A Note on the Regularized Approach to Biased 2SLS Estimation with Weak Instruments

The presence of weak instruments is translated into a nearly singular problem in a control function representation. Therefore, the math formula-norm type of regularization is proposed to implement the 2SLS estimation for addressing the weak instrument problem. The math formula-norm regularization with a regularized parameter O(n) allows us to obtain the Rothenberg (1984) type of higher-order approximation of the 2SLS estimator in the weak instrument asymptotic framework. The proposed regularized parameter yields the regularized concentration parameter O(n), which is used as a standardized factor in the higher-order approximation. We also show that the proposed math formula-norm regularization consequently reduces the finite sample bias. A number of existing estimators that address finite sample bias in the presence of weak instruments, especially Fuller's limited information maximum likelihood estimator, are compared with our proposed estimator in a simple Monte Carlo exercise.