Using D-Separation to Calculate Zero Partial Correlations in Linear Models with Correlated Errors
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 ρ 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. In a linear structural equation model (SEM) some partial correlations may be equal to zero for all values of the model’s free parameters (for which the partial correlation is defined). (When we refer to “all values” of the free parameters, we assume that there are no constraints upon the models parameters except for the coefficients and the correlations among the error variables that are fixed at zero.) In this case we will say that the SEM linearly entails that the partial correlation is zero. 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 nonrecursive linear model with uncorrelated errors if and only if the model linearly entails that ρ 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 linearly entails that a particular partial