The HS-SAS and GSH-SAS Distribution as Model for Unconditional and Conditional Return Distributions

  • Matthias Fischer University of Erlangen-Nürnberg, Germany
  • Klaus Herrmann University of Erlangen-Nürnberg, Germany

Abstract

We introduce two new skewed and leptokurtic distributions derived from the hyperbolic secant distribution and from Vaughan (2002)’s generalized hyperbolic distribution by use of the sinh-arcsinh transformation introduced in Jones and Pewsey (2009). Properties of these new distribution are given. Their flexibility for modeling financial return data is comparable to that of their most advanced peers. Contrary to the latter for both distributions a closed-form solution for the density, cumulative distribution and quantile function can be given.

References

Bowley, A. L. (1920). Elements of Statistics. New York: Charles Scribner’s Sons.

Crow, E. L., and Siddiqui, M. M. (1967). Robust estimation of location. Journal of the American Statistical Association, 62, 353-389.

Fischer, M. (2004). Skew generalized secant hyperbolic distributions: Unconditional and conditional fit to asset returns. Austrian Journal of Statistics, 33, 293-304.

Fischer, M. (2006). The skew generalized secant hyperbolic family. Austrian Journal of Statistics, 35, 437-443.

Fischer, M. (2011). Hyperbolic secant distributions and generalizations. In M. Lovric (Ed.), International Encyclopedia of Statistical Sciences. Springer.

Fischer, M., Gao, Y., and Herrmann, K. (2010). Volatility models with innovations from new maximum entropy densities at work. IWQW Discussion Paper Series, 03.

Fischer, M., and Vaughan, D. (2010). The beta-hyperbolic secant distribution. Austrian Journal of Statistics, 39, 245-258.

Grottke, M. (2001). Die t-Verteilung und ihre Verallgemeinerungen als Modell für Finanzmarktdaten. Köln.

Harkness, W. L., and Harkness, M. L. (1968). Generalized hyperbolic secant distributions. Journal of the American Statistical Association, 63, 329-337.

Herrmann, K. (2011). Maximum Entropy Models for Time-Varying Moments Applied to Daily Financial Returns. Unpublished doctoral dissertation, University of Erlangen Nürnberg.

Jones, M. C., and Pewsey, A. (2009). Sinh-arcsin distributions. Biometrika, 96, 761-780.

Komunjer, I. (2007). Asymmetric power distribution: Theory and applications to risk management. Journal of Applied Econometrics, 22, 821-921.

Nolan, J. P. (2010). Stable distributions - models for heavy tailed data. (To be published)

Rosco, J. F., Jones, M. C., and Pewsey, A. (2010). Skew t distributions via the sinh-arcsinh transformation. Test. (10.1007/s11749-010-0222-2)

van Zwet, W. R. (1964). Convex Transformations of Random Variables. Unpublished doctoral dissertation, Amsterdam University.

Vaughan, D. C. (2002). The generalized secant hyperbolic distribution and its properties. Communications in Statististics – Theory and Method, 31, 219-238.

Zhu, D., and Galbraith, J. W. (2010). A generalized asymmetric Student t-distribution with application to financial econometrics. Journal of Econometrics, 157, 297-305.
Published
2016-02-24
How to Cite
Fischer, M., & Herrmann, K. (2016). The HS-SAS and GSH-SAS Distribution as Model for Unconditional and Conditional Return Distributions. Austrian Journal of Statistics, 42(1), 33–45. https://doi.org/https://doi.org/10.17713/ajs.v42i1.164
Section
Articles