Alternative Approximations of the Bias and MSE of the IV Estimator under Weak Identification with an Application to Bias Correction

We provide analytical formulae for the asymptotic bias (ABIAS) and mean squared error (AMSE) of the IV estimator, and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage F-statistic to converge to a finite (possibly small) positive limit as the number of instruments approaches infinity. The approximations so obtained are shown, via regression analysis, to yield good approximations for ABIAS and AMSE functions, and the AMSE approximation is shown to perform well relative to the approximation of Donald and Newey (2001). Additionally, the manner in which our framework generalizes that of Richardson and Wu (1971) is discussed. One consequence of the asymptotic framework adopted here is that consistent estimators for the ABIAS and AMSE can be obtained. As a result, we are able to construct a number of bias corrected OLS and IV estimators, which we show to be consistent under a sequential asymptotic scheme. These bias-corrected estimators are also robust, in the sense that they remain consistent in a conventional asymptotic setup, where the model is fully identified. A small Monte Carlo experiment documents the relative performance of our bias adjusted estimators versus standard IV, OLS, LIML estimators, and it is shown that our estimators have lower bias than LIML for various levels of endogeneity and instrument relevance.