Markov Properties for Linear Causal Models with Correlated Errors

Changsung Kang; Jin Tian

A linear causal model with correlated errors, represented by a DAG with bi-directed edges, can be tested by the set of conditional independence relations implied by the model. A global Markov property specifies, by the d-separation criterion, the set of all conditional independence relations holding in any model associated with a graph. A local Markov property specifies a much smaller set of conditional independence relations which will imply all other conditional independence relations which hold under the global Markov property. For DAGs with bi-directed edges associated with arbitrary probability distributions, a local Markov property is given in Richardson (2003) which may invoke an exponential number of conditional independencies. In this paper, we show that for a class of linear structural equation models with correlated errors, there is a local Markov property which will invoke only a linear number of conditional independence relations. For general linear models, we provide a local Markov property that often invokes far fewer conditional independencies than that in Richardson (2003). The results have applications in testing linear structural equation models with correlated errors.

Markov Properties for Linear Causal Models with Correlated Errors

Abstract