  While causal models are robust in that they are prediction optimal under
arbitrarily strong interventions, they may not be optimal when the
interventions are bounded. We prove that the classical K-class estimator
satisfies such optimality by establishing a connection between K-class
estimators and anchor regression. This connection further motivates a novel
estimator in instrumental variable settings that minimizes the mean squared
prediction error subject to the constraint that the estimator lies in an
asymptotically valid confidence region of the causal coefficient. We call this
estimator PULSE (p-uncorrelated least squares estimator), relate it to work on
invariance, show that it can be computed efficiently as a data-driven K-class
estimator, even though the underlying optimization problem is non-convex, and
prove consistency. We evaluate the estimators on real data and perform
simulation experiments illustrating that PULSE suffers from less variability.
There are several settings including weak instrument settings, where it
outperforms other estimators.
