This paper considers the instrument selection problem in instrumental variable (IV) regression model when there is a large set of instruments with potential invalidity. I derive higher-order mean square error (MSE) approximation of two-stage least squares (2SLS), limited information maximum likelihood (LIML), Fuller (FULL) and bias-adjusted 2SLS (B2SLS) estimators with allowing for local violation of the instrument-exogeneity conditions. Based on the approximation to the higher-order MSE, I propose instrument selection criteria that are robust to potential invalidity of instruments. Furthermore, I also show the optimality results of instrument selection criteria in Donald and Newey (2001, Econometrica) under faster than N^(-1/2) locally invalid instruments specication
