On The Bias and MSE of The IV Estimator Under Weak Identi ̄cation¤

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 identi ̄cation 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 ̄ndings 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. We also show that in the special case where all available instruments are used and where the underlying simultaneous equations model has an orthonormal canonical structure, the bias and MSE formulae which we obtain are identical to the exact bias and MSE of the 2SLS estimator obtained by Richardson and Wu (1971) under Gaussian error assumptions. This result gives a partial con ̄rmation to the Staiger-Stock assertion, based on intuitive arguments, that the limiting distribution of the 2SLS estimator derived under the more general assumptions of the Staiger-Stock local-to-zero asymptotic framework coincides with the exact distribution of the same estimator derived under the more restrictive assumptions of a ̄xed instrument/Gaussian model. Because our analytical formulae for bias and MSE are complex functionals of con°uent hypergeometric functions, we also derive approximations for these formulae which are based on an expansion that allows the number of instruments to grow to in ̄nity while keeping the population analogue of the ̄rst stage F-statistic ̄xed. 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 ̄rst stage F-statistic is kept constant. One feature of our approach which ties our ̄ndings to the earlier IV literature is that our results apply not only to the weak instrument case asymptotically, but also to the ̄nite sample case with ̄xed (possibly good) instruments and Gaussian errors, since our formulae correspond to the exact bias and MSE functionals when a ̄xed instrument/Gaussian model is assumed. JEL classi ̄cation: C12, C22.