On Endogeneity and Shape Invariance in Extended Partially Linear Single Index Models

Abstract

In this paper, the important (but so far unrevealed) usefulness of the extended generalized partially linear single-index (EGPLSI) model introduced by Xia et al. (1999) in its ability to model a flexible shape-invariant specification is elaborated. More importantly, a control function approach is proposed to address the potential endogeneity problems in the EGPLSI model in order to enhance its applicability to empirical studies. In the process, it is shown that the attractive asymptotic features of the single-index type of a semiparametric model are still valid in our proposed estimation procedure given intrinsic generated covariates. Our newly developed method is then applied to address the endogeneity of expenditure in the semiparametric analysis of a system of empirical Engel curves by using the British data, highlights the convenient applicability of our proposed method.