Unobserved confounding is the main obstacle to causal effect estimation from observational data. Instrumental variables (IVs) are widely used for causal effect estimation when there exist latent confounders. With the standard IV method, when a given IV is valid, unbiased estimation can be obtained, but the validity requirement of standard IV is strict and untestable. Conditional IV has been proposed to relax the requirement of standard IV by conditioning on a set of observed variables (known as a conditioning set for a conditional IV). However, the criterion for finding a conditioning set for a conditional IV needs complete causal structure knowledge or a directed acyclic graph (DAG) representing the causal relationships of both observed and unobserved variables. This makes it impossible to discover a conditioning set directly from data. In this paper, by leveraging maximal ancestral graphs (MAGs) in causal inference with latent variables, we propose a new type of IV, ancestral IV in MAG, and develop the theory to support data-driven discovery of the conditioning set for a given ancestral IV in MAG. Based on the theory, we develop an algorithm for unbiased causal effect estimation with an ancestral IV in MAG and observational data. Extensive experiments on synthetic and real-world datasets have demonstrated the performance of the algorithm in comparison with existing IV methods.

Instrumental variables (IVs) are widely used to identify causal effects. For this purpose IVs have to be exogenous, i.e., causally unrelated to all variables in the model except the explanatory variable X . It can be hard to find such variables. A generalized IV method has been proposed that only requires exogeneity conditional on a set of covariates. This leads to a wider choice of potential IVs, but is rarely used yet. Here we address two issues with conditional IVs. First, they are conceptually rather distant to standard IVs; even variables that are independent of X could qualify as conditional IVs. We propose a new concept called ancestral IV , which interpolates between the two existing notions. Second, so far only exponential-time algorithms are known to find conditional IVs in a given causal diagram. Indeed, we prove that this problem is NP-hard. Nevertheless, we show that whenever a conditional IV exists, so does an ancestral IV, and ancestral IVs can be found in polynomial time. Together this implies a complete and constructive solution to causal effect identification using IVs in linear causal models.