Constructing Optimal Instruments by First-Stage Prediction Averaging

This paper considers model averaging as a way to construct optimal instruments for the two-stage least squares (2SLS), limited information maximum likelihood (LIML), and Fuller estimators in the presence of many instruments. We propose averaging across least squares predictions of the endogenous variables obtained from many different choices of instruments and then use the average predicted value of the endogenous variables in the estimation stage. The weights for averaging are chosen to minimize the asymptotic mean squared error of the model averaging version of the 2SLS, LIML, or Fuller estimator. This can be done by solving a standard quadratic programming problem.