Using d-separation to calculate zero partial correlations in linear models with correlated errors

Abstract: "It has been shown in Spirtes(1995) that X and Y are d- separated given Z in a directed graph associated with a recursive or non- recursive linear model without correlated errors if and only if the model entails that [rho][subscript xy.z] = 0. This result cannot be directly applied to a linear model with correlated errors, however, because the standard graphical representation of a linear model with correlated errors is not a directed graph. The main result of this paper is to show how to associate a directed graph with a linear model L with correlated errors, and then use d-separation in the associated directed graph to determine whether L entails that a particular partial correlation is zero."