  We study causal effect estimation from a mixture of observational and
interventional data in a confounded linear regression model with multivariate
treatments. We show that the statistical efficiency in terms of expected
squared error can be improved by combining estimators arising from both the
observational and interventional setting. To this end, we derive methods based
on matrix weighted linear estimators and prove that our methods are
asymptotically unbiased in the infinite sample limit. This is an important
improvement compared to the pooled estimator using the union of interventional
and observational data, for which the bias only vanishes if the ratio of
observational to interventional data tends to zero. Studies on synthetic data
confirm our theoretical findings. In settings where confounding is substantial
and the ratio of observational to interventional data is large, our estimators
outperform a Stein-type estimator and various other baselines.
