Choosing a Set of Instruments among Many and Possibly Invalid Instruments

We consider the instrument selection problem in instrumental variable regression model when there is a large set of instruments with potential invalidity. Existing methods for instrument selection are based on a priori assumptions of an instrument's validity. Moreover, existing approaches based on consistent moment selection or first-order asymptotics can lead to a large finite sample bias with many instruments. We derive higher-order mean square error (MSE) approximation of a two-stage least squares (2SLS) estimator allowing locally invalid instruments. Based on the approximation to the higher-order MSE, we propose an invalidity-robust instrument selection criterion (IRC) that captures two sources of finite sample bias at the same time: bias from using many instruments and bias from invalid instruments. We show that our criterion is an asymptotically unbiased estimator of the higher-order MSE approximation. In various simulation experiments, our criterion performs well in terms of lower bias and better coverage properties of the estimator compare with other existing selection methods. We also apply our criterion to the empirical application in Card (1995)