Optimal Smoothing for aComputationally andStatistically Efficient SingleIndex Estimator

In semiparametric models it is a common approach to under-smooth the nonparametric functions inorder that estimators of the finite dimensional parameters can achieve root-n consistency. The requirementof under-smoothing may result as we show from inefficient estimation methods or technical diffculties.Based on local linear kernel smoother, we propose an estimation method to estimate the single-index modelwithout under-smoothing. Under some conditions, our estimator of the single-index is asymptoticallynormal and most effcient in the semi-parametric sense. Moreover, we derive higher expansions for ourestimator and use them to define an optimal bandwidth for the purposes of index estimation. As aresult we obtain a practically more relevant method and we show its superior performance in a variety ofapplications.