Before revision: In this note we prove Pearl's conjecture , showing that the exclusion restriction of an instrument  cannot be tested without structural assumptions in general instrumental variable models with a continuously distributed endogenous variable . This stands in contrast to models with a discretely distributed endogenous variable, where tests have been devised in the literature, and shows that there is a fundamental difference between the continuous and the discrete case .
After revision: In this note we prove Pearl's conjecture , showing that the exclusion restriction of an instrument cannot be tested in the most general case when the endogenous variable is continuous. However, as a second contribution, it shows that already weak restrictions on the instrumental variable model reestablish theoretical testability. Continuity is already enough. Monotonicity introduces further testable implications .