Whenever treatment effects are heterogeneous, and there is sorting into treatment based on the gain, monotonicity is a condition that both instrumental variable (IV) and fuzzy regression discontinuity (RD) designs must satisfy for their estimate to be interpretable as a local average treatment effect. However, applied economic work often omits a discussion of this important assumption. A possible explanation for this missing step is the lack of a clear framework to think about monotonicity in practice. In this paper, we use an extended Roy model to provide insights into the interpretation of IV and fuzzy RD estimates under various degrees of treatment effect heterogeneity, sorting on gain and violation of monotonicity. We then extend our analysis to two applied settings to illustrate how monotonicity can be investigated using a mix of economic insights, data patterns and formal tests. For both settings, we use a Roy model to interpret the estimate even in the absence of monotonicity. We conclude with a set of recommendations for the applied researcher.

中文翻译：

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仔细检查 IV 和模糊 RD 设计中的单调性假设*

每当治疗效果是异质的，并且存在基于增益的治疗分类时，单调性是工具变量 (IV) 和模糊回归不连续性 (RD) 设计必须满足的条件，才能将它们的估计解释为局部平均治疗效果. 然而，应用经济学工作经常忽略对这一重要假设的讨论。对这一缺失步骤的一个可能解释是缺乏一个清晰的框架来考虑实践中的单调性。在本文中，我们使用扩展的 Roy 模型来深入了解在不同程度的治疗效果异质性、增益排序和单调性破坏下对 IV 和模糊 RD 估计的解释。然后，我们将分析扩展到两个应用设置，以说明如何使用经济见解、数据模式和正式测试的组合来研究单调性。对于这两种设置，即使在没有单调性的情况下，我们也使用 Roy 模型来解释估计。最后，我们为应用研究人员提出了一系列建议。