  Economists are often interested in the mechanisms by which a particular
treatment affects an outcome. This paper develops tests for the ``sharp null of
full mediation'' that the treatment $D$ operates on the outcome $Y$ only
through a particular conjectured mechanism (or set of mechanisms) $M$. A key
observation is that if $D$ is randomly assigned and has a monotone effect on
$M$, then $D$ is a valid instrumental variable for the local average treatment
effect (LATE) of $M$ on $Y$. Existing tools for testing the validity of the
LATE assumptions can thus be used to test the sharp null of full mediation when
$M$ and $D$ are binary. We develop a more general framework that allows one to
test whether the effect of $D$ on $Y$ is fully explained by a potentially
multi-valued and multi-dimensional set of mechanisms $M$, allowing for
relaxations of the monotonicity assumption. We further provide methods for
lower-bounding the size of the alternative mechanisms when the sharp null is
rejected. An advantage of our approach relative to existing tools for mediation
analysis is that it does not require stringent assumptions about how $M$ is
assigned; on the other hand, our approach helps to answer different questions
than traditional mediation analysis by focusing on the sharp null rather than
estimating average direct and indirect effects. We illustrate the usefulness of
the testable implications in two empirical applications.
