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Endogeneity in ultrahigh dimension

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  • Fan, Jianqing
  • Liao, Yuan

Abstract

Most papers on high-dimensional statistics are based on the assumption that none of the regressors are correlated with the regression error, namely, they are exogenous. Yet, endogeneity arises easily in high-dimensional regression due to a large pool of regressors and this causes the inconsistency of the penalized least-squares methods and possible false scientic discoveries. A necessary condition for model selection of a very general class of penalized regression methods is given, which allows us to prove formally the inconsistency claim. To cope with the possible endogeneity, we construct a novel penalized focussed generalized method of moments (FGMM) criterion function and oer a new optimization algorithm. The FGMM is not a smooth function. To establish its asymptotic properties, we rst study the model selection consistency and an oracle property for a general class of penalized regression methods. These results are then used to show that the FGMM possesses an oracle property even in the presence of endogenous predictors, and that the solution is also near global minimum under the over-identication assumption. Finally, we also show how the semi-parametric efficiency of estimation can be achieved via a two-step approach.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 38698.

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Date of creation: 2012
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Handle: RePEc:pra:mprapa:38698

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Related research

Keywords: Focused GMM; Sparsity recovery; Endogenous variables; Oracle property; Conditional moment restriction; Estimating equation; Over identi cation; Global minimization; Semi-parametric efficiency;

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  1. P. B├╝hlmann & M. Kalisch & M. H. Maathuis, 2010. "Variable selection in high-dimensional linear models: partially faithful distributions and the pc -simple algorithm," Biometrika, Biometrika Trust, Biometrika Trust, vol. 97(2), pages 261-278.
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  12. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, Econometric Society, vol. 50(4), pages 1029-54, July.
  13. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320.
  14. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
  15. Horowitz, Joel L, 1992. "A Smoothed Maximum Score Estimator for the Binary Response Model," Econometrica, Econometric Society, Econometric Society, vol. 60(3), pages 505-31, May.
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