Testability of instrument validity under continuous endogenous variables

Instrumental variables have informed the research on causal inference in economics over the last century. Despite their ubiquity and decided usefulness, the current consensus in the literature is that the validity of an instrument cannot be tested, particularly when the treatment is continuous. This note addresses this issue in two ways. As a first contribution, it presents a proof confirming the conjecture in Pearl (1995), showing that the validity of an instrument indeed 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.