The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Extremal Quantiles and Value-at-Risk

  • Victor Chernozhukov
  • Songzi Du
Reference work entry


This article examines the theory and empirics of extremal quantiles in economics, in particular value-at-risk. The theory of extremes has gone through remarkable developments and produced valuable empirical findings since the late 1980s. We emphasize conditional extremal quantile models and methods, which have applications in many areas of economic analysis. Examples of applications include the analysis of factors of high risk in finance and risk management, the analysis of socio-economic factors that contribute to extremely low infant birthweights, efficiency analysis in industrial organization, the analysis of reservation rules in economic decisions, and inference in structural auction models.


Bootstrap Extremal conditional quantiles Extremal quantiles Gamma variables Inference Maximum likelihood Production frontiers Quantile regression function Value-at-risk 

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We thank Emily Gallagher, Greg Fischer and Raymond Guiteras for their help and valuable comments.


  1. Abrevaya, J. 2001. The effect of demographics and maternal behavior on the distribution of birth outcomes. Empirical Economics 26: 247–259.CrossRefGoogle Scholar
  2. Aigner, D.J., and S.F. Chu. 1968. On estimating the industry production function. American Economic Review 58: 826–839.Google Scholar
  3. Aigner, D.J., T. Amemiya, and D.J. Poirier. 1976. On the estimation of production frontiers: Maximum likelihood estimation of the parameters of a discontinuous density function. International Economic Review 17: 377–396.CrossRefGoogle Scholar
  4. Arrow, K., T. Harris, and J. Marschak. 1951. Optimal inventory policy. Econometrica 19: 205–272.Google Scholar
  5. Bickel, P., and D. Freedman. 1981. Some asymptotic theory for the bootstrap. Annals of Statistics 9: 1196–1217.CrossRefGoogle Scholar
  6. Caballero, R., and E. Engel. 1999. Explaining investment dynamics in U.S. manufacturing: A generalized (S,s) approach. Econometrica 67: 783–826.CrossRefGoogle Scholar
  7. Chernozhukov, V. 2005. Extremal quantile regression. Annals of Statistics 33: 806–839.CrossRefGoogle Scholar
  8. Chernozhukov, V. 2006. Inference for extremal conditional quantile models, with an application to birthweights. Working paper, Department of Economics, Massachusetts Institute of Technology.Google Scholar
  9. Chernozhukov, V., and L. Umantsev. 2001. Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics 26: 271–293.CrossRefGoogle Scholar
  10. de Haan, L. 1970. On regular variation and its applications to the weak convergence, Tract no. 2. Amsterdam: Mathematical Centre.Google Scholar
  11. Dekkers, A., and L. de Haan. 1989. On the estimation of the extreme-value index and large quantile estimation. Annals of Statistics 17: 1795–1832.CrossRefGoogle Scholar
  12. Donald, S.G., and H.J. Paarsch. 2002. Superconsistent estimation and inference in structural econometric models using extreme order statistics. Journal of Econometrics 109: 305–340.CrossRefGoogle Scholar
  13. Embrechts, P., Klüppelberg, C. and Mikosch, T. 1997. Modelling extremal events, vol. 33 of Applications of mathematics. Berlin: Springer.Google Scholar
  14. Fama, E.F. 1965. The behavior of stock market prices. Journal of Business 38: 34–105.CrossRefGoogle Scholar
  15. Fox, M., and H. Rubin. 1964. Admissibility of quantile estimates of a single location parameter. Annals of Mathematics and Statistics 35: 1019–1030.CrossRefGoogle Scholar
  16. Gnedenko, B. 1943. Sur la distribution limité du terme d’ une série alétoire. Annals of Mathematics 44: 423–453.CrossRefGoogle Scholar
  17. Hill, B.M. 1975. A simple general approach to inference about the tail of a distribution. Annals of Statistics 3: 1163–1174.CrossRefGoogle Scholar
  18. Jansen, D.W., and C.G. de Vries. 1991. On the frequency of large stock returns: Putting booms and busts into perspective. Review of Economics and Statistics 73: 18–24.CrossRefGoogle Scholar
  19. Knight, K. 2001. Limiting distributions of linear programming estimators. Extremes 4(2): 87–103.CrossRefGoogle Scholar
  20. Koenker, R. 2006. Quantreg: Quantile regression. R package version 3.90. Online. Available at Accessed 23 April 2007.
  21. Koenker, R., and G.S. Bassett. 1978. Regression quantiles. Econometrica 46: 33–50.CrossRefGoogle Scholar
  22. Laplace, P.-S. 1818. Théorie analytique des probabilités. Paris: Éditions Jacques Gabay, 1995.Google Scholar
  23. Leadbetter, M.R., G. Lindgren, and H. Rootzén. 1983. Extremes and related properties of random sequences and processes. New York/Berlin: Springer.CrossRefGoogle Scholar
  24. Longin, F.M. 1996. The asymptotic distribution of extreme stock market returns. Journal of Business 69: 383–408.CrossRefGoogle Scholar
  25. Mandelbrot, M. 1963. The variation of certain speculative prices. Journal of Business 36: 394–419.CrossRefGoogle Scholar
  26. Meyer, R.M. 1973. A poisson-type limit theorem for mixing sequences of dependent ‘rare’ events. Annals of Probability 1: 480–483.CrossRefGoogle Scholar
  27. Pareto, V. 1964. Cours d’économie politique. Genéve: Droz.CrossRefGoogle Scholar
  28. Pickands, J. III. 1975. Statistical inference using extreme order statistics. Annals of Statistics 3: 119–131.CrossRefGoogle Scholar
  29. Portnoy, S., and J. Jurečková 1999. On extreme regression quantiles. Extremes 2: 227–243 (2000).Google Scholar
  30. Praetz, V. 1972. The distribution of share price changes. Journal of Business 45: 49–55.CrossRefGoogle Scholar
  31. Quetelet, A. 1871. Anthropométrie. Brussels: Muquardt.Google Scholar
  32. Roy, A.D. 1952. Safety first and the holding of assets. Econometrica 20: 431–449.CrossRefGoogle Scholar
  33. Sen, A. 1973. On economic inequality. New York: Oxford University Press.CrossRefGoogle Scholar
  34. Timmer, C.P. 1971. Using a probabilistic frontier production function to measure technical efficiency. Journal of Political Economy 79: 776–794.CrossRefGoogle Scholar
  35. Zipf, G. 1949. Human behavior and the principle of last effort. Cambridge, MA: Addison-Wesley.Google Scholar

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Victor Chernozhukov
    • 1
  • Songzi Du
    • 1
  1. 1.