Let **m***x*
be a vector of manifest exogenous variables. Partition the variables into
vectors **s**1 and **s**2 as follows:

(34)

and

(35)

Then one may write

**s**1
= **Bs**1
+ **Gs**2 (36)

where

(37)

and

(38)

Assuming a nonsingular **I -
B, **Equation 23 may be rewritten as

**s**1
= (**I** - **B**)-1**Gs**2. (39)

Let **G** be a filter matrix which extracts
the manifest variables from **s**1,
and let **X = ***E*(**s**2**s**2¢ )
be the covariance matrix for **s**2.

Then

(40)

and one obtains the following model for covariance structure:

**S**
= **G**(**B** - **I**)-1**GXG¢ **(**B¢ **-
**I**)-1**G¢ ** (41)

The covariance matrix **Cov **(**s**1) = **Y**
for manifest exogenous, manifest endogenous, and latent endogenous variables
may be computed as

**Y**
= (**B** - **I**)-1**GXG¢ **(**B¢ **-
**I**)-1 (42)

The model of Equation 41 allows direct correspondence
between all permissible PATH1 statements and the algebraic model. There
is no need to concoct dummy latent variables. All possible types of relationships
among manifest and latent variables are accounted for. After a model is
complete, all variables can immediately be assigned to one of the 4 vectors
**m***n*, **m***x*,
**l***n*, or **l***x*.
All coefficients (for arrows) are then assigned to the matrices **F**1 through **F**8.
The column index for a variable (in any of these 8 matrices) represents
the variable from which the arrow points, the row index the variable to
which the arrow points. Coefficients for wires are represented in a similar
manner in the matrix **X.**

The model of Equation 41 sacrifices some of
the simplicity of the RAM model, because variables must be assigned to
4 types before the location of model coefficients can be determined*.
*However, in our typology and with the *SEPATH*
diagramming rules the typing of each variable into one of 4 categories
can be determined by looking *only at that variable in the path diagram.
*Because two headed arrows are eliminated, a variable is endogenous
if and only if it has an arrowhead directed toward it. A variable is latent
if and only if it appears in an oval or circle. (If it is not already
obvious, let us note that with two headed arrows one must look away from
the variable of interest to determine if the variable is endogenous, because
an arrowhead attached to the variable and pointing to it might be two-headed!
*Not only is the SEPATH system less cluttered, but it is also visually
more efficient.)*

Two final points should be emphasized. First,
it is not clear which of the above models is, in any overall sense, "superior"
to the others. The *SEPATH* model of
Equation 41 was chosen primarily because it offered a good trade-off between
certain conceptual and computational advantages. However, there are also
definite advantages, both conceptual and computational, in each of the
other model formulations.

Second, it is possible to express some of the
models as special cases of the others. For example, the LISREL model can
be written easily as a COSAN model. To see why, suppose that the manifest
and latent variables were ordered in the **v** of Equation 10 so that

(43)

Then it follows immediately that one may write

v =**F*****v** + **r*** where

(44)

and

(45)

If **P* **is defined as the covariance matrix
of **r***, then clearly one can test any LISREL model as a COSAN model
of the form

**S**
= **G **(**F*** - **I**)-1
**P **(**F¢
*** - **I**)-1**G¢ ** (46)

where **G** is a matrix which filters **x**
and **y** from **v**.