Scrutinizing the Monotonicity Assumption in IV and fuzzy RD designs

Whenever treatment effects are heterogeneous, and there is sorting into treatment based on the gain, monotonicity is a condition that both Instrumental Variable and fuzzy Regression Discontinuity designs must satisfy for their estimate to be interpretable as a LATE. 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.