Monotone single index models

Monotone Single Index Models
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Unformatted text preview: Simple Estimators for Monotone Index Models Hyungtaik Ahn Dongguk University, Hidehiko Ichimura University College London, James L. Powell University of California, Berkeley (powell@econ.berkeley.edu) June 2004 Abstract In this paper, estimation of the coe¢ cients in a &single-index¡regression model is considered under the assumption that the regression function is a smooth and strictly monotonic function of the index. The estimation method follows a &two-step¡ approach, where the £rst step uses a nonparametric regression estimator for the dependent variable, and the second step estimates the unknown index coe¢ cients (up to scale) by an eigenvector of a matrix de£ned in terms of this £rst- step estimator. The paper gives conditions under which the proposed estimator is root-n-consistent and asymptotically normal. JEL Classi&cation: C24, C14, C13. Acknowledgements This research was supported by the National Science Foundation. Hyungtaik Ahn¤s research was supported by Dongguk Research Fund. We are grateful to Bo HonorØ, Ekaterini Kyriazidou, Robin Lumsdaine, Thomas Rothenberg, Paul Ruud, and Mark Watson for their helpful comments. 1. Introduction Estimation of the unknown coe¢ cients & in the single index regression model E ( y i j x i ) = G ( x i & ) ; (1.1) where y i and x i are observable and G ( & ) is an unknown function, has been investigated in a number of papers in the econometric literature on semiparametric estimation. (A survey of these estima- tors is given in Powell (1994).) Some estimation methods, like the &average derivative¡approach of H£rdle and Stoker (1989) and Powell, Stock, and Stoker (1989) and the &density-weighted least squares¡estimator of Ruud (1986) and Newey and Ruud (1991) exploit an assumption of smooth- ness (continuity and di/erentiability) of the unknown function G , but require all components of the regressor vector x to be jointly continuously distributed, which rarely applies in practice. H£rdle and Horowitz (1996) has extended the average derivative estimator to allow for discrete regressors at the expense of introducing four additional nuisance parameters to be chosen by users of their estimator in addition to the standard smoothing parameter choice required in all nonparametric estimators. Other estimation methods which assume smoothness of G include the &single-index regression¡estimators of Ichimura (1993a), Ichimura and Lee (1991), and, for the special case of a binary dependent variable, Klein and Spady (1993); these estimation methods permit general distributions of the regressors, but can be computationally burdensome, since they involve min- imization problems with nonparametric estimators of G whose solutions cannot be written in a simple closed form. Still other estimators of were proposed for the &generalized regression model¡ proposed by Han (1987), y i = T ( x i & ;" i ) ; (1.2) where the unknown transformation T ( & ) is assumed to be monotonic in its ¤rst argument, and where the unobservable error term...
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