By Grenander U.
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See Hanushek and Jackson (1977) for a relatively simple introduction, Kmenta (1971) for a more thorough and demanding treatment, and Malinvaud (1970) for a very demanding treatment that approaches estimation in the manner closest to what is used for the covariance structure model. These models can be estimated with a variety of commonly available software packages. It is beyond the scope of this discussion to review these methods. Rather, the focus is on applications that cannot be easily incorporated into standard statistical packages and that have been relatively neglected in the econometrics literature.
In the structural equation model the assumption that the variables are measured without error is often unrealistic, requiring the introduction of errors in variables or measurement error. Those using the factor model, in which errors in variables are of major concern, are often interested in making statements about the structural relationships among the unobserved factors. That is, there is a need for a structural equation model relating the factors. The result of the converging needs of those using structural equation models and factor models is the covariance structure model.
Exogenous variables and errors in equations are uncorrelated: E(xz') = 0 or E(zx') = 0. c. None of the structural equations is redundant: 1 (I-B)-1exists. , that 1 exists). This assumption is not very restrictive, simply meaning that none of the equations in the model is redundant. 1, a number of covariance matrices can be defined. The covariance among the errors in equations, the zi's, are contained in the symmetric matrix Y of dimension (r × r). , E(zi) = 0), Y can be defined as: Y = E(xz'). Values of Y are generally unknown, although off-diagonal elements can be restricted to zero to indicate that errors in equations are uncorrelated across two equations.