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 \citet*{pearl1995testability}  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 modelreestablish theoretical testability . Continuity is already enough. Monotonicity introduces further testable implications .
This note presents a proof of the conjecture in \citet*{pearl1995testability} about testing the validity of an instrumental variable in hidden variable models. It implies that instrument validity cannot be tested in the case where the endogenous treatment is continuously distributed. This stands in contrast to the classical testability results for instrument validity when the treatment is discrete. However, imposing weak structural assumptions on the model, such as continuity between the observable variables, can re-establish theoretical testability in the continuous setting .
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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
This note
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confirming
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Pearl (1995), showing that
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about testing
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instrument indeed
instrumental variable in hidden variable models. It implies that instrument validity
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most general case when the endogenous variable is continuous. However, as a second contribution, it shows that already weak restrictions on the instrumental variable modelreestablish theoretical testability . Continuity is already enough. Monotonicity introduces further testable implications
case where the endogenous treatment is continuously distributed. This stands in contrast to the classical testability results for instrument validity when the treatment is discrete. However, imposing weak structural assumptions on the model, such as continuity between the observable variables, can re-establish theoretical testability in the continuous setting