On endogeneity and shape invariance in extended partially linear single index models
阐述了扩展广义部分线性单指数模型在灵活建模形状不变规范方面的作用，并提出控制函数方法解决其内生性问题，以增强实证应用性，最后用英国数据分析了支出内生性。
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.