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Sparse Linear Models and l1−Regularized 2SLS with High-Dimensional Endogenous Regressors and Instruments

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  • Zhu, Ying

Abstract

We explore the validity of the 2-stage least squares estimator with l_{1}-regularization in both stages, for linear regression models where the numbers of endogenous regressors in the main equation and instruments in the first-stage equations can exceed the sample size, and the regression coefficients are sufficiently sparse. For this l_{1}-regularized 2-stage least squares estimator, finite-sample performance bounds are established. We then provide a simple practical method (with asymptotic guarantees) for choosing the regularization parameter. We show that this practical method can produce an l_{2}-consistent 2SLS estimator whose rate of convergence can be made as arbitrarily close as the scaling of our finite-sample performance bounds under quite standard conditions.

Suggested Citation

  • Zhu, Ying, 2015. "Sparse Linear Models and l1−Regularized 2SLS with High-Dimensional Endogenous Regressors and Instruments," MPRA Paper 81217, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:81217
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    File URL: https://mpra.ub.uni-muenchen.de/82184/1/MPRA_paper_82184.pdf
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    References listed on IDEAS

    as
    1. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(03), pages 497-521, June.
    2. Carrasco, Marine, 2012. "A regularization approach to the many instruments problem," Journal of Econometrics, Elsevier, vol. 170(2), pages 383-398.
    3. Jeffrey M Wooldridge, 2010. "Econometric Analysis of Cross Section and Panel Data," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262232588.
    4. Fan, Jianqing & Liao, Yuan, 2012. "Endogeneity in ultrahigh dimension," MPRA Paper 38698, University Library of Munich, Germany.
    5. 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.
    6. Amemiya, Takeshi, 1974. "The nonlinear two-stage least-squares estimator," Journal of Econometrics, Elsevier, vol. 2(2), pages 105-110, July.
    7. 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.
    8. 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.
    9. 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.
    10. Caner, Mehmet, 2009. "Lasso-Type Gmm Estimator," Econometric Theory, Cambridge University Press, vol. 25(01), pages 270-290, February.
    11. 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.
    Full references (including those not matched with items on IDEAS)

    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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