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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.<\/jats:p>","DOI":"10.1214\/13-aos1202","type":"journal-article","created":{"date-parts":[[2014,5,20]],"date-time":"2014-05-20T09:33:08Z","timestamp":1400578388000},"page":"872-917","source":"Crossref","is-referenced-by-count":86,"title":["Endogeneity in high dimensions"],"prefix":"10.1214","volume":"42","author":[{"given":"Jianqing","family":"Fan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuan","family":"Liao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"5","doi-asserted-by":"publisher","unstructured":"Belloni, A. and Chernozhukov, V. (2013). 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