Testing recursive path models with correlated errors using D-separation

This article shows how to extend the inferential test of Shipley (2000b), which is applicable to recursive path models without correlated errors (a directed acyclic graph [DAG] model), to a class of recursive path models that include correlated errors (a semi-Markov model). The path model is first converted to a partial ancestral graph (PAG) and then, for PAGs that do not require latent variables, an inducing path DAG is obtained that is equivalent in its conditional independence relations to the original path model. The null probabilities of the k tests of independence that are implied by this DAG are combined using Fisher's test statistic C = -2SigmaLn(p(i)), which is distributed as a chi-square variate with 2k degrees of freedom.