A regularization approach to the many instruments problem

This paper focuses on the estimation of a …nite dimensional parameter in a linear model where the number of instruments is very large or in…nite. In order to improve the small sample properties of standard instrumental variables (IV) estimators, we propose three modi…ed IV estimators based on three di¤erent ways of inverting the covariance matrix of the instruments. These inverses involve a regularization or smoothing parameter. It should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation. We show that the three estimators are asymptotically normal and attain the semiparametric e¢ ciency bound. Higher-order analysis of the MSE reveals that the bias of the modi…ed estimators do not depend on the number of instruments. Finally, we suggest a data-driven method for selecting the regularization parameter. Interestingly, our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument. Key Words: Factor model, many instruments, mean square error, regularization methods. An earlier version of this paper was entitled “Instrumental variables estimator based on principal components”. The author thanks the participants of CEME (Cambridge, 2005), ESEM (Vienna, 2006), CESG (Niagara Falls, 2006), Econometric Society Meeting (Chicago, 2007), CIREQ GMM conference (Montreal, November 2007), and the seminar participants at Boston University, the University of Chicago, Harvard-MIT, and HEC Genève for helpful comments. She also gratefully aknowledges partial …nancial support from the National Science Foundation. Address: Université de Montréal, Departement de Sciences Economiques, CP 6128, succ Centre Ville, Montreal, QC H3C3J7, Canada. Email: [email protected], http://www.econ.rochester.edu/Faculty/Carrasco.html