Scrutinizing the Monotonicity Assumption in IV and fuzzy RD designs*
用扩展的Roy模型分析工具变量和模糊断点回归在违背单调性假设时的估计量含义，并结合两个应用案例展示如何用经济直觉、数据模式和正式检验来考察单调性，为实证研究者提供建议。
Abstract 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.