Structural equation modeling (SEM) has dominated causal analysis in the social and behavioral sciences since the 1960s. Currently, many SEM practitioners are having difficulty articulating the causal content of SEM and are seeking foundational answers.

Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultimately fallacious interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to awed methods for applying background knowledge to model selection. We present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated as the equilibrium distribution of dynamic models with feedback. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for DAGs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feedback in econometrics. Keywords: Causal model; cha...

We consider acyclic directed mixed graphs, in which directed edges (x → y) and bi-directed edges (x ↔ y) may occur. A simple extension of Pearl’s d-separation criterion, called m-separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m-separation criterion.

This article establishes a new criterion for the identification of recursive linear models in which some errors are correlated. We show that identification is ensured as long as error correlation does not exist between a cause and its direct effect; no restrictions are imposed on errors associated with indirect causes. Before structural equation models (SEM) can be estimated and evaluated against data, a researcher must make sure that the parameters of the estimated model are identified, namely, that they can be determined uniquely from the population covariance matrix. The importance of testing identification prior to data analysis is summarized succinctly by Rigdon (1995): To avoid devoting research resources toward a hopeless cause (and to avoid ignoring productive research avenues out of an unfounded fear of underidentification), researchers need a way to quickly evaluate a model's identification status before data are collected. Furthermore, because models are often altered in the course of research (Joreskog, 1993), researchers need a technique that helps them understand the impact of potential structural changes on the identification status of the model, (p. 359) It is well known that, in recursive path models with correlated errors, the identification problem is unsolved. In other words, we are not in possession of a necessary and sufficient criterion for deciding whether the parameters in such a model can be computed from the population covariance matrix of the observed variables. Certain restricted classes of models are nevertheless known to be identifiable, and

Partial Least Squares (PLS) is a class of techniques for modeling the association between blocks of observed variables by means of latent variables. Originated by Herman Wold in the 1970's, PLS is important in many scientific disciplines, including psychology, economics, chemistry, medicine and the pharmaceutical sciences, and process modelling (Rannar et al. [10]). PLS has many variants. The algorithm can be run in two modes, called A and B. It can be applied to data that are divided into two or more blocks. The general algorithm due to Wold can be followed, or it can be modified. Wold stated his general algorithm in terms different from those customarily used by statisticians. In the current work the algorithm is placed into a more familiar notation, and the two-block case is discussed. Canonical two-block Mode A PLS (PLS-C2A) is stated. Its properties, and the properties of the coefficients it computes, are examined in detail. In particular PLS-C2A is shown to be a special case of Wold's ...

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This paper deals with the problem of identifying direct causal effects in recursive linear structural equation models. The paper establishes a sufficient criterion for identifying individual causal effects and provides a procedure computing identified causal effects in terms of observed covariance matrix.

The local Markov condition for a DAG to be an independence map of a probability distribution is well known. For DAGs with latent variables, represented as bi-directed edges in the graph, the local Markov property may invoke exponential number of conditional independencies. This paper shows that the number of conditional independence relations required may be reduced if the probability distributions satisfy the composition axiom. In certain types of graphs, only linear number of conditional independencies are required. The result has applications in testing linear structural equation models with correlated errors. 1

In recursive linear models, the multivariate normal joint distribution of all variables exhibits a dependence structure induced by recursive systems of linear structural equations. Such models appear in particular in seemingly unrelated regressions, structural equation modelling, simultaneous equation systems, and in Gaussian graphical modelling. We show that recursive linear models that are ‘bow-free’ are well-behaved statistical models, namely, they are everywhere identifiable and form curved exponential families. Here, ‘bow-free ’ refers to models satisfying the condition that if a variable x occurs in the structural equation for y, then the errors for x and y are uncorrelated. For the computation of maximum likelihood estimates in ‘bow-free ’ recursive linear models we introduce the Residual Iterative Conditional Fitting (RICF) algorithm. Compared to existing algorithms RICF is easily implemented requiring only least squares computations, has clear convergence properties, and finds parameter estimates in closed form whenever possible. KEY WORDS: Linear structural equation model; curved exponential family; maximum likelihood estimation; residual iterative conditional fitting; bow-free acyclic path diagrams; BAP. 1

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 the local Markov property will invoke only 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.