ML and GMM with concentrated instruments in the static panel data model

We study the asymptotic behavior of instrumental variable estimators in the static panel model under many-instruments asymptotics. We provide new estimators and standard errors based on concentrated instruments as alternatives to an estimator based on maximum likelihood. We prove that the latter estimator is consistent under many-instruments asymptotics only if the starting value in an iterative procedure is root-N consistent. A similar approach for continuous updating GMM shows the derivation is nontrivial. For the standard cross-sectional case (T = 1), the simple formulation of standard errors offer an alternative to earlier formulations.