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ASYMPTOTIC THEORY FOR NONLINEAR QUANTILE REGRESSION UNDER WEAK DEPENDENCE

Published online by Cambridge University Press:  23 March 2015

Walter Oberhofer  and
Harry Haupt
Walter Oberhofer
Affiliation:
University of Regensburg
Harry Haupt*
Affiliation:
University of Passau
*
*Address correspondence to Harry Haupt, Department of Statistics, University of Passau, 94030 Passau, Germany; e-mail: [email protected].
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Abstract

This paper studies the asymptotic properties of the nonlinear quantile regression model under general assumptions on the error process, which is allowed to be heterogeneous and mixing. We derive the consistency and asymptotic normality of regression quantiles under mild assumptions. First-order asymptotic theory is completed by a discussion of consistent covariance estimation.

Type
ARTICLES
Information
Econometric Theory , Volume 32 , Issue 3 , June 2016 , pp. 686 - 713
Copyright
Copyright © Cambridge University Press 2015 

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References

REFERENCES

Andrews, D.W.K. (1987) Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica 55, 1465–1471.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.Google Scholar
Andrews, D.W.K. (1994a) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62, 43–72.Google Scholar
Andrews, D.W.K. (1994b) Empirical process methods in econometrics. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 2247–2294. Elsevier Science.Google Scholar
Buchinsky, M. (1995) Estimating the asymptotic covariance matrix for quantile regression models. Journal of Econometrics 68, 303–338.CrossRefGoogle Scholar
Buchinsky, M. (1998) Recent advances in quantile regression. Journal of Human Resources 33, 88–126.Google Scholar
Cai, Z. (2002) Regression quantiles for time series. Econometric Theory 18, 169–192.Google Scholar
Chamberlain, G. (1994) Quantile regression, censoring, and the structure of wages. In Sims, C. (ed.), Advances in Econometrics: Sixth World Congress, vol. 1, pp. 171–208. Cambridge University Press.CrossRefGoogle Scholar
Chen, X., Koenker, R., & Xiao, Z. (2009) Copula-based nonlinear quantile autoregression. Econometrics Journal 12, S50–S67.CrossRefGoogle Scholar
Chen, L.-A., Tran, L.T., & Lin, L.-C. (2004) Symmetric regression quantile and its application to robust estimation for the nonlinear regression model. Journal of Statistical Planning and Inference 126, 423–440.Google Scholar
Chen, N. & Zhou, S. (2010) Simulation-based estimation of cycle time using quantile regression. IIE Transactions 43, 176–191.CrossRefGoogle Scholar
Chernozhukov, V. & Umantsev, L. (2001) Conditional value-at-risk: Aspects of modeling and estimation. Empirical Economics 26, 271–292.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.Google Scholar
Dedecker, J., Doukhan, P., Lang, G., Leon, J.R., Louhichi, S., & Prieur, C. (2007) Weak Dependence: With Examples and Applications. Springer-Verlag.CrossRefGoogle Scholar
De Gooijer, J.G. & Zerom, D. (2003) On additive conditional quantiles with high-dimensional covariates. Journal of the American Statistical Association 98, 135–146.CrossRefGoogle Scholar
Dominicy, Y., Hörmann, S., Ogata, H., & Veredas, D. (2012) On sample marginal quantiles for stationary processes. Statistics and Probability Letters 83, 28–36.Google Scholar
Doukhan, P. (1994) Mixing. Springer-Verlag.Google Scholar
Doukhan, P. & Louhichi, S. (1999) A new weak dependence condition and applications to moment inequalities. Stochastic Processes and their Applications 84, 313–342.Google Scholar
El Ghouch, A. & Genton, M.G. (2009) Local polynomial quantile regression with parametric features. Journal of the American Statistical Association 104, 1416–1429.Google Scholar
Engle, R.F. & Manganelli, S. (2004) CAViaR: Conditional autoregressive value at risk by regression quantiles. Journal of Business and Economics Statistics 22, 367–381.Google Scholar
Fitzenberger, B. (1997) The moving blocks bootstrap and robust inference for linear least squares and quantile regressions. Journal of Econometrics 82, 235–287.Google Scholar
