Shrinkage methods for instrumental variable estimation∗

This paper proposes shrinkage methods for instrumental variable estimation 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. That is, we reconstruct the estimating equation of an instrumental variable estimator, which is a weighted sum of sample moment conditions, by shrinking some elements of the weighting vector. This procedure can also be understood by a two-step process of 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. We find 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.