Optimal smoothing in semiparametric index approximation of regression functions

Michel Delecroix, Marian Hristache and Valentin Patilea

No 2000,4, SFB 373 Discussion Papers from Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes

Abstract: The problem of approximating a general regression function m(x) = E (Y IX = x) is addressed. As in the case of the c1assical L2-type projection pursuit regression considered by Hall (1989), we propose to approximate m(x) through a regression of Y given an index, that is a unidimensional projection of X. The orientation vector defining the projection of X is taken to be the optimum of a Kullback-Leibler type criterion. The first step of the c1assical projection pursuit regression and the single-index models (SIM) are obtained as particular cases. We define a kernel-based estimator of the 'optimal' orientation vector and we suggest a simple empirical bandwidth selection rule. Finally, the true regression function m(•) is approximated through a kernel regression of Y given the estimated index. Our procedure extends the idea of Härdle, Hall and Ichimura (1993) which propose, in the case of SIM, to minimize an empirical L2-type criterion simultaneously with respect to the orientation vector and the bandwidth. We show that a same bandwidth of order n - 1/5 can be used for the root-n estimation of the orientation and for the kernel approximation of the true regression function. Our methodology could be extended to more accurate multi-index approximations.

Date: 2000
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