Endogeneity in Ultrahigh Dimension

53 Pages Posted: 25 Apr 2012

See all articles by Jianqing Fan

Jianqing Fan

Princeton University - Bendheim Center for Finance

Yuan Liao

Rutgers, The State University of New Jersey - New Brunswick/Piscataway

Date Written: April 24, 2012

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 exogeneous. 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 scientific 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 offer a new optimization algorithm. The FGMM is not a smooth function. To establish its asymptotic properties, we first 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-identification assumption. Finally, we also show how the semi-parametric efficiency of estimation can be achieved via a two-step approach.

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

JEL Classification: C13, C14

Suggested Citation

Fan, Jianqing and Liao, Yuan, Endogeneity in Ultrahigh Dimension (April 24, 2012). Available at SSRN: https://ssrn.com/abstract=2045864 or http://dx.doi.org/10.2139/ssrn.2045864

Jianqing Fan

Princeton University - Bendheim Center for Finance ( email )

26 Prospect Avenue
Princeton, NJ 08540
United States
609-258-7924 (Phone)
609-258-8551 (Fax)

HOME PAGE: http://orfe.princeton.edu/~jqfan/

Yuan Liao (Contact Author)

Rutgers, The State University of New Jersey - New Brunswick/Piscataway ( email )

94 Rockafeller Road
New Brunswick, NJ 08901
United States

HOME PAGE: http://rci.rutgers.edu/~yl1114

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