Appendix 1.A Expressing Variances and Covariances Among Observed Variables as Functions of Model Parameters
Let us denote 
the population variance/covariance matrix of variables y and x, then
(1.22) 
where the diagonal elements are variances of the variables y and x, respectively; and the off-diagonal elements are covariances among y and x. In SEM it is hypothesized that the population variance/covariance matrix of y and x can be expressed as a function of the model parameters 
, that is:
(1.23) 
where 
is called the model implied variance/covariance matrix.
Based on the three basic SEM equations [Equation (1.1)], we can derive that 
can be expressed as functions of the parameters in the eight fundamental SEM matrices. Let us start with the variance/covariance matrix of y, then the variance/covariance matrix of x and the variance/covariance matrix of y and x, and then finally assemble them together.
The variance/covariance matrix of y can be expressed as:
were 
is the variance/covariance matrix of the error term 
.
(1.25) 
Assuming that 
is independent of 
, then
where 
is the variance/covariance matrix of the latent variable 
; 
is the variance/covariance matrix of the residual 
. Substituting Equation (1.26) into Equation (1.24), we have:
(1.27) 
This equation implies that variances/covariances of the observed y variables are a function of model parameters such as factor loadings 
, path coefficients 
and 
, the variances/covariances 
of the exogenous latent variables, residual variances/covariances matrix 
, and the error variances/covariances 
.
The variance/covariance matrix of x can be expressed as:
(1.28) 
Assuming that 
is independent of 
, then
where 
is the variance/covariance matrix of the error term 
. Equation (1.29) implies that variances/covariances of the observed x variables are a function of model parameters, such as the loadings 
, the variances/covariances 
of the exogenous latent variables, and the error variances/covariances 
.
The covariance matrix among x and y can be expressed as:
(1.30) 
Assuming that 
and 
are independent of each other and independent of the latent variables, then
(1.31) 
Thus, the variances and covariances among the observed variables x and y can be expressed as in terms of the model parameters:
(1.32) 
where the upper right part of the matrix is the transpose of the covariance matrix among x and y. Each element in the model implied variance/covariance matrix 
is a function of model parameters. For a set of specific model parameters from the eight SEM fundamental matrices that constitute a SEM model, there is one and only one corresponding model implied variance/covariance matrix 
(Hayduk, 1987).