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SMOOTHED EMPIRICAL LIKELIHOOD METHODS FOR QUANTILE REGRESSION MODELS

Published online by Cambridge University Press:  09 February 2006

Yoon-Jae Whang
Yoon-Jae Whang
Affiliation:
Seoul National University and Cowles Foundation for Research in Economics, Yale University
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Abstract

This paper considers an empirical likelihood method to estimate the parameters of the quantile regression (QR) models and to construct confidence regions that are accurate in finite samples. To achieve the higher order refinements, we smooth the estimating equations for the empirical likelihood. We show that the smoothed empirical likelihood estimator is first-order asymptotically equivalent to the standard QR estimator and establish that confidence regions based on the smoothed empirical likelihood ratio have coverage errors of order n−1 and may be Bartlett corrected to produce regions with errors of order n−2, where n denotes the sample size. Our result is an extension of the previous result of Chen and Hall (1993, Annals of Statistics 21, 1166–1181) to the regression context. Monte Carlo experiments suggest that the smoothed empirical likelihood confidence regions may be more accurate in small samples than the confidence regions that can be constructed from the smoothed bootstrap method recently suggested by Horowitz (1998, Econometrica 66, 1327–1351).I thank the co-editor Bruce Hansen and anonymous referees for valuable suggestions and comments. I also thank Song X. Chen, Joel Horowitz, Yuichi Kitamura, Oliver Linton, Whitney Newey, Peter Phillips, and Richard Smith for helpful comments. Parts of this paper were written while I was visiting the Cowles Foundation at Yale University, whose hospitality is gratefully acknowledged. Young-Hyun Cho has provided excellent research assistance. This work was supported by the Korea Research Foundation grant KRF 2003-041-B00072.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 2 , April 2006 , pp. 173 - 205
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Baggerly, K.A. (1998) Empirical likelihood as a goodness-of-fit measure. Biometrika 85, 535–547.CrossRefGoogle Scholar
Beran, R. (1987) Prepivoting to reduce level error of confidence sets. Biometrika 74, 457–468.CrossRefGoogle Scholar
Beran, R. (1988) Prepivoting test statistics: A bootstrap view of asymptotic refinements. Journal of the American Statistical Association 83, 687–697.CrossRefGoogle Scholar
Bertail, P. (2003) Empirical Likelihood in Some Semiparametric Models. Manuscript, Laboratoire de Statistique, CREST.
Bhattacharya, R.N. & J.K. Ghosh (1978) On the validity of the formal Edgeworth expansion. Annals of Statistics 6, 434–451.CrossRefGoogle Scholar
Bhattacharya, R.N. & R.R. Rao (1976) Normal Approximation and Asymptotic Expansions. Wiley.
Bravo, F. (2002) Testing linear restriction in linear models with empirical likelihood. Econometrics Journal 5, 104–130.CrossRefGoogle Scholar
Bravo, F. (2004) Empirical likelihood based inference with applications to some econometric models. Econometric Theory 20, 231–264.Google Scholar
Buchinsky, M. (1995) Estimating the asymptotic covariance matrix for quantile regression models: A Monte Carlo study. Journal of Econometrics 68, 303–338.CrossRefGoogle Scholar
Buchinsky, M. (2000) Recent advances in quantile regression models: A practical guideline for empirical research. Journal of Human Resources 33, 88–126.Google Scholar
Chen, S.X. & H. Cui (2002) On Bartlett Correction of Empirical Likelihood in the Presence of Nuisance Parameters. Manuscript, Department of Statistics, Iowa State University.
Chen, S.X. & H. Cui (2003) On the Second Order Properties of Empirical Likelihood for Generalized Estimating Equations. Manuscript, Department of Statistics, Iowa State University.
Chen, S.X. & P. Hall (1993) Smoothed empirical likelihood confidence intervals for quantiles. Annals of Statistics 21, 1166–1181.CrossRefGoogle Scholar
De Angelis, D., P. Hall, & G.A. Young (1993) Analytical and bootstrap approximations to estimator distributions in L1 regressions. Journal of the American Statistical Association 88, 1310–1316.Google Scholar
DiCiccio, T.J., P. Hall, & J.P. Romano (1991) Empirical likelihood is Bartlett-correctable. Annals of Statistics 19, 1053–1061.CrossRefGoogle Scholar
Donald, S.G., G.W. Imbens, & W.K. Newey (2003) Empirical likelihood estimation and consistent tests with conditional moment restrictions. Journal of Econometrics 117, 55–93.CrossRefGoogle Scholar
