The control function approach which employs an instrumental variable excluded from the outcome equation is a very common solution to deal with the problem of endogeneity in nonseparable models. Exclusion restrictions, however, are frequently controversial. We first argue that, in a nonparametric triangular structure typical of the control function literature, one can actually test this exclusion restriction provided the instrument satisfies a local irrelevance condition. Second, we investigate identification without such exclusion restrictions, i.e., if the “instrument” that is independent of the unobservables in the outcome equation also directly affects the outcome variable. In particular, we show that identification of average causal effects can be achieved in the two most common special cases of the general nonseparable model: linear random coefficients models and single index models.

中文翻译：

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测试并放宽控制函数方法中的排除限制

采用从结果方程中排除工具变量的控制函数方法是处理不可分离模型中内生性问题的一种非常常见的解决方案。然而，排除限制经常引起争议。我们首先认为，在控制函数文献中典型的非参数三角形结构中，只要仪器满足局部不相关条件，就可以实际测试这种排除限制。其次，我们研究没有这种排除限制的识别，即独立于结果方程中不可观察量的“工具”是否也直接影响结果变量。特别是，我们表明，在一般不可分离模型的两种最常见的特殊情况下，可以实现平均因果效应的识别：线性随机系数模型和单指数模型。