Optimal Estimation with Complete Subsets of Instruments

In this paper we propose a two-stage least squares (2SLS) estimator whose first stage is based on the equal-weight average over a complete subset. We derive the approximate mean squared error (MSE) that depends on the size of the complete subset and characterize the proposed estimator based on the approximate MSE. The size of the complete subset is chosen by minimizing the sample counterpart of the approximate MSE. We show that this method achieves the asymptotic optimality. To deal with weak or irrelevant instruments, we generalize the approximate MSE under the presence of a possibly growing set of irrelevant instruments, which provides useful guidance under weak IV environments. The Monte Carlo simulation results show that the proposed estimator outperforms alternative methods when instruments are correlated with each other and there exists high endogeneity. As an empirical illustration, we estimate the logistic demand function in Berry, Levinsohn, and Pakes (1995).