  Instrumental variable analysis is a widely used method to estimate causal
effects in the presence of unmeasured confounding. When the instruments,
exposure and outcome are not measured in the same sample, Angrist and Krueger
(1992) suggested to use two-sample instrumental variable (TSIV) estimators that
use sample moments from an instrument-exposure sample and an instrument-outcome
sample. However, this method is biased if the two samples are from
heterogeneous populations so that the distributions of the instruments are
different. In linear structural equation models, we derive a new class of TSIV
estimators that are robust to heterogeneous samples under the key assumption
that the structural relations in the two samples are the same. The widely used
two-sample two-stage least squares estimator belongs to this class. It is
generally not asymptotically efficient, although we find that it performs
similarly to the optimal TSIV estimator in most practical situations. We then
attempt to relax the linearity assumption. We find that, unlike one-sample
analyses, the TSIV estimator is not robust to misspecified exposure model.
Additionally, to nonparametrically identify the magnitude of the causal effect,
the noise in the exposure must have the same distributions in the two samples.
However, this assumption is in general untestable because the exposure is not
observed in one sample. Nonetheless, we may still identify the sign of the
causal effect in the absence of homogeneity of the noise.
