This article reviews several frameworks commonly used in modelling heavy-tailed densities and distributions in economics, finance, risk management, econometrics and statistics. The results and conclusions discussed in the article indicate that the presence of heavy tails can either reinforce or reverse the implications of a number of models in these fields, depending on the degree of heavy-tailedness.
KeywordsHeavy-tailed densities Heavy-tailed distributions Power laws Stable distributions Mean Variance Moments Semi-heavy tails Diversification Portfolio choice Value at risk Linear estimators Sample mean Efficiency Regression Robustness Dependence α-symmetric distributions Models with common shocks
The author gratefully acknowledges partial support by NSF grant SES-0821024, a Harvard Academy Junior Faculty Department grant and the Warburg Research Funds (Department of Economics, Harvard University).
- Anderson, C. 2006. The long tail. New York: Hyperion.Google Scholar
- Bouchard, J.-P., and M. Potters. 2004. Theory of financial risk and derivative pricing: From statistical physics to risk management, 2nd ed. Cambridge: Cambridge University Press.Google Scholar
- Embrechts, P., A. McNeil, and D. Straumann. 2002. Correlation and dependence in risk management: Properties and pitfalls. In Risk management: Value at risk and beyond, ed. M.A.H. Dempster, 176–223. Cambridge: Cambridge University Press.Google Scholar
- Granger, C.W.J., and D. Orr. 1972. Infinite variance and research strategy in time series analysis. Journal of the American Statistical Association 67: 275–285.Google Scholar
- Ibragimov, R. 2005a. New majorization theory in economics and martingale convergence results in econometrics. Ph.D. dissertation, Yale University, New Haven.Google Scholar
- Ibragimov, R. 2005b. Portfolio diversification and value at risk under thicktailedness, Harvard University Research Discussion Paper 2086. Available at: http://www.economics.harvard.edu/pub/hier/2005/HIER2086.pdf. Accessed 13 Jan 2009. Forthcoming in Quantitative Finance.
- Ibragimov, R., D. Jaffee, and J. Walden. 2008. Nondiversification traps in catastrophe insurance markets. Review of Financial Studies. Available at: http://rfs.oxfordjournals.org/cgi/content/abstract/hhn021v1. Accessed 13 Jan 2009.
- McCulloch, J.H. 1997. Measuring tail thickness to estimate the stable index alpha: A critique. Journal of Business and Economic Statistics 15: 74–81.Google Scholar
- McNeil, A.J., R. Frey, and P. Embrechts. 2005. Quantitative risk management: Concepts, techniques, and tools. Princeton: Princeton University Press.Google Scholar
- Rachev, S.T., and S. Mittnik. 2000. Stable Paretian models in finance. New York: Wiley.Google Scholar
- Rachev, S.T., C. Menn, and F.J. Fabozzi. 2005. Fat-tailed and skewed asset return distributions: Implications for risk management, portfolio selection, and option pricing. Hoboken: Wiley.Google Scholar
- Zolotarev, V.M. 1986. One-dimensional stable distributions. Providence: American Mathematical Society.Google Scholar