Instrumental variables estimation and inference in the presence of many exogenous regressors

We consider a standard instrumental variables model contaminated by the presence of a large number of exogenous regressors. In an asymptotic framework where this number is proportional to the sample size, we study the impact of their ratio on the validity of existing estimators and tests. When the instruments are few, the inference using the conventional 2SLS estimator and associated t and J statistics, as well as the AndersonRubin and Kleibergen tests, is still valid. When the instruments are many, the LIML estimator remains consistent, but the presence of many exogenous regressors changes its asymptotic variance. Moreover, the conventional bias correction of the 2SLS estimator is no longer appropriate. We provide asymptotically correct versions of bias correction for the 2SLS estimator, derive its asymptotically correct variance estimator, extend the HansenHausmanNewey LIML variance estimator to the case of many exogenous regressors, and propose asymptotically valid modifications of the J overidentification tests based on the LIML and bias-corrected 2SLS estimators.