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Semiparametric Estimation of Index Coefficients

James L. Powell
James H. Stock
Thomas M. Stoker

Abstract

This paper gives a solution to the problem of estimating coefficients of index models, through the estimation of the density-weighted average derivative of a general regression function. We show how a normalized version of the density-weighted average derivatives can be estimated by certain linear instrumental variables coefficients. Both of the estimators are computationally simple, root-N-consistent and asymptotically normal; their statistical properties do not rely on functional form assumptions on the regression function or the distribution of the data. The estimators, based on sample analogues of the product moment representation of the average derivative, are constructed using nonparametric kernel estimators of the density of the regressors. Asymptotic normality is established using extensions of classical U-statistic theorems, and asymptotic bias is reduced through use of a higher-order kernel. Consistent estimators of the asymptotic variance-covariance matrices of the estimators are given, and a limited Monte Carlo simulation is used to study the practical performance of the procedures.