On the Bias and MSE of the IV Estimator Under Weak Identification

In this paper we provide further results on the properties of the IV estimator in the presence of weak instruments. We begin by formalizing the notion of weak identification within the local-to-zero asymptotic framework of Staiger and Stock (1997), and deriving explicit analytical formulae for the asymptotic bias and mean square error (MSE) of the IV estimator. These results generalize earlier findings by Staiger and Stock (1997), who give an approximate measure for the asymptotic bias of the two-stage least squares (2SLS) estimator relative to that of the OLS estimator. Because our analytical formulae for bias and MSE are complex functionals of confluent hypergeometric functions, we also derive approximations for these formulae which are based on an expansion that allows the number of instruments to grow to infinity while keeping the population analogue of the first stage F-statistic fixed. In addition, we provide a series of regression results that show this expansion to give excellent approximations for the bias and MSE functions in general. These approximations allow us to make several interesting additional observations. For example, when the approximation method is applied to the bias, the lead term of the expansion, when appropriately standardized by the asymptotic bias of the OLS estimator, is exactly the relative bias measure given in Staiger and Stock (1997) in the case where there is only one endogenous regressor. In addition, the lead term of the MSE expansion is the square of the lead term of the bias expansion, implying that the variance component of the MSE is of a lower order relative to the bias component in a scenario where the number of instruments used is taken to be large while the population analogue of the first stage F-statistic is kept constant. One feature of our approach which ties our findings to the earlier IV literature is that our results apply not only to the weak instrument case asymptotically, but also to the finite sample case with fixed (possibly good) instruments and Gaussian errors, since our formulae correspond to the exact bias and MSE functionals when a fixed instrument/Gaussian model is assumed.