Structural Equation Modeling

The General Model

The general structural equation model, as outlined by Jöreskog (1973), consists of two parts: (a) the structural part linking latent variables to each other via systems of simultaneous equations, and (b) the measurement part linking latent variables to observed variables via a restricted (confirmatory) factor model. The structural part of the model can be written as

where η is a vector of endogenous (criterion) latent variables, α is a vector of structural intercepts, ξ is a vector of exogenous (predictor) latent variables, B is a matrix of regression coefficients relating the latent endogenous variables to each other, is a matrix of regression coefficients relating endogenous variables to exogenous variables, and ς is a vector of disturbance terms.

The latent variables are linked to observable variables via measurement equations for the endogenous variables and exogenous variables. These equations are defined as

and

where Ay and Ax are matrices of factor loadings, respectively, and ε and δ are vectors of uniqueness, respectively. In addition, the general model specifies variances and covariances for ξ, ς, ε, and δ, denoted as, Θε, and Θδ, respectively. A path diagram of a prototypical structural equation model is given in Figure 1.

Structural equation modeling is a methodology designed primarily to test substantive theories. As such, a theory might be sufficiently developed to suggest that certain constructs do not affect other constructs, that certain variables do not load on certain factors, and that certain disturbances and measurement errors do not covary. This implies that some of the elements of B, y, x, Θε, and Θδ are fixed to zero by hypothesis. The remaining parameters are free to be estimated. The pattern of fixed and free parameters implies a specific structure for the covariance matrix of the observed variables. In this way, structural equation modeling can be seen as a special case of a more general covariance structure model defined as = (Ω), where is the population covariance matrix, and (Ω) is a matrix-valued function of the parameter vector Ω containing all of the parameters of the model.

Identification of Structural Equation Models

A prerequisite for the estimation of the parameters of the model is to establish whether the parameters are identified. A definition of identification can be given by considering the problem from the covariance structure modeling perspective. The population covariance matrix contains population variances and covariances that are assumed to follow a model characterized by a set of parameters contained in Ω. We know that the variances and covariances in can be estimated by their sample counterparts in the sample covariance matrix S using straightforward formulae for the calculation of sample variances and covariances. Thus, the parameters in are identified. What needs to be [Page 1089]established prior to estimation is whether the unknown parameters in Ω are identified from the elements of .

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