Near Exogeneity and Weak Identification in Generalized Empirical Likelihood Estimators : Fixed and Many Moment Asymptotics

This paper analyzes near exogeneity and weak identification in Generalized Empirical Likelihood Estimators. Near exogeneity and weak identification are related to the exogeneity and relevance of the instruments, respectively. These two issues are important from an applied perspective, such as empirical growth theory and labor economics. In the case of empirical growth and institutional economics literature a small number of moments/instruments are used in studies. First, we analyze the limit behavior of estimators and tests under fixed number of weak moments and near exogeneity. We show that Anderson-Rubin (1949) and Kleibergen (2002) type of tests' limits change when there is small correlation between the instruments and the structural equation error. The new limits are obtained under the null hypothesis at the true vale of the parameter. The test statistics are no longer asymptotically pivotal in the joint case of near exogeneity and weak instruments compared to the weak identification case. We also show that when used with the x2 critical values, which are not valid in the case of near exogeneity and weak instruments, the tests show very large size distortions. This is an important warning to applied researchers who may use these tests without taking into account the near exogeneity problem. We try subsampling and delete-d jackknife methods to recover asymptotic limits. Both of these methods are inconsistent. However, we show that the asymptotic limit of delete-d jackknife is arbitrarily close to true limit and only slightly liberal. In simulations, exponential tilting based tests with delete-d jackknife method have good size compared to the others. Then we develop the limits of estimators and tests under many weak moments with near exogeneity. The results are different from the fixed moments case. Estimators are consistent, and test limits are simple, noncentral x2