  Instrumental variable models allow us to identify a causal function between
covariates $X$ and a response $Y$, even in the presence of unobserved
confounding. Most of the existing estimators assume that the error term in the
response $Y$ and the hidden confounders are uncorrelated with the instruments
$Z$. This is often motivated by a graphical separation, an argument that also
justifies independence. Positing an independence restriction, however, leads to
strictly stronger identifiability results. We connect to the existing
literature in econometrics and provide a practical method called HSIC-X for
exploiting independence that can be combined with any gradient-based learning
procedure. We see that even in identifiable settings, taking into account
higher moments may yield better finite sample results. Furthermore, we exploit
the independence for distribution generalization. We prove that the proposed
estimator is invariant to distributional shifts on the instruments and
worst-case optimal whenever these shifts are sufficiently strong. These results
hold even in the under-identified case where the instruments are not
sufficiently rich to identify the causal function.