Efron, B. (1979) Bootstrap methods: Another look at the jackknife. Annals of Statistics 7, 1–26.CrossRefGoogle Scholar
Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. Society for Industrial Applied Mathematics.
Guggenberger, P. & R.J. Smith (2003) Generalized empirical likelihood estimators and tests under partial, weak and strong identification. Manuscript, Department of Economics, University of California at Los Angeles.
Hahn, J. (1995) Bootstrapping quantile regression estimators. Econometric Theory 11, 105–121.CrossRefGoogle Scholar
Hall, P. (1986) On the bootstrap and confidence intervals. Annals of Statistics 14, 1431–1452.CrossRefGoogle Scholar
Hall, P. (1987) On the bootstrap and likelihoood-based confidence regions. Biometrika 74, 481–493.CrossRefGoogle Scholar
Hall, P. (1992) The Bootstrap and Edgeworth Expansions. Springer-Verlag.
Hall, P. & J.L. Horowitz (1990) Bandwidth selection in semiparametric estimation of censored linear regression models. Econometric Theory 6, 123–150.CrossRefGoogle Scholar
Hall, P. & B. LaScalar (1990) Methodology and algorithms of empirical likelihood. International Statistical Review 58, 109–127.CrossRefGoogle Scholar
Horowitz, J.L. (1997) Bootstrap methods in econometrics: Theory and numerical performance. In D. Kreps & K.W. Wallis (eds.), Advances in Economics and Econometrics: Theory and Applications, vol. 3, pp. 188–222. Cambridge University Press.
Horowitz, J.L. (1998) Bootstrap methods for median regression models. Econometrica 66, 1327–1351.CrossRefGoogle Scholar
Horowitz, J.L. (2001) The bootstrap. In J.J. Heckman & E. Leamer (eds.), Handbook of Econometrics, vol. 5, pp. 3159–3228. North-Holland.
Imbens, G.W., R.H. Spady, & P. Johnson (1998) Information theoretic approaches to inference in moment condition models. Econometrica 66, 333–357.CrossRefGoogle Scholar
Jing, B.-Y. & A.T.A. Wood (1996) Exponential empirical likelihood is not Bartlett correctable. Annals of Statistics 24, 365–369.CrossRefGoogle Scholar
Kitamura, Y. (1997) Empirical likelihood methods with weakly dependent processes. Annals of Statistics 25, 2084–2102.CrossRefGoogle Scholar
Kitamura, Y. (2001) Asymptotic optimality of empirical likelihood for testing moment restrictions. Econometrica 69, 1661–1672.CrossRefGoogle Scholar
Kitamura, Y. & M. Stutzer (1997) An information-theoretic alternative to generalized method of moment estimation. Econometrica 65, 861–874.CrossRefGoogle Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.
Koenker, R. & G. Bassett (1978) Regression quantiles. Econometrica 46, 33–50.CrossRefGoogle Scholar
Koenker, R. & G. Bassett (1982) Robust tests for heteroskedasticity based on regression quantiles. Econometrica 50, 43–61.CrossRefGoogle Scholar
Monti, A.C. (1997) Empirical likelihood confidence regions in time series models. Biometrika 84, 395–405.CrossRefGoogle Scholar
Moon, H.R. & F. Schorfheide (2003) Empirical Likelihood Estimation Based on Inequality Constraints. Manuscript, Department of Economics, University of Southern California.
Newey, W.K. & R.J. Smith (2004) Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72, 219–256.CrossRefGoogle Scholar
Otsu, T. (2003) Empirical Likelihood for Quantile Regression. Manuscript, Department of Economics, University of Wisconsin.
Owen, A. (1988) Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75, 237–249.CrossRefGoogle Scholar
Owen, A.B. (1990) Empirical likelihood ratio confidence regions. Annals of Statistics 18, 90–120.CrossRefGoogle Scholar
Owen, A.B. (1991) Empirical likelihood for linear models. Annals of Statistics 19, 1725–1747.CrossRefGoogle Scholar
Owen, A.B. (2001) Empirical Likelihood. Chapman and Hall.
Powell, J.L. (1984) Least absolute deviations estimation for the censored regression model. Journal of Econometrics 25, 303–325.CrossRefGoogle Scholar
Powell, J.L. (1986) Censored regression quantiles. Journal of Econometrics 32, 143–155.CrossRefGoogle Scholar
Qin, J. & J. Lawless (1994) Empirical likelihood and general estimating equations. Annals of Statistics 22, 300–325.CrossRefGoogle Scholar
Su, L. & H. White (2003) Testing Conditional Independence via Empirical Likelihood. Manuscript, Department of Economics, University of California at San Diego.
van der Vaart, A.W. & J.A. Wellner (1996) Weak Convergence and Empirical Processes. Springer-Verlag.
Wilks, S.S. (1938) The large-sample distribution of the likelihood ratio for testing composite hypothesis. Annals of Mathematical Statistics 9, 60–62.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap and other resampling methods in regression analysis (with discussion). Annals of Statistics 14, 1261–1350.CrossRefGoogle Scholar