A is pxp, and by convention has zero elements on its principal diagonal. A contains the coefficients indicating how much unit change in some endogenous variables produces change in other endogenous variables. The model also provides for some of the Y's and some of the X's to be latent variables as in common factor analysis. The model is generally used to test substantive hypotheses involving causal relations of a linear nature.
Researchers specify a priori certain parameters of the matrices A and G to overidentify the model.
As long as the unspecified parameters are identified, these are estimated either by ML or GLS methods. It is assumed that the disturbance random variables in E are unrelated to the exogenous variables in X and to disturbance variables of other endogenous variables that are causal determinants of the endogenous variable to which the disturbance variable in question is assigned. Exogenous variables may be correlated. Because the covariance matrix among the manifest variables can be derived as functions of the A and G matrices, the model is generally tested by how well it reproduces the covariance matrix among the manifest variables. This happens, for instance, in mechanics and control theory.