Accurately predicting conditional average treatment effects (CATEs) is crucial in personalized medicine and digital platform analytics. Since often the treatments of interest cannot be directly randomized, observational data is leveraged to learn CATEs, but this approach can incur significant bias from unobserved confounding. One strategy to overcome these limitations is to seek latent quasi-experiments in instrumental variables (IVs) for the treatment, for example, a randomized intent to treat or a randomized product recommendation. This approach, on the other hand, can suffer from low compliance, i.e., IV weakness. Some subgroups may even exhibit zero compliance meaning we cannot instrument for their CATEs at all. In this paper we develop a novel approach to combine IV and observational data to enable reliable CATE estimation in the presence of unobserved confounding in the observational data and low compliance in the IV data, including no compliance for some subgroups. We propose a two-stage framework that first learns biased CATEs from the observational data, and then applies a compliance-weighted correction using IV data, effectively leveraging IV strength variability across covariates. We characterize the convergence rates of our method and validate its effectiveness through a simulation study. Additionally, we demonstrate its utility with real data by analyzing the heterogeneous effects of 401(k) plan participation on wealth.
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