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High-dimensional instrumental variables regression and confidence sets

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  • Eric Gautier

    () (CREST - Centre de Recherche en Économie et Statistique - INSEE - ENSAE ParisTech - École Nationale de la Statistique et de l'Administration Économique, ENSAE ParisTech - École Nationale de la Statistique et de l'Administration Économique)

  • Alexandre Tsybakov

    () (CREST - Centre de Recherche en Économie et Statistique - INSEE - ENSAE ParisTech - École Nationale de la Statistique et de l'Administration Économique, ENSAE ParisTech - École Nationale de la Statistique et de l'Administration Économique)

Abstract

We propose an instrumental variables method for inference in high-dimensional structural equations with endogenous regressors. The number of regressors K can be much larger than the sample size. A key ingredient is sparsity, i.e., the vector of coefficients has many zeros, or approximate sparsity, i.e., it is well approximated by a vector with many zeros. We can have less instruments than regressors and allow for partial identification. Our procedure, called STIV (Self Tuning Instrumental Variables) estimator, is realized as a solution of a conic program. The joint confidence sets can be obtained by solving K convex programs. We provide rates of convergence, model selection results and propose three types of joint confidence sets relying each on different assumptions on the parameter space. Under the stronger assumption they are adaptive. The results are uniform over a wide classes of distributions of the data and can have finite sample validity. When the number of instruments is too large or when one only has instruments for an endogenous regressor which are too weak, the confidence sets can have infinite volume with positive probability. This provides a simple one-stage procedure for inference robust to weak instruments which could also be used for low dimensional models. In our IV regression setting, the standard tools from the literature on sparsity, such as the restricted eigenvalue assumption are inapplicable. Therefore we develop new sharper sensitivity characteristics, as well as easy to compute data-driven bounds. All results apply to the particular case of the usual high-dimensional regression. We also present extensions to the high-dimensional framework of the two-stage least squares method and method to detect endogenous instruments given a set of exogenous instruments.

Suggested Citation

  • Eric Gautier & Alexandre Tsybakov, 2014. "High-dimensional instrumental variables regression and confidence sets," Working Papers hal-00591732, HAL.
  • Handle: RePEc:hal:wpaper:hal-00591732 Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-00591732v4
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    References listed on IDEAS

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    1. Alastair R. Hall & Fernanda P. M. Peixe, 2003. "A Consistent Method for the Selection of Relevant Instruments," Econometric Reviews, Taylor & Francis Journals, vol. 22(3), pages 269-287, January.
    2. Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
    3. Okui, Ryo, 2011. "Instrumental variable estimation in the presence of many moment conditions," Journal of Econometrics, Elsevier, vol. 165(1), pages 70-86.
    4. Caner, Mehmet, 2009. "Lasso-Type Gmm Estimator," Econometric Theory, Cambridge University Press, vol. 25(01), pages 270-290, February.
    5. Jerry A. Hausman & Whitney K. Newey & Tiemen Woutersen & John C. Chao & Norman R. Swanson, 2012. "Instrumental variable estimation with heteroskedasticity and many instruments," Quantitative Economics, Econometric Society, vol. 3(2), pages 211-255, July.
    6. Carrasco, Marine & Florens, Jean-Pierre, 2000. "Generalization Of Gmm To A Continuum Of Moment Conditions," Econometric Theory, Cambridge University Press, vol. 16(06), pages 797-834, December.
    7. A. Belloni & D. Chen & V. Chernozhukov & C. Hansen, 2012. "Sparse Models and Methods for Optimal Instruments With an Application to Eminent Domain," Econometrica, Econometric Society, vol. 80(6), pages 2369-2429, November.
    8. Amemiya, Takeshi, 1974. "The nonlinear two-stage least-squares estimator," Journal of Econometrics, Elsevier, vol. 2(2), pages 105-110, July.
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    Citations

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    Cited by:

