Is Monotonicity in an IV and RD design testable? No, but you can still check it

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 have to satisfy for their estimand to be interpretable as a LATE. Angrist and Imbens (1995) argue that the monotonicity assumption is testable whenever the treatment is multivalued. We show that their test is informative if counterfactuals are observed. Yet applying the test without observing counterfactuals, as it is generally done, is not. Nevertheless, we argue that monotonicity can and should be investigated using a mix of economic intuition and data patterns, just like other untestable assumptions in an IV or RD design. We provide examples in a variety of settings as a guide to practice.