Ultra High-Dimensional Variable Selection Consistency with Endogenous Covariates

• Activity Number: 83
• Type: Contributed
• Date/Time: Sunday, July 31, 2011 : 4:00 PM to 5:50 PM
• Sponsor: IMS
• Author(s): Yuan Liao*+ and Jianqing Fan
• Companies: Princeton University and Princeton University
• Keywords: penalized GMM ; penalized empirical likelihood ; penalty function ; sparsity recovery ; oracle property ; conditional moment restriction

Abstract

High dimensional models have gained considerable importance in several areas of applications. We consider ultra high dimensional variable selection problem in which the number of regressors grows exponentially fast with the sample size. Sparse modeling has been widely used to deal with high dimensionality, which assumes that many components of the parameter vector are exactly zero. The variable selection is carried out by penalized optimization. We give sufficient and necessary conditions for a general penalized optimization to achieve the consistency for both variable selection and estimation. We then study a conditional moment restricted model, and verify that under mild assumptions, both penalized GMM (PGMM) and penalized empirical likelihood (PEL) satisfy the proposed sufficient conditions. An interesting finding is, when there exists an endogenous variable whose true regression coefficient is zero, the penalized OLS does not satisfy the necessary condition of variable selection regardless of the penalty function used.