Many policy evaluations occur in settings where treatment is randomized at the cluster level, and there is treatment noncompliance within each cluster. For example, villages might be assigned to treatment and control, but residents in each village may choose to comply or not with their assigned treatment status. When noncompliance is present, the instrumental variables framework can be used to identify and estimate causal effects. While a large literature exists on instrumental variables estimation methods, relatively little work has been focused on settings with clustered treatments. Here, we review extant methods for instrumental variable estimation in clustered designs and derive both the finite and asymptotic properties of these estimators. We prove that the properties of current estimators depend on unrealistic assumptions. We then develop a new IV estimation method for cluster randomized trials and study its formal properties. We prove that our IV estimator allows for possible treatment effect heterogeneity that is correlated with cluster size and is robust to low compliance rates within clusters. We evaluate these methods using simulations and apply them to data from a randomized intervention in India.
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