Koster, J. T. A. Markov properties of nonrecursive causal models. (English) Zbl 0867.62056 Ann. Stat. 24, No. 5, 2148-2177 (1996). Summary: This paper aims to solve an often noted incompatibility between graphical chain models which elucidate the conditional independence structure of a set of random variables and simultaneous equations systems which focus on direct linear interactions and correlations between random variables. Various authors have argued that the incompatibility arises mainly from the fact that in a simultaneous equations system (e.g., a LISREL model) reciprocal causality is possible whereas this is not so in the case of graphical chain models. It is shown that this view is not correct. In fact, the definition of the Markov property embodied in a graph can be generalized to a wider class of graphs which includes certain nonrecursive graphs. The resulting class of reciprocal graph probability models strictly includes the class of chain graph probability models. The class of lattice conditional independence probability models is also strictly included. It is shown that the resulting methodology is directly applicable to quite general simultaneous equations systems that are subject to mild restrictions only. Provided some adjustments are made, general simultaneous equations systems can be handled as well. In all cases, consistency with the LISREL methodology is maintained. Cited in 1 ReviewCited in 19 Documents MSC: 62H99 Multivariate analysis 62J99 Linear inference, regression 05C90 Applications of graph theory 62P99 Applications of statistics Keywords:undirected graph; global Markov property; Gibbs factorization; finite distributive lattice; incompatibility; graphical chain models; conditional independence structure; simultaneous equations systems; nonrecursive graphs; reciprocal graph probability models; chain graph probability models; lattice conditional independence probability models; LISREL Software:LISREL PDFBibTeX XMLCite \textit{J. T. A. Koster}, Ann. Stat. 24, No. 5, 2148--2177 (1996; Zbl 0867.62056) Full Text: DOI