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SPECIFICATION TESTING IN NONPARAMETRIC INSTRUMENTAL QUANTILE REGRESSION

Published online by Cambridge University Press:  07 January 2020

Christoph Breunig
Christoph Breunig*
Affiliation:
Emory University
*
Address correspondence to Christoph Breunig, Department of Economics, Emory University, Rich Memorial Building, Atlanta, GA30322, USA; e-mail: christoph.breunig@emory.edu.
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Abstract

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There are many environments in econometrics which require nonseparable modeling of a structural disturbance. In a nonseparable model with endogenous regressors, key conditions are validity of instrumental variables and monotonicity of the model in a scalar unobservable variable. Under these conditions the nonseparable model is equivalent to an instrumental quantile regression model. A failure of the key conditions, however, makes instrumental quantile regression potentially inconsistent. This article develops a methodology for testing the hypothesis whether the instrumental quantile regression model is correctly specified. Our test statistic is asymptotically normally distributed under correct specification and consistent against any alternative model. In addition, test statistics to justify the model simplification are established. Finite sample properties are examined in a Monte Carlo study and an empirical illustration is provided.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 4 , August 2020 , pp. 583 - 625
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Footnotes

Parts of this article derive from my doctoral dissertation, completed under the guidance of Enno Mammen. I would like to thank Liangjun Su and four anonymous referees for excellent comments and suggestions that greatly improved the article. I also thank seminar participants at Boston College, Bristol, Mannheim, Toulouse School of Economics, University College London, WIAS Berlin, and Yale. I am also grateful for the support and hospitality of the Cowles Foundation.

