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HONEST CONFIDENCE SETS IN NONPARAMETRIC IV REGRESSION AND OTHER ILL-POSED MODELS

Published online by Cambridge University Press:  05 March 2020

Andrii Babii
Andrii Babii*
Affiliation:
University of North Carolina at Chapel Hill
*
Address correspondence to Andrii Babii, University of North Carolina at Chapel Hill, Gardner Hall, CB 3305 Chapel Hill, NC 27599-3305, USA; [email protected]
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Abstract

This article develops inferential methods for a very general class of ill-posed models in econometrics encompassing the nonparametric instrumental variable regression, various functional regressions, and the density deconvolution. We focus on uniform confidence sets for the parameter of interest estimated with Tikhonov regularization, as in Darolles et al. (2011, Econometrica 79, 1541–1565). Since it is impossible to have inferential methods based on the central limit theorem, we develop two alternative approaches relying on the concentration inequality and bootstrap approximations. We show that expected diameters and coverage properties of resulting sets have uniform validity over a large class of models, that is, constructed confidence sets are honest. Monte Carlo experiments illustrate that introduced confidence sets have reasonable width and coverage properties. Using U.S. data, we provide uniform confidence sets for Engel curves for various commodities.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 4 , August 2020 , pp. 658 - 706
Copyright
© Cambridge University Press 2020

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Footnotes

First draft: March 2016. This article is a revised first chapter of my Ph.D. thesis. I’m grateful to the Co-Editor and valuable referees for constructive criticism and suggestions of how to improve the article. I’m deeply indebted to my advisor Jean-Pierre Florens and other members of my Ph.D. committee: Eric Gautier, Ingrid Van Keilegom, and Timothy Christensen for helpful suggestions and insightful conversations. This article also benefited from discussions with Christian Bontemps, Samuele Centorrino, Jasmin Fliegner, Emanuele Guerre, Vitalijs Jascisens, Jihyun Kim, Rohit Kumar, Elia Lapenta, Pascal Lavergne, Thierry Magnac, André Mas, Nour Meddahi, Markus Reiss, and Shruti Sinha.

