Semiparametric Efficiency in Irregularly Identified Models∗

This paper considers efficient estimation of structural parameters in a class of semiparametric models where these parameters are irregularly identified. For such models conventional semiparametric efficiency bound calculations of Stein(1956) are of little use as they do not result in finite bounds. Thus the notion of efficiency has to be reconsidered for this class of models and we attempt to address this gap in the literature. Along the lines of the minimax bounds attained in Ibragimov and Has’minskii(1981) we attain not only minimax from below orders of n−1/2, but qualitative bounds as well. We focus on such minimax bounds for three examplesthe regression coefficients in binary choice models under median and mean restrictions (Horowitz(1992,1993) and Lewbel(1998,2000)), the randomly censored regression model considered in Koul et al.(1981), and the ATE under unconfoundedness (Hahn(1998))), noting that under stated conditions, regular rates of convergence are unattainable. ∗We thank H. Hong for helpful comments. †Department of Economics, Duke University, 213 Social Sciences Building, Durham, NC 27708. Phone: (919) 660-1873. ‡Department of Economics, University of California Berkeley, 508-1 Evans Hall, #3880, Berkeley, CA 94720. Support from the National Science Foundation is gratefully acknowledged.