References

REFERENCES

Adams, R.A. & Fournier, J.J. (2003) Sobolev Spaces. Pure and Applied Mathematics, vol. 140. Elsevier/Academic Press.Google Scholar
Ait-Sahalia, Y., Bickel, P.J., & Stoker, T.M. (2001) Goodness-of-fit tests for kernel regression with an application to option implied volatilities. Journal of Econometrics 105(2), 363–412.CrossRefGoogle Scholar
Angrist, J.D. & Lavy, V.C. (1999) Using Maimonides’ rule to estimate the effect of class size on scholastic achievement. The Quarterly Journal of Economics 114(2), 533–575.CrossRefGoogle Scholar
Awad, A.M. (1981) Conditional central limit theorems for martingales and reversed martingales. The Indian Journal of Statistics, Series A 43, 10–106.Google Scholar
Blundell, R. & Horowitz, J. (2007) A nonparametric test of exogeneity. Review of Economic Studies 74(4), 1035–1058.CrossRefGoogle Scholar
Breunig, C. (2015) Goodness-of-fit tests based on series estimators in nonparametric instrumental regression. Journal of Econometrics 184(2), 328–346.CrossRefGoogle Scholar
Chen, X. (2007) Large sample sieve estimation of semi-nonparametric models. In Heckman, J.J., Learner, E. (eds.), Handbook of Econometrics, pp. 5549–5632. Elsevier.CrossRefGoogle Scholar
Chen, X. & Christensen, T.M. (2015) Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions. Journal of Econometrics 188(2), 447–465.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. (2012) Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals. Econometrica 80(1), 277–321.Google Scholar
Chen, X. & Pouzo, D. (2015) Sieve Wald and QLR inferences on semi/nonparametric conditional moment models. Econometrica 83(3), 1013–1079.CrossRefGoogle Scholar
Chen, X. & Reiß, M. (2011) On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27(03), 497–521.CrossRefGoogle Scholar
Chen, X. & Santos, A. (2018) Overidentification in regular models. Econometrica 86(5), 1771–1817.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Chen, X., Chernozhukov, V., Lee, S., & Newey, W.K. (2014) Local identification of nonparametric and semiparametric models. Econometrica 82(2), 785–809.Google Scholar
Chernozhukov, V. & Hansen, C. (2005) An IV model of quantile treatment effects. Econometrica 73, 245–261.CrossRefGoogle Scholar
Chernozhukov, V., Imbens, G., & Newey, W.K. (2007) Instrumental variable estimation of nonseparable models. Journal of Econometrics 139(1), 4–14.CrossRefGoogle Scholar
Chesher, A. (2003) Identification in nonseparable models. Econometrica 71(5), 1405–1441.CrossRefGoogle Scholar
Chetverikov, D. & Wilhelm, D. (2017) Nonparametric instrumental variable estimation under monotonicity. Econometrica 85(4), 1303–1320.CrossRefGoogle Scholar
Chiappori, P.-A., Komunjer, I., & Kristensen, D. (2015) Nonparametric identification and estimation of transformation models. Journal of Econometrics 188(1), 22–39.CrossRefGoogle Scholar
Dunker, F., Florens, J.-P., Hohage, T., Johannes, J., & Mammen, E. (2014) Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression. Journal of Econometrics 178, 444–455.CrossRefGoogle Scholar
Escanciano, J.C. & Goh, S.-C. (2014). Specification analysis of linear quantile models. Journal of Econometrics 178, 495–507.CrossRefGoogle Scholar
Escanciano, J.C. & Velasco, C. (2010) Specification tests of parametric dynamic conditional quantiles. Journal of Econometrics 159(1), 209–221.CrossRefGoogle Scholar
Fève, F., Florens, J.-P., & Van Keilegom, I. (2018) Estimation of conditional ranks and tests of exogeneity in nonparametric nonseparable models. Journal of Business & Economic Statistics 36(2), 334–345.CrossRefGoogle Scholar
Gagliardini, P. & Scaillet, O. (2012) Nonparametric instrumental variable estimation of structural quantile effects. Econometrica 80(4), 1533–1562.Google Scholar
Gagliardini, P. & Scaillet, O. (2017) A specification test for nonparametric instrumental variable regression. Annals of Economics and Statistics/Annales d’Économie et de Statistique 128, 151–202.Google Scholar
Hanke, M., Neubauer, A., & Scherzer, O. (1995) A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numerische Mathematik 72(1), 21–37.CrossRefGoogle Scholar
He, X. & Shi, P. (1994) Convergence rate of b-spline estimators of nonparametric conditional quantile functions. Journal of Nonparametric Statistics 3(3–4), 299–308.CrossRefGoogle Scholar
Hoderlein, S. & Mammen, E. (2007) Identification of marginal effects in nonseparable models without monotonicity. Econometrica 75(5), 1513–1518.CrossRefGoogle Scholar
Hoderlein, S., Su, L., White, H., & Yang, T.T. (2016) Testing for monotonicity in unobservables under unconfoundedness. Journal of Econometrics 193(1), 183–202.CrossRefGoogle Scholar
Hong, Y. & White, H. (1995) Consistent specification testing via nonparametric series regression. Econometrica 63, 1133–1159.CrossRefGoogle Scholar
Horowitz, J.L. (2011) Applied nonparametric instrumental variables estimation. Econometrica 79(2), 347–394.Google Scholar
Horowitz, J.L. (2012) Specification testing in nonparametric instrumental variables estimation. Journal of Econometrics 167, 383–396.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. (2005) Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association 100(472), 1238–1249.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. (2007) Nonparametric instrumental variables estimation of a quantile regression model. Econometrica 75, 1191–1208.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. (2009) Testing a parametric quantile-regression model with an endogenous explanatory variable against a nonparametric alternative. Journal of Econometrics 152(2), 141–152.CrossRefGoogle Scholar
Koenker, R., Ng, P., & Portnoy, S. (1994) Quantile smoothing splines. Biometrika 81(4), 673–680.CrossRefGoogle Scholar
Kress, R. (1989) Linear Integral Equations, 2nd ed. Applied Mathematical Sciences, vol. 82. Springer.CrossRefGoogle Scholar
Lewbel, A., Lu, X., & Su, L. (2015) Specification testing for transformation models with an application to generalized accelerated failure-time models. Journal of Econometrics 184(1), 81–96.CrossRefGoogle Scholar
Newey, W.K. (1997) Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79(1), 147–168.CrossRefGoogle Scholar
Santos, A. (2012) Inference in nonparametric instrumental variables with partial identification. Econometrica 80(1), 213–275.Google Scholar
van der Vaart, A. & Wellner, J. (2000) Weak Convergence and Empirical Processes: With Applications to Statistics (Springer Series in Statistics). Springer.Google Scholar