References

REFERENCES

Adusumilli, K. & Otsu, T. 2018 Nonparametric instrumental regression with errors in variables. Econometric Theory 34, 1256–1280.CrossRefGoogle Scholar
Almås, I. 2012 International income inequality: Measuring PPP bias by estimating Engel curves for food. The American Economic Review 102, 1093–1117.CrossRefGoogle Scholar
Babii, A. 2017 Identification and estimation in the functional linear instrumental regression. UNC Working Paper.Google Scholar
Babii, A. & Florens, J.-P. 2018 Is completeness necessary? Estimation and inference in non-identified models. UNC Working Paper.CrossRefGoogle Scholar
Banks, J., Blundell, R., & Lewbel, A. 1997 Quadratic Engel curves and consumer demand. Review of Economics and Statistics 79, 527–539.CrossRefGoogle Scholar
Belloni, A., Chernozhukov, V., Chetverikov, D., & Kato, K. 2015 Some new asymptotic theory for least squares series: Pointwise and uniform results. Journal of Econometrics 186, 345–366.CrossRefGoogle Scholar
Benatia, D., Carrasco, M., & Florens, J.-P. 2017 Functional linear regression with functional response. Journal of Econometrics 201 269–291.CrossRefGoogle Scholar
Bissantz, N., Dümbgen, L., Holzmann, H., & Munk, A. 2007 Non-parametric confidence bands in deconvolution density estimation. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69, 483–506.CrossRefGoogle Scholar
Blundell, R., Chen, X., & Kristensen, D. 2007 Semi-nonparametric IV estimation of shape-invariant engel curves. Econometrica 75, 1613–1669.CrossRefGoogle Scholar
Bonhomme, S. & Robin, J.-M. 2010 Generalized non-parametric deconvolution with an application to earnings dynamics. The Review of Economic Studies 77, 491–533.CrossRefGoogle Scholar
Boucheron, S., Lugosi, G., & Massart, P. 2013 Concentration inequalities: A nonasymptotic theory of independence. OUP, Oxford, UK.CrossRefGoogle Scholar
Cardot, H., Mas, A., & Sarda, P. 2007 CLT in functional linear regression models. Probability Theory and Related Fields 138, 325–361.CrossRefGoogle Scholar
Cardot, H., Ferraty, F., Mas, A., & Sarda, P. 2003 Testing hypotheses in the functional linear model. Scandinavian Journal of Statistics 30, 241–255.CrossRefGoogle Scholar
Carrasco, M. & Florens, J.-P. 2011 A spectral method for deconvolving a density. Econometric Theory 27, 546–581.CrossRefGoogle Scholar
Carrasco, M., Florens, J.-P., & Renault, E. 2007 Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization. Handbook of Econometrics 6, 5633–5751.CrossRefGoogle Scholar
Carrasco, M, Florens, J.-P., & Renault, E. 2014 Asymptotic normal inference in linear inverse problems. Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics, Oxford University Press.Google Scholar
Centorrino, S. 2016 Data driven selection of the regularization parameter in additive nonparametric instrumental regressions. Working Paper, Stony Brook University.CrossRefGoogle Scholar
Chen, X. & Pouzo, D. 2015 Sieve Wald and QLR inferences on semi/nonparametric conditional moment models. Econometrica 83, 1013–1079.CrossRefGoogle Scholar
Chen, X. & Reiss, M. 2011 On rate optimality for ill-posed inverse problems in econometrics. Econometric Theory 27, 497–521.CrossRefGoogle Scholar
Chen, X. & Christensen, T.M. 2018 Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression. Quantitative Economics 9, 39–84.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. 2014. Gaussian approximation of suprema of empirical processes. Annals of Statistics 42, 1564–1597.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. 2016 Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings. Stochastic Processes and Their Applications 126, 3632–3651.CrossRefGoogle Scholar
Darolles, S., Fan, Y., Florens, J.-P., & Renault, E. 2011 Nonparametric instrumental regression. Econometrica 79, 1541–1565.Google Scholar
Dudley, R.M 2014 Uniform Central Limit Theorems, vol. 142, Cambridge University Press.Google Scholar
Dudley, R.M 2016 V.N. Sudakov’s work on expected suprema of Gaussian processes. High Dimensional Probability VII. pp. 37–43. Springer.CrossRefGoogle Scholar
Evdokimov, K. 2010 Identification and estimation of a nonparametric panel data model with unobserved heterogeneity. Princeton University Working Paper.Google Scholar
Fève, F. & Florens, J.-P. 2010 The practice of non-parametric estimation by solving inverse problems: The example of transformation models. The Econometrics Journal 13, S1–S27.CrossRefGoogle Scholar
Florens, J.-P. 2003 Inverse problems and structural econometrics. Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress 2, 46–85.Google Scholar
Florens, J.-P. & Van Bellegem, S. 2015 Instrumental variable estimation in functional linear models. Journal of Econometrics 186, 465–476.CrossRefGoogle Scholar
Florens, J.-P., Johannes, J., & Van Bellegem, S. 2011 Identification and estimation by penalization in nonparametric instrumental regression. Econometric Theory 27, 472–496.CrossRefGoogle Scholar
Florens, J.-P., Horowitz, J.L., & Van Keilegom, I. 2017 Bias-corrected confidence intervals in a class of linear inverse problems. Annals of Economics and Statistics 128, 203–228.CrossRefGoogle Scholar
Gagliardini, P. & Scaillet, O. 2012a Nonparametric instrumental variable estimation of structural quantile effects. Econometrica 80, 1533–1562.Google Scholar
Gagliardini, P. & Scaillet, O. 2012b Tikhonov regularization for nonparametric instrumental variable estimators. Journal of Econometrics 167, 61–75.CrossRefGoogle Scholar
Gautier, E. & Kitamura, Y. 2013 Nonparametric estimation in random coefficients binary choice models. Econometrica 81, 581–607.Google Scholar
Giné, E. & Guillou, A. 2002 Rates of strong uniform consistency for multivariate kernel density estimators. Annales de l’Institut Henri Poincare (B) Probability and Statistics 38, 907–921.CrossRefGoogle Scholar
Giné, E. & Nickl, R. 2016 Mathematical foundations of infinite-dimensional statistical models. Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.Google Scholar
Hall, P. & Horowitz, J.L. 2005 Nonparametric methods for inference in the presence of instrumental variables. Annals of Statistics 33, 2904–2929.CrossRefGoogle Scholar
Hall, P. & Horowitz, J.L. 2007 Methodology and convergence rates for functional linear regression. Annals of Statistics 35, 70–91.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. 2012 Uniform confidence bands for functions estimated nonparametrically with instrumental variables. Journal of Econometrics 168, 175–188.CrossRefGoogle Scholar
Kato, K. & Sasaki, Y. 2018 Uniform confidence bands in deconvolution with unknown error distribution. Journal of Econometrics 207, 129–161.CrossRefGoogle Scholar
Koltchinskii, V. 2001 Rademacher penalties and structural risk minimization. Information Theory, IEEE Transactions on 47, 1902–1914.CrossRefGoogle Scholar
Koltchinskii, V. 2006 Local Rademacher complexities and oracle inequalities in risk minimization. Annals of Statistics 34, 2593–2656.CrossRefGoogle Scholar
Koltchinskii, V. 2011 Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems: Ecole d’Eté de Probabilités de Saint-Flour XXXVIII-2008, vol. 2033, Springer.CrossRefGoogle Scholar
Li, K.-C. 1989 Honest confidence regions for nonparametric regression. Annals of Statistics 17, 1001–1008.CrossRefGoogle Scholar
Lounici, K. & Nickl, R. 2011 Global uniform risk bounds for wavelet deconvolution estimators. Annals of Statistics 39, 201–231.CrossRefGoogle Scholar
Nair, M.T. 2009 Linear Operator Equations: Approximation and Regularization. World Scientific.CrossRefGoogle Scholar
Nakamura, E., Steinsson, J., & Liu, M. 2016 Are Chinese growth and inflation too smooth? Evidence from Engel curves. American Economic Journal: Macroeconomics 8, 113–44.Google Scholar
Newey, W.K. & Powell, J.L. 2003 Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578.CrossRefGoogle Scholar
Ruymgaart, F.H 1998 A note on weak convergence of density estimators in Hilbert spaces. Statistics 30, 331–343.CrossRefGoogle Scholar
Tao, Jing 2014 Inference for point and partially identified semi-nonparametric conditional moment models. Working Paper, University of Wisconsin-Madison.Google Scholar
Tikhonov, A. N. 1963 On the Solution of Ill-Posed Problems and the Method of Regularization, vol. 151, 501–504, Russian Academy of Sciences.Google Scholar
Tsybakov, A. 2009 Introduction to Nonparametric Estimation. Springer.CrossRefGoogle Scholar
Van Der Vaart, A.W. & Wellner, J.A. 2000 Weak Convergence. Springer.Google Scholar
Yatchew, A. & No, J.A. 2001 Household gasoline demand in Canada. Econometrica 69, 1697–1709.CrossRefGoogle Scholar