Regularized LIML for many instruments

The use of many moment conditions improves the asymptotic efficiency of the instrumental variables estimators. However, in finite samples, the inclusion of an excessive number of moments increases the bias. To solve this problem, we propose regularized versions of the limited information maximum likelihood (LIML) based on three different regularizations: Tikhonov, Landweber Fridman, and principal components. Our estimators are consistent and reach the semiparametric efficiency bound under some standard assumptions. We show that the regularized LIML estimator based on principal components possesses finite moments when the sample size is large enough. The higher order expansion of the mean square error (MSE) shows the dominance of regularized LIML over regularized two-staged least squares estimators. We devise a data driven selection of the regularization parameter based on the approximate MSE. A Monte Carlo study shows that the regularized LIML works well and performs better in many situations than competing methods. Two empirical applications illustrate the relevance of our estimators: one regarding the return to schooling and the other regarding the elasticity of intertemporal substitution.