Recently, in causal discovery, invariance properties such as the moment
criterion which two-stage least square estimator leverage have been exploited
for causal structure learning: e.g., in cases, where the causal parameter is
not identifiable, some structure of the non-zero components may be identified,
and coverage guarantees are available. Subsequently, anchor regression has been
proposed to trade-off invariance and predictability. The resulting estimator is
shown to have optimal predictive performance under bounded shift interventions.
In this paper, we show that the concepts of anchor regression and K-class
estimators are closely related. Establishing this connection comes with two
benefits: (1) It enables us to prove robustness properties for existing K-class
estimators when considering distributional shifts. And, (2), we propose a novel
estimator in instrumental variable settings by minimizing the mean squared
prediction error subject to the constraint that the estimator lies in an
asymptotically valid confidence region of the causal parameter. We call this
estimator PULSE (p-uncorrelated least squares estimator) and show that it can
be computed efficiently, even though the underlying optimization problem is
non-convex. We further prove that it is consistent. We perform simulation
experiments illustrating that there are several settings including weak
instrument settings, where PULSE outperforms other estimators and suffers from
less variability.Comment: 85 pages, 15 figure