Fitzenberger, B., Wilke, R., & Zhang, X. (2010) Implementing Box-Cox quantile regression. Econometric Reviews 29, 158–181.CrossRefGoogle Scholar
Gallant, A.R. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Basil Blackwell.Google Scholar
Haupt, H. & Oberhofer, W. (2009) On asymptotic normality in nonlinear regression. Statistics and Probability Letters 79, 848–849.Google Scholar
He, X. & Shao, Q.-M. (1996) A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs. Annals of Statistics 24, 2608–2630.Google Scholar
Hendricks, W. & Koenker, R. (1992) Hierarchical spline models for conditional quantiles and the demand for electricity. Journal of the American Statistical Association 87, 58–68.Google Scholar
Huber, P.J. (1967) Behavior of maximum likelihood estimates under nonstandard conditions. In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 129–156, Berkeley. UC Press.Google Scholar
Ioannides, D.A. (2004) Fixed design regression quantiles for time series. Statistics and Probability Letters 68, 235–245.CrossRefGoogle Scholar
Jureckova, J. & Prochazka, B. (1994) Regression quantiles and trimmed least squares estimators in nonlinear regression models. Journal of Nonparametric Statistics 3, 201–222.Google Scholar
Karlsson, A. (2007) Nonlinear quantile regression estimation of longitudinal data. Communications in Statistics – Simulation and Computation 37, 114–131.Google Scholar
Karlsson, A. (2009) Bootstrap methods for bias correction and confidence interval estimation for nonlinear quantile regression of longitudinal data. Journal of Statistical Computation and Simulation 79, 1205–1218.CrossRefGoogle Scholar
Kim, T.S., Kim, H.K., & Hur, S. (2002) Asymptotic properties of a particular nonlinear regression quantile estimation. Statistics and Probability Letters 60, 387–394.Google Scholar
Knight, K. (1998) Limiting distributions for L 1 regression estimators under general conditions. Annals of Statistics 26, 755–770.Google Scholar
Knight, K. (1999) Asympotics for L 1-estimators of regression parameters under heteroscedasticity. Canadian Journal of Statistics 27, 497–507.Google Scholar
Koenker, R. (2005) Quantile Regression. Econometric Society Monographs No. 38. Cambridge University Press.Google Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 33–50.Google Scholar
Koenker, R. & Bassett, G. (1982) Robust tests for heteroscedasticity based on regression quantiles. Econometrica 50, 43–61.Google Scholar
Koenker, R. & Hallock, K.F. (2001) Quantile regression. Journal of Economic Perspectives 15, 143–156.Google Scholar
Koenker, R. & Park, B. (1994) An interior point algorithm for nonlinear quantile regression. Journal of Econometrics 71, 265–283.Google Scholar
Komunjer, I. (2005) Quasi-maximum likelihood estimation for conditional quantiles. Journal of Econometrics 128, 137–164.Google Scholar
Li, T.-H. (2012) Quantile periodograms. Journal of the American Statistical Association 107, 765–776.Google Scholar
Liese, F. & Vajda, I. (2003) A general asymptotic theory of M-estimators. I. Mathematical Methods of Statistics 12, 454–477.Google Scholar
Liese, F. & Vajda, I. (2004) A general asymptotic theory of M-estimators. II. Mathematical Methods of Statistics 13, 82–95.Google Scholar
Mukherjee, K. (1999) Asymptotics of quantiles and rank scores in nonlinear time series. Journal of Time Series Analysis 20, 173–192.Google Scholar
Mukherjee, K. (2000) Linearization of randomly weighted empiricals under long range dependence with applications to nonlinear regression quantiles. Econometric Theory 16, 301–323.Google Scholar
Newey, W.K. & McFadden, D.L. (1994) Large sample estimation and hypothesis testing. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 2111–2245. Elsevier Science.Google Scholar
Nze, P.A. & Doukhan, P. (2004) Weak dependence: Models and applications to econometrics. Econometric Theory 20, 169–192.Google Scholar
Oberhofer, W. (1982) The consistency of nonlinear regression minimizing the L 1 norm. Annals of Statistics 10, 316–319.Google Scholar
Oberhofer, W. & Haupt, H. (2005) The asymptotic distribution of the unconditional quantile estimator under dependence. Statistics and Probability Letters 73, 243–250.Google Scholar
Peracchi, F. (2001) Econometrics. Wiley.Google Scholar
