Theory Review

Chapter

Abstract

In modeling data in the performing arts, the presence of highly paid ‘superstars’—earning the lion’s share of performance royalty for musical compositions or receipts from live concerts—causes the earnings distribution to be highly skewed. Incorrect probability judgments can be made in the analysis of economic data when normality is assumed, but asymmetry is present. The purpose of this chapter is to review the skew-normal and skew-t statistical distributions theory and present a model that can be used to estimate regression models when the distribution is highly skewed. To correct for asymmetrical forms in econometric data modeling, flexible forms of both the univariate and multivariate skew-normal and skew-t distributions have been developed. Walls (2005) suggests two reasons why the log-skew-t is appealing in economic modeling. First, it is easier—computationally—to implement the skew-t than some other distributions (like the stable Paretian model or the ’eevy stable regression model) using standard maximum likelihood statistical techniques that are within reach of applied researchers. Second, the skew-t extends the normal distribution by permitting tails that are heavy and asymmetric. The log-normal is just a special case of the log-skew-normal when α=0.

Keywords

Covariance 

References

  1. Arellano-Valle, R. and Azzalini, A. (2006). On the unification of families of skew-normal distributions. Scandinavian Journal of Statistics, 33:561–574.CrossRefGoogle Scholar
  2. Azzalini, A. (1985). A class of distribution which includes the normal ones. Scandinavian Journal of Statistics, 12:171–178.Google Scholar
  3. Azzalini, A. (1986). Further results on a class of distribution which includes the normalones. Statistica, 46:199–208.Google Scholar
  4. Azzalini, A. (2005). The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics, 32:159–188.CrossRefGoogle Scholar
  5. Azzalini, A. (2008). A very brief introduction to the skew-normal distribution. http://azzalini.stat.unipd.it/SN/intro.html .
  6. Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew-normaldistribution. Journal of the Royal Statistical Society, B61:579–602.Google Scholar
  7. Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis ona multivariate skew-t distribution. Journal of the Royal Statistical Society, B65:367–389.Google Scholar
  8. Azzalini, A., DalCappello, T., and Kotz, S. (2003). Log-skew-normal and log-skew-t distributions as models for familyincome data. Journal of Income Distribution, 11(3–4):12–20.Google Scholar
  9. Azzalini, A. and Genton, M. (2008). Robust likelihood methods based on the skew-t and relateddistributions. International Statistical Review, 76:106–129.CrossRefGoogle Scholar
  10. Dalla-Valle, A. (2007). A test for the hypothesis of skew-normality in a population. Journal of Statistical Computation and Simulation,77(1):63–77.CrossRefGoogle Scholar
  11. Genton, M., editor (2004). Skew-Elliptical Distributions and Their Applications: A JourneyBeyond Normality. Chapman & Hall/CRC.Google Scholar
  12. Halvorsen, R. and Palmquist, R. (1980). The interpretation of dummy variables in semi-logarithmic equations. The American Economic Review, 70(4):474–475.Google Scholar
  13. Jarque, C. and Bera, A. (1980). Efficient tests for normality, homoscedasticity and serialindependence of regression residuals. Econometric Letters, 6:255–259.CrossRefGoogle Scholar
  14. Jarque, C. and Bera, A. (1987). A test for normality of observations and regression residuals. International Statistical Review, 55:163–172.CrossRefGoogle Scholar
  15. Kennedy, P. (1981). Estimation with correctly interpreted dummy variables insemilogarithmic equations. The American Economic Review, 71(4):801.Google Scholar
  16. NIST (2008). Dataplot Reference Manual, Chapter 8. http://www.itl.nist.gov/.
  17. Shapiro, S. and Wilk, M. (1965). An analysis of variance test for normality. Biometrika, 52: 591–611.Google Scholar
  18. Thadewald, T. and Büning, H. (2007). Jarque–Bera test and its competitors for testing normality – a powercomparision. Journal of Applied Statistics, 34(1):87–105.CrossRefGoogle Scholar
  19. Walls, W. (2005). Modeling heavy tails and skewness in film returns. Applied Financial Economics, 15:1181–1188.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.American Society of Composers, Authors and PublishersNew YorkUSA

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