• 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 can arise incidentally from a large pool of regressors in a high-dimensional regression. This causes the inconsistency of the penalized least-squares method and possible false scientific discoveries. A necessary condition for model selection consistency of a general class of penalized regression methods is given, which allows us to prove formally the inconsistency claim. To cope with the incidental endogeneity, we construct a novel penalized focused generalized method of moments (FGMM) criterion function. The FGMM effectively achieves the dimension reduction and applies the instrumental variable methods. We show that it possesses the oracle property even in the presence of endogenous predictors, and that the solution is also near global minimum under the over-identification assumption. Finally, we also show how the semi-parametric efficiency of estimation can be achieved via a two-step approach.

• Paper: pdf file, remaining proofs

• Earlier version: pdf file. When the important covariates are known to be exogenous but not unimportant covariates, no instrumental variables are needed. However, the regular penalized least squares do not work.

• Matlab functions: zip , main program

• Slides: pdf file

• Related literature:

  Gautier and Tsybakov (2011).
  High dimensional instrumental variables regression and confidence sets.
  Manuscript.

  Belloni, and Chernozhukov (2011).
  Least squares after model selection in high-dimensional sparse models.
  Forthcoming in Bernoulli.

  Fan and Lv (2011)
  Non-concave penalized likelihood with NP-dimensionality.
  IEEE Transactions on Information Theory, 57 5467-5484.

• Presentations:

  2011

  • Midwest Econometric Group Meeting, Chicago
  • University of Maryland
  • JSM, Miami
  • Econometric Society Summer Meeting, St. Louis