    1. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney K. Newey & James Robins, 2017. "Double/debiased machine learning for treatment and structural parameters," CeMMAP working papers CWP28/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Achim Ahrens & Arnab Bhattacharjee, 2015. "Two-Step Lasso Estimation of the Spatial Weights Matrix," Econometrics, MDPI, Open Access Journal, vol. 3(1), pages 1-28, March.
    3. Hansen, Christian & Liao, Yuan, 2016. "The Factor-Lasso and K-Step Bootstrap Approach for Inference in High-Dimensional Economic Applications," MPRA Paper 75313, University Library of Munich, Germany.
    4. Yoonseok Lee & Mehmet Caner & Xu Han, 2015. "Adaptive Elastic Net GMM Estimation with Many Invalid Moment Conditions: Simultaneous Model and Moment Selection," Center for Policy Research Working Papers 177, Center for Policy Research, Maxwell School, Syracuse University.
    5. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2014. "High-Dimensional Methods and Inference on Structural and Treatment Effects," Journal of Economic Perspectives, American Economic Association, vol. 28(2), pages 29-50, Spring.
    6. Cheng, Xu & Liao, Zhipeng, 2015. "Select the valid and relevant moments: An information-based LASSO for GMM with many moments," Journal of Econometrics, Elsevier, vol. 186(2), pages 443-464.
    7. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Inference on Treatment Effects After Selection Amongst High-Dimensional Controls," Papers 1201.0224, arXiv.org, revised May 2012.
    8. Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach," Annual Review of Economics, Annual Reviews, vol. 7(1), pages 649-688, August.
    9. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2016. "Double/Debiased Machine Learning for Treatment and Causal Parameters," Papers 1608.00060, arXiv.org, revised Dec 2017.
    10. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Central limit theorems and multiplier bootstrap when p is much larger than n," CeMMAP working papers CWP45/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. Michal Kolesár & Raj Chetty & John Friedman & Edward Glaeser & Guido W. Imbens, 2015. "Identification and Inference With Many Invalid Instruments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(4), pages 474-484, October.
    12. Hansen, Christian & Kozbur, Damian, 2014. "Instrumental variables estimation with many weak instruments using regularized JIVE," Journal of Econometrics, Elsevier, vol. 182(2), pages 290-308.
    13. Fan, Jianqing & Liao, Yuan, 2012. "Endogeneity in ultrahigh dimension," MPRA Paper 38698, University Library of Munich, Germany.
    14. Áureo de Paula, 2015. "Econometrics of network models," CeMMAP working papers CWP52/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    15. Victor Chernozhukov & Christian Hansen & Martin Spindler, 2015. "Post-Selection and Post-Regularization Inference in Linear Models with Many Controls and Instruments," American Economic Review, American Economic Association, vol. 105(5), pages 486-490, May.
    16. repec:bla:jorssb:v:79:y:2017:i:3:p:939-956 is not listed on IDEAS
    17. Aman Ullah & Huansha Wang, 2013. "Parametric and Nonparametric Frequentist Model Selection and Model Averaging," Econometrics, MDPI, Open Access Journal, vol. 1(2), pages 1-23, September.
    18. Eric Gautier & Alexandre B, Tsybakov, 2013. "Pivotal Estimation in High-Dimensional Regression via Linear Programming," Working Papers 2013-40, Center for Research in Economics and Statistics.
    19. Alexandre Belloni & Victor Chernozhukov & Christian Hansen, 2011. "Inference for High-Dimensional Sparse Econometric Models," Papers 1201.0220, arXiv.org.
    20. Alexandre Belloni & Victor Chernozhukov & Lie Wang, 2013. "Pivotal estimation via square-root lasso in nonparametric regression," CeMMAP working papers CWP62/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    21. Zhu, Ying, 2013. "Sparse Linear Models and Two-Stage Estimation in High-Dimensional Settings with Possibly Many Endogenous Regressors," MPRA Paper 49846, University Library of Munich, Germany.
    22. Chatterjee, A. & Gupta, S. & Lahiri, S.N., 2015. "On the residual empirical process based on the ALASSO in high dimensions and its functional oracle property," Journal of Econometrics, Elsevier, vol. 186(2), pages 317-324.
    23. Alexis Le Chapelain, 2014. "Market for Education and Student Achievement," Sciences Po publications info:hdl:2441/1jgbspo1909, Sciences Po.
    24. Christian Hansen & Yuan Liao, 2016. "The Factor-Lasso and K-Step Bootstrap Approach for Inference in High-Dimensional Economic Applications," Departmental Working Papers 201610, Rutgers University, Department of Economics.
    25. Xu Cheng & Zhipeng Liao, 2012. "Select the Valid and Relevant Moments: A One-Step Procedure for GMM with Many Moments," PIER Working Paper Archive 12-045, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.

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