Pfanzagl, J. (1969) On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–272.Google Scholar
Phillips, P.C.B. (1991) A shortcut to LAD estimator asymptotics. Econometric Theory 7, 450–463.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1989) A uniform law of large numbers for dependent and heterogeneous data processes. Econometrica 57, 675–683.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1994) Generic uniform convergence and equicontinuity concepts for random functions. Journal of Econometrics 60, 23–63.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1997) Dynamic Nonlinear Econometric Models: Asymptotic Theory. Springer-Verlag.Google Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag.Google Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–199.Google Scholar
Portnoy, S. (1991) Asymptotic behavior of regression quantiles in non-stationary, dependent cases. Journal of Multivariate Analysis 38, 100–113.Google Scholar
Powell, J.L. (1991) Estimation of monotonic regression models under quantile restrictions. In Barnett, W., Powell, J.L., & Tauchen, G. (eds.), Nonparametric and Semiparametric Methods in Econometrics and Statistics, pp. 357–384. Cambridge University Press.Google Scholar
Powell, J.L. (1994) Estimation of semiparametric models. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 2443–2521. Elsevier Science.Google Scholar
Prakasa Rao, B.L.S. (1984) The rate of convergence of the least squares estimator in a nonlinear regression model with dependent errors. Journal of Multivariate Analysis 14, 315–322.Google Scholar
Prakasa Rao, B.L.S. (1987) Asymptotic Theory of Statistical Inference. Wiley.Google Scholar
Richardson, G.D. & Bhattacharyya, B.B. (1987) Consistent L 1-estimators in nonlinear regression for a noncompact parameter space. Sankhya Series A 49, 377–387.Google Scholar
Robinson, P.M. (1997) Large-sample inference for nonparametric regression with dependent errors. Annals of Statistics 25, 2054–2083.Google Scholar
Rogers, A.J. (2001) Least absolute deviations regression under nonstandard conditions. Econometric Theory 17, 820–852.Google Scholar
Roussas, G.G., Tran, L.T., & Ioannides, D.A. (1992) Fixed design regression for time series: Asymptotic normality. Journal of Multivariate Analysis 40, 262–291.Google Scholar
Tran, L.T., Roussas, G.G., Yakowitz, S., & Van, B.T. (1996) Fixed-design regression for linear time series. Annals of Statistics 24, 975–991.Google Scholar
Wang, J. (1995) Asymptotic normality of L 1-estimators in nonlinear regression. Journal of Multivariate Analysis 54, 227–238.Google Scholar
Wang, J. (1996) Asymptotics of least-squares estimators for constrained nonlinear regression. Annals of Statistics 4, 1316–1326.Google Scholar
Weiss, A.A. (1991) Estimating nonlinear dynamic models using least absolute error estimation. Econometric Theory 7, 46–68.Google Scholar
White, H. (1994) Estimation, Inference and Specification Analysis. Econometric Society Monographs No. 22. Cambridge University Press.Google Scholar
Withers, C.S. (1981) Central limit theorems for dependent variables. I. Zeitschrift fr Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 509–534.Google Scholar
Wooldridge, J.M. (2010) Econometric Analysis of Cross Section and Panel Data, 2nd ed.MIT Press.Google Scholar
Wu, W.B. & Mielniczuk, J. (2010) A new look at measuring dependence. In Doukhan, P. et al. . (eds.), Dependence in Probability and Statistics. Lecture Notes in Statistics, vol. 200, pp. 123–142. Springer.Google Scholar
Yu, K., Lu, Z., & Stander, J. (2003) Quantile regression: Applications and current research areas. The Statistician 52, 331–350.Google Scholar
Zhao, Q. (2001) Asymptotic efficient median regression in the presence of heteroscedasticity of unknown form. Econometric Theory 17, 765–784.Google Scholar
Zhao, Z. & Xiao, Z. (2014) Efficient regressions via optimally combining quantile information. Econometric Theory 30, 1272–1314.Google Scholar
Zheng, J.X. (1998) A consistent nonparametric test of parametric regression models under conditional quantile restrictions. Econometric Theory 14, 123–138.Google Scholar
Zhou, Z. & Shao, X. (2013) Inference for linear models with dependent errors. Journal of the Royal Statistical Society B 75, 323–343.Google Scholar