We define heterogeneous endogeneity as the case where a potentially endogenous regressor is endogenous for some sub-groups of the data but exogenous for other subgroups. We derive an estimator and test procedure based on the control function approach to deal with the phenomenon. We show that accounting for heterogeneous endogeneity can greatly increase the power of endogeneity tests and increase the precision of our estimator over traditional IV. While the gains get larger as the instrument gets weaker and as the relative size of the non-endogenous subgroup gets larger, we find efficiency gains even when the underlying instrument is very strong. We illustrate our approach with an example using data from Abaide et al. (Econometrica 70:91–117).

中文翻译：

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异质内生性

我们将异质内生性定义为这样一种情况，即潜在的内生回归量对于数据的某些子组是内生的，但对于其他子组是外生的。我们基于控制函数方法推导出一个估计器和测试程序来处理这种现象。我们表明，对异质内生性的考虑可以大大提高内生性检验的能力，并提高我们估计量相对于传统 IV 的精度。虽然收益随着工具变弱和非内生子组的相对规模变大而变得更大，但我们发现即使基础工具非常强大，效率也会提高。我们用来自 Abaide 等人的数据的例子来说明我们的方法。（计量经济学 70：91-117）。