The Annals of Statistics

Extremal quantile regression

Victor Chernozhukov

Full-text: Open access

Abstract

Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. It can be used to measure the effect of covariates not only in the center of a distribution, but also in the upper and lower tails. This paper develops a theory of quantile regression in the tails. Specifically, it obtains the large sample properties of extremal (extreme order and intermediate order) quantile regression estimators for the linear quantile regression model with the tails restricted to the domain of minimum attraction and closed under tail equivalence across regressor values. This modeling setup combines restrictions of extreme value theory with leading homoscedastic and heteroscedastic linear specifications of regression analysis. In large samples, extreme order regression quantiles converge weakly to arg min functionals of stochastic integrals of Poisson processes that depend on regressors, while intermediate regression quantiles and their functionals converge to normal vectors with variance matrices dependent on the tail parameters and the regressor design.

Article information

Source
Ann. Statist., Volume 33, Number 2 (2005), 806-839.

Dates
First available in Project Euclid: 26 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1117114337

Digital Object Identifier
doi:10.1214/009053604000001165

Mathematical Reviews number (MathSciNet)
MR2163160

Zentralblatt MATH identifier
1068.62063

Subjects
Primary: 62G32: Statistics of extreme values; tail inference 62G30: Order statistics; empirical distribution functions 62P20: Applications to economics [See also 91Bxx]
Secondary: 62E30 62J05: Linear regression

Keywords
Conditional quantile estimation regression extreme value theory

Citation

Chernozhukov, Victor. Extremal quantile regression. Ann. Statist. 33 (2005), no. 2, 806--839. doi:10.1214/009053604000001165. https://projecteuclid.org/euclid.aos/1117114337


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References

  • Abrevaya, J. (2001). The effects of demographics and maternal behavior on the distribution of birth outcomes. Empirical Economics 26 247–257.
  • Buchinsky, M. (1994). Changes in the U.S. wage structure 1963–1987: Application of quantile regression. Econometrica 62 405–458.
  • Cade, B. (2003). Quantile regression models of animal habitat relastionships. Ph.D. dissertation, Colorado State Univ. Available at www.fort.usgs.gov.
  • Chamberlain, G. (1994). Quantile regression, censoring, and the structure of wages. In Advances in Econometrics: Sixth World Congress (C. Sims, ed.). Cambridge Univ. Press.
  • Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760–777.
  • Chaudhuri, P., Doksum, K. and Samarov, A. (1997). On average derivative quantile regression. Ann. Statist. 25 715–744.
  • Chernozhukov, V. (1998). Nonparametric extreme regression quantiles. Working paper, Stanford Univ. Presented at Princeton Econometrics Seminar, December 1998.
  • Chernozhukov, V. (1999). Conditional extremes and near-extremes: Estimation, inference, and economic applications. Ph.D. dissertation, Dept. Economics, Stanford Univ. Available at www.mit.edu/~vchern.
  • Chernozhukov, V. and Umantsev, L. (2001). Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics 26 271–292.
  • Davidson, J. (1994). Stochastic Limit Theory. Oxford Univ. Press, New York.
  • de Haan, L. (1984). Slow variation and characterization of domains of attraction. In Statistical Extremes and Applications (I. Tiago de Oliveira, ed.) 31–48. Reidel, Dordrecht.
  • de Haan, L. and Rootzén, H. (1993). On the estimation of high quantiles. J. Statist. Plann. Inference 35 1–13.
  • Dekkers, A. and de Haan, L. (1989). On the estimation of the extreme-value index and large quantile estimation. Ann. Statist. 17 1795–1832.
  • Doksum, K. A. and Gasko, M. (1990). On a correspondence between models in binary regression analysis and in survival analysis. Internat. Statist. Rev. 58 243–252.
  • Donald, S. G. and Paarsch, H. J. (1993). Piecewise pseudo-maximum likelihood estimation in empirical models of auctions. Internat. Econom. Rev. 34 121–148.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
  • Feigin, P. D. and Resnick, S. I. (1994). Limit distributions for linear programming time series estimators. Stochastic Process. Appl. 51 135–165.
  • Geyer, C. J. (1996). On the asymptotics of convex stochastic optimization. Technical report, Dept. Statistics, Univ. Minnesota.
  • Gutenbrunner, C. and Jurečková, J. (1992). Regression rank scores and regression quantiles. Ann. Statist. 20 305–330.
  • He, X. (1997). Quantile curves without crossing. Amer. Statist. 51 186–192.
  • Hendricks, W. and Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity. J. Amer. Statist. Assoc. 87 58–68.
  • Huber, P. J. (1973). Robust regression: Asymptotics, conjectures and Monte Carlo. Ann. Statist. 1 799–821.
  • Knight, K. (1998). Limiting distributions for $L\sb 1$ regression estimators under general conditions. Ann. Statist. 26 755–770.
  • Knight, K. (1999). Epi-convergence and stochastic equisemicontinuity. Technical report, Dept. Statistics, Univ. Toronto, Available at www.utstat.toronto.edu/keith/papers/.
  • Knight, K. (2001). Limiting distributions of linear programming estimators. Extremes 4 87–103.
  • Koenker, R. and Bassett, G. S. (1978). Regression quantiles. Econometrica 46 33–50.
  • Koenker, R. and Bassett, G. S. (1982). Robust tests for heteroscedasticity based on regression quantiles. Econometrica 50 43–61.
  • Koenker, R. and Geling, O. (2001). Reappraising medfly longevity: A quantile regression survival analysis. J. Amer. Statist. Assoc. 96 458–468.
  • Koenker, R. and Portnoy, S. (1987). ${L}$-estimation for linear models. J. Amer. Statist. Assoc. 82 851–857.
  • Korostelëv, A. P., Simar, L. and Tsybakov, A. B. (1995). Efficient estimation of monotone boundaries. Ann. Statist. 23 476–489.
  • Laplace, P.-S. (1818). Théorie Analytique des Probabilités. Éditions Jacques Gabay (1995), Paris.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Meyer, R. M. (1973). A Poisson-type limit theorem for mixing sequences of dependent “rare” events. Ann. Probab. 1 480–483.
  • Pickands, III, J. (1975). Statistical inference using extreme order statistics. Ann. Statist. 3 119–131.
  • Portnoy, S. (1991a). Asymptotic behavior of regression quantiles in nonstationary, dependent cases. J. Multivariate Anal. 38 100–113.
  • Portnoy, S. (1991b). Asymptotic behavior of the number of regression quantile breakpoints. SIAM J. Sci. Statist. Comput. 12 867–883.
  • Portnoy, S. and Jurečková, J. (1999). On extreme regression quantiles. Extremes 2 227–243.
  • Portnoy, S. and Koenker, R. (1997). The Gaussian hare and the Laplacian tortoise (with discussion). Statist. Sci. 12 279–300.
  • Powell, J. L. (1986). Censored regression quantiles. J. Econometrics 32 143–155.
  • Rao, C. R. (1965). Linear Statistical Inference and Its Applications. Wiley, New York.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Robinson, P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185–207.
  • Smith, R. L. (1994). Nonregular regression. Biometrika 81 173–183.
  • Tsay, R. S. (2002). Analysis of Financial Time Series. Wiley, New York.
  • Watts, V., Rootzén, H. and Leadbetter, M. R. (1982). On limiting distributions of intermediate order statistics from stationary sequences. Ann. Probab. 10 653–662.