The Estimation Accuracy of the Conditional Instrumental Variable (CIV) Estimator in Non-recursive Structural Equation Models

Abstract

This paper considers a situation in which the cause-effect relationships among random variables can be described by a non-recursive linear structural causal model (SCM). We then extend the applicability of the conditional instrumental variable (CIV) method—one of the most powerful tools for estimating the direct effect based on statistical data—from recursive linear SCMs to a broader class that includes both recursive and non-recursive linear SCMs. In addition, we formulate the asymptotic variance of the estimated direct effect when the direct effect is identifiable using the CIV method. Furthermore, when there are several IVs, noting that the selection of CIVs is not unique, we show that, in some situations, the differences among them can be explained by the graph structure from the viewpoint of asymptotic variance. This paper also proposes a unified CIV estimator for the direct effect with improved estimation accuracy. Overall, our work establishes an IV-based framework that unifies recursive and non-recursive linear structural equation models.