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.
Abrevaya, J. 2001. The effect of demographics and maternal behavior on the distribution of birth outcomes. Empirical Economics 26: 247–259.CrossRefGoogle Scholar
Aigner, D.J., and S.F. Chu. 1968. On estimating the industry production function. American Economic Review 58: 826–839.Google Scholar
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
Arrow, K., T. Harris, and J. Marschak. 1951. Optimal inventory policy. Econometrica 19: 205–272.Google Scholar
Bickel, P., and D. Freedman. 1981. Some asymptotic theory for the bootstrap. Annals of Statistics 9: 1196–1217.CrossRefGoogle Scholar
Caballero, R., and E. Engel. 1999. Explaining investment dynamics in U.S. manufacturing: A generalized (S,s) approach. Econometrica 67: 783–826.CrossRefGoogle Scholar
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
Chernozhukov, V., and L. Umantsev. 2001. Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics 26: 271–293.CrossRefGoogle Scholar
de Haan, L. 1970. On regular variation and its applications to the weak convergence, Tract no. 2. Amsterdam: Mathematical Centre.Google Scholar
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
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
Embrechts, P., Klüppelberg, C. and Mikosch, T. 1997. Modelling extremal events, vol. 33 of Applications of mathematics. Berlin: Springer.Google Scholar