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Sparse Linear Models and Two-Stage Estimation in High-Dimensional Settings with Possibly Many Endogenous Regressors

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  • Zhu, Ying

Abstract

This paper explores the validity of the two-stage estimation procedure for sparse linear models in high-dimensional settings with possibly many endogenous regressors. In particular, the number of endogenous regressors in the main equation and the instruments in the first-stage equations can grow with and exceed the sample size n. The analysis concerns the exact sparsity case, i.e., the maximum number of non-zero components in the vectors of parameters in the first-stage equations, k1, and the number of non-zero components in the vector of parameters in the second-stage equation, k2, are allowed to grow with n but slowly compared to n. I consider the high-dimensional version of the two-stage least square estimator where one obtains the fitted regressors from the first-stage regression by a least square estimator with l_1-regularization (the Lasso or Dantzig selector) when the first-stage regression concerns a large number of instruments relative to n, and then construct a similar estimator using these fitted regressors in the second-stage regression. The main theoretical results of this paper are non-asymptotic bounds from which I establish sufficient scaling conditions on the sample size for estimation consistency in l_2-norm and variable-selection consistency (i.e., the two-stage high-dimensional estimators correctly select the non-zero coefficients in the main equation with high probability). A technical issue regarding the so-called "restricted eigenvalue (RE) condition" for estimation consistency and the "mutual incoherence (MI) condition" for selection consistency arises in the two-stage estimation from allowing the number of regressors in the main equation to exceed n and this paper provides analysis to verify these RE and MI conditions. Depending on the underlying assumptions that are imposed, the upper bounds on the l_2-error and the sample size required to obtain these consistency results differ by factors involving k1 and/or k2. Simulations are conducted to gain insight on the finite sample performance of the high-dimensional two-stage estimator.

Suggested Citation

  • 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.
  • Handle: RePEc:pra:mprapa:49846
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    References listed on IDEAS

    as
    1. Carrasco, Marine, 2012. "A regularization approach to the many instruments problem," Journal of Econometrics, Elsevier, vol. 170(2), pages 383-398.
    2. Nevo, Aviv, 2001. "Measuring Market Power in the Ready-to-Eat Cereal Industry," Econometrica, Econometric Society, vol. 69(2), pages 307-342, March.
    3. 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.
    4. 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.
    5. Sala-i-Martin, Xavier, 1997. "I Just Ran Two Million Regressions," American Economic Review, American Economic Association, vol. 87(2), pages 178-183, May.
    6. Eric Gautier & Alexandre Tsybakov, 2011. "High-Dimensional Instrumental Variables Regression and Confidence Sets," Working Papers 2011-13, Center for Research in Economics and Statistics.
    7. Amemiya, Takeshi, 1974. "The nonlinear two-stage least-squares estimator," Journal of Econometrics, Elsevier, vol. 2(2), pages 105-110, July.
    8. C. Lanier Benkard & Patrick Bajari, 2005. "Hedonic Price Indexes With Unobserved Product Characteristics, and Application to Personal Computers," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 61-75, January.
    9. Pradeep Ravikumar & John Lafferty & Han Liu & Larry Wasserman, 2009. "Sparse additive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 1009-1030.
    10. Hansen, Christian & Hausman, Jerry & Newey, Whitney, 2008. "Estimation With Many Instrumental Variables," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 398-422.
    11. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    12. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    13. Garen, John, 1984. "The Returns to Schooling: A Selectivity Bias Approach with a Continuous Choice Variable," Econometrica, Econometric Society, vol. 52(5), pages 1199-1218, September.
    14. Joshua D. Angrist & Alan B. Keueger, 1991. "Does Compulsory School Attendance Affect Schooling and Earnings?," The Quarterly Journal of Economics, Oxford University Press, vol. 106(4), pages 979-1014.
    15. Caner, Mehmet, 2009. "Lasso-Type Gmm Estimator," Econometric Theory, Cambridge University Press, vol. 25(01), pages 270-290, February.
    16. Jianqing Fan & Jinchi Lv & Lei Qi, 2011. "Sparse High-Dimensional Models in Economics," Annual Review of Economics, Annual Reviews, vol. 3(1), pages 291-317, September.
    17. Berry, Steven & Levinsohn, James & Pakes, Ariel, 1995. "Automobile Prices in Market Equilibrium," Econometrica, Econometric Society, vol. 63(4), pages 841-890, July.
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    More about this item

    Keywords

    High-dimensional statistics; Lasso; sparse linear models; endogeneity; two-stage estimation;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation

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