Constructing Optimal Instruments by First Stage Prediction Averaging
This paper considers model averaging as a way to select instruments for the two stage least squares and limited information maximum likelihood 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. This can be done by solving a standard quadratic programming problem and, in some cases, closed form solutions for the optimal weights are available. We demonstrate both theoretically and in Monte Carlo experiments that our method nests and dominates existing number-of-instrument-selection procedures.