  Suppose one is interested in estimating causal effects in the presence of
potentially unmeasured confounding with the aid of a valid instrumental
variable. This paper investigates the problem of making inferences about the
average treatment effect when data are fused from two separate sources, one of
which contains information on the treatment and the other contains information
on the outcome, while values for the instrument and a vector of baseline
covariates are recorded in both. We provide a general set of sufficient
conditions under which the average treatment effect is nonparametrically
identified from the observed data law induced by data fusion, even when the
data are from two heterogeneous populations, and derive the efficiency bound
for estimating this causal parameter. For inference, we develop both parametric
and semiparametric methods, including a multiply robust and locally efficient
estimator that is consistent even under partial misspecification of the
observed data model. We illustrate the methods through simulations and an
application on public housing projects.
