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Unformatted text preview: Simple Estimators for Semiparametric Multinomial Choice Models James L. Powell and Paul A. Ruud University of California, Berkeley March 2008 Preliminary and Incomplete Comments Welcome Abstract This paper considers estimation of the coe cients in a semiparametric multinomial choice model with linear indirect utility functions (with common coe cients but di/ering regressors) and errors that are assumed to be independent of the regressors. This implies that the conditional mean of the vector of dependent indicator variables is a smooth and invertible function of a corresponding vector of linear indices. The estimation method is an extension of an approach proposed by Ahn, Ichimura, and Powell (2004) for monotone single-index regression models to a multi-index setting, estimating the unknown index coe cients (up to scale) by an eigenvector of a matrix de&ned in terms of a &rst- step nonparametric estimator of the conditional choice probabilities. Under suitable conditions, the proposed estimator is root-n-consistent and asymptotically normal. JEL Classi&cation: C24, C14, C13. 1. Introduction While a large literature exists for estimation of single-index regression and semiparametric binary response models &examples include Ahn, Ichimura, and Powell (1996), Han (1987), Hrdle and Stoker (1989), Hrdle and Horowitz (1996), Ichimura (1993), Klein and Spady (1993), Manski (1975, 1985), Newey and Ruud (1991), and Powell, Stock and Stoker (1989), among many others &there are fewer results available on estimation of multiple-index regression models and semiparametric multinomial choice models. Lee (1995) constructs a multinomial analogue to Klein and Spadys (1993) estimator of the semiparametric binary choice model, estimating the index coe cients by minimizing a semipara- metric "prole likelihood" constructed using nonparametric estimators of the choice probabilities as functions of the indices. Lee demonstrates semiparametric e ciency of the estimator under the "dis- tributional index" restriction that the conditional distribution of the errors depends on the regressors only through the indices. As Thompson (1993) shows, though, the semiparametric e ciency bound for multinomial choice under the assumption of independence of the errors &which coincides with the bound the weaker distributional index restriction for binary choice &di/ers when the number of choices exceeds two, indicating possible e ciency improvements from the stronger independence restriction. Ruud (2000) shows that the stronger independence restrictions yield choice probabilities that are in- vertible functions of the indices and whose derivative matrix (with respect to the indices) is symmetric, neither of which needs hold under only the distributional index restriction....
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- Fall '08