Shrinkage methods for instrumental variable estimation

This paper proposes shrinkage methods in instrumental variable estimations to solve the ``many instruments'' problem. Even though using a large number of instruments reduces the asymptotic variances of the estimators, it has been observed both in theoretical works and in practice that in finite samples the estimators may behave very poorly if the number of instruments is large. This problem can be addressed by shrinking the influence of a subset of instrumental variables. An instrumental variable estimator is the solution to an equation which is a weighted sum of sample moment conditions; We reconstruct the estimating equation by shrinking some elements of that weighting vector. This idea can also be interpreted as shrinking some of the OLS coefficient estimates from the regression of the endogenous variables on the instruments then using the predicted values of the endogenous variables based on the shrunk coefficient estimates as the instruments. The shrinkage parameter is chosen to minimize the asymptotic MSE. It is found that the optimal shrinkage parameter has a closed form which leads to easy implementation. The Monte Carlo result shows that the shrinkage methods work well and moreover perform better than the instrument selection procedure in Donald and Newey (2001) in several situations relevant to applications