Monte Carlo indique généralement une bonne conformité des échantillons finis avec ces propriétés asymptotiques.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 2009s-13.

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Date of creation: 01 Apr 2009
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Handle: RePEc:cir:cirwor:2009s-13

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Keywords: asymmetric distribution; expected shortfall; maximum likelihood estimation; distribution asymétrique; manque à gagner prévu; estimation du maximum de vraisemblance;

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  1. Dima Alberg & Haim Shalit & Rami Yosef, 2008. "Estimating stock market volatility using asymmetric GARCH models," Applied Financial Economics, Taylor & Francis Journals, vol. 18(15), pages 1201-1208.
  2. Hansen, B.E., 1992. "Autoregressive Conditional Density Estimation," RCER Working Papers 322, University of Rochester - Center for Economic Research (RCER).
  3. Panayiotis Theodossiou, 1998. "Financial Data and the Skewed Generalized T Distribution," Management Science, INFORMS, vol. 44(12-Part-1), pages 1650-1661, December.
  4. Philip Hans Franses & Marco van der Leij & Richard Paap, 2008. "A Simple Test for GARCH Against a Stochastic Volatility Model," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 6(3), pages 291-306, Summer.
  5. 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.
  6. John Galbraith & Dongming Zhu, 2009. "Forecasting Expected Shortfall With A Generalized Asymmetric Student-T Distribution," Departmental Working Papers 2009-01, McGill University, Department of Economics.
  7. Mittnik, Stefan & Paolella, Marc S., 2003. "Prediction of Financial Downside-Risk with Heavy-Tailed Conditional Distributions," CFS Working Paper Series 2003/04, Center for Financial Studies (CFS).
  8. Bollerslev, Tim, 1987. "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economics and Statistics, MIT Press, vol. 69(3), pages 542-47, August.
  9. Kjersti Aas & Ingrid Hobaek Haff, 2006. "The Generalized Hyperbolic Skew Student's t-Distribution," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 4(2), pages 275-309.
  10. 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).
  11. 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.
  12. Fernández, C. & Steel, M.F.J., 1996. "On Bayesian Modelling of Fat Tails and Skewness," Discussion Paper 1996-58, Tilburg University, Center for Economic Research.
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Cited by:
  1. Carol Alexander & Jose Maria Sarabia, 2010. "Generalized Beta-Generated Distributions," ICMA Centre Discussion Papers in Finance icma-dp2010-09, Henley Business School, Reading University.
  2. Nadarajah, Saralees & Chan, Stephen & Afuecheta, Emmanuel, 2013. "On the characteristic function for asymmetric Student t distributions," Economics Letters, Elsevier, vol. 121(2), pages 271-274.
  3. Rubio, Francisco Javier & Steel, Mark F. J., 2014. "Bayesian modelling of skewness and kurtosis with two-piece scale and shape transformations," MPRA Paper 57102, University Library of Munich, Germany.
  4. Harvey, A. & Sucarrat, G., 2012. "EGARCH models with fat tails, skewness and leverage," Cambridge Working Papers in Economics 1236, Faculty of Economics, University of Cambridge.
  5. Stavros Degiannakis & Pamela Dent & Christos Floros, 2014. "A Monte Carlo Simulation Approach to Forecasting Multi-period Value-at-Risk and Expected Shortfall Using the FIGARCH-skT Specification," Manchester School, University of Manchester, vol. 82(1), pages 71-102, 01.
  6. John Galbraith & Dongming Zhu, 2009. "Forecasting Expected Shortfall With A Generalized Asymmetric Student-T Distribution," Departmental Working Papers 2009-01, McGill University, Department of Economics.
  7. Colletaz, Gilbert & Hurlin, Christophe & Pérignon, Christophe, 2013. "The Risk Map: A new tool for validating risk models," Journal of Banking & Finance, Elsevier, vol. 37(10), pages 3843-3854.
  8. Zhu, Dongming & Galbraith, John W., 2011. "Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions," Journal of Empirical Finance, Elsevier, vol. 18(4), pages 765-778, September.
  9. Daniel T. Cassidy & Michael J. Hamp & Rachid Ouyed, 2010. "Student's t-Distribution Based Option Sensitivities: Greeks for the Gosset Formulae," Papers 1003.1344, arXiv.org, revised Jul 2010.

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