On endogeneity and shape invariance in extended partially linear single index models

In this article, the usefulness of the extended generalized partially linear single-index (EGPLSI) model introduced by Xia et al. 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 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 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.