Nonparametric Estimation of Triangular Simultaneous Equations Models under Weak Identification

This paper analyzes the problem of weak instruments on identification, estimation, and inference in a simple nonparametric model of a triangular system. The paper derives a necessary and sufficient rank condition for identification, based on which weak identification is established. Then, nonparametric weak instruments are defined as a sequence of reduced-form functions where the associated rank shrinks to zero. The problem of weak instruments is characterized as concurvity and to be similar to the ill-posed inverse problem, which motivates the introduction of a regularization scheme. The paper proposes a penalized series estimation method to alleviate the effects of weak instruments and shows that it achieves desirable asymptotic properties. Monte Carlo results are presented, and an empirical example is given in which the effect of class size on test scores is estimated nonparametrically. ∗I am very grateful to my advisors, Donald Andrews and Edward Vytlacil, and committee members, Xiaohong Chen and Yuichi Kitamura for their inspiration, guidance and support. I am deeply indebted to Donald Andrews for his thoughtful advice throughout the project. The earlier version of this paper has benefited from discussions with Joseph Altonji, Ivan Canay, Philip Haile, Keisuke Hirano, Han Hong, Joel Horowitz, Seokbae Simon Lee, Oliver Linton, Whitney Newey, Byoung Park, Peter Phillips, Andres Santos, and Alex Torgovitsky. I gratefully acknowledge financial support from a Carl Arvid Anderson Prize from the Cowles Foundation. I also thank the seminar participants at Yale, UT Austin, Chicago Booth, Notre Dame, SUNY Albany, Sogang, SKKU, and Yonsei, as well as the participants at NASM and Cowles Summer Conference.