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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
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    More about this item

    Keywords

    High-dimensional statistics; Lasso; sparse linear models; endogeneity; two-stage estimation;
    All these keywords.

    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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