Culling the herd of moments with penalized empirical likelihood

Models defined by moment conditions are at the center of structural econometric estimation, but economic theory is mostly silent about moment selection. A large pool of valid moments can potentially improve estimation efficiency, whereas a few invalid ones may undermine consistency. This paper investigates the empirical likelihood estimation of these moment-defined models in high-dimensional settings. We propose a penalized empirical likelihood (PEL) estimation and show that it achieves the oracle property under which the invalid moments can be consistently detected. While the PEL estimator is asymptotically normally distributed, a projected PEL procedure can further eliminate its asymptotic bias and provide more accurate normal approximation to the finite sample distribution. Simulation exercises are carried out to demonstrate excellent numerical performance of these methods in estimation and inference.