Matrix () : Effects of Column Variables on Row Variables

The row and column variables of matrix are the set of manifest and latent variables in the RAM model. Unlike the LINEQS model, the set of latent variables in the RAM model matrix does not include the error or disturbance variables. Each entry or element in the matrix represents an effect of the associated column variable on the associated row variable or a path coefficient from the associated column variable to the associated row variable. A zero entry means an absence of a path or an effect.

The pattern of matrix determines whether a variable is endogenous or exogenous. A variable in the RAM model is endogenous if its associated row in the matrix has at least one nonzero entry. Any other variable in the RAM model is exogenous.

Mathematically, you do not need to arrange the set of variables for matrix in a particular order, as long as the order of variables is the same for the rows and the columns. However, arranging the variables according to whether they are endogenous or exogenous is useful for showing the partitions of the model matrices and certain mathematical properties. See the section Partitions of the RAM Model Matrices and Some Restrictions for details.

Matrix (): Variances, Covariances, Partial Variances, and Partial Covariances

The row and column variables of matrix refer to the same set of manifest and latent variables that are defined in the RAM model matrix . The diagonal entries of contain variances or partial variances of variables. If a variable is exogenous, then the corresponding diagonal element in the matrix represents its variance. Otherwise, the corresponding diagonal element in the matrix represents its partial variance. This partial variance is an unsystematic source of variance that is not explained by the interrelationships of variables in the model. In most cases, you can interpret a partial variance as the error variance for an endogenous variable.

The off-diagonal elements of contain covariances or partial covariances among variables. An off-diagonal element in that is associated with exogenous row and column variables represents covariance between the two exogenous variables. An off-diagonal element in that is associated with endogenous row and column variables represents partial covariance between the two variables. This partial covariance is unsystematic, in the sense that it is not explained by the interrelationships of variables in the model. In most cases, you can interpret a partial covariance as the error covariance between the two endogenous variables involved. An off-diagonal element in that is associated with one exogenous variable and one endogenous variable in the row and column represents the covariance between the exogenous variable and the error of the endogenous variable. While this interpretation sounds a little awkward and inelegant, this kind of covariance, fortunately, is rare in most applications.

Vector (): Intercepts and Means

The row variables of vector refer to the same set of manifest and latent variables that are defined in the RAM model matrix . Elements in represent either intercepts or means. An element in that is associated with an exogenous row variable represents the mean of the variable. An element in that is associated with an endogenous row variable represents the intercept term for the variable.

Model Restrictions on the Matrix

The matrix is partitioned as

where is an matrix for paths or effects from (column) endogenous variables to (row) endogenous variables and is an matrix for paths (effects) from (column) exogenous variables to (row) endogenous variables.

As shown in the matrix partitions, there are four submatrices. The two submatrices at the lower parts are seemingly structured to zeros. However, this should not be interpreted as restrictions imposed by the model. The zero submatrices are artifacts created by the exogenous-endogenous arrangement of the row and column variables. The only restriction on the matrix is that the diagonal elements must all be zeros. This implies that the diagonal elements of the submatrix are also zeros. This restriction prevents a direct path from any endogenous variable to itself. There are no restrictions on the pattern of .

It is useful to denote the lower partitions of the matrix by (lower left) and (lower right) so that

Although they are zero matrices in the initial model specification, their entries could become non-zero (paths) in an improved model when you modify your model by using the Lagrange multiplier statistics (see the section Modification Indices or the MODIFICATION option). Hence, you might need to reference these two submatrices when you apply the customized LM tests on them during the model modification process (see the LMTESTS statement).

For the purposes of defining specific parameter regions in customized LM tests, you might also partition the matrix in other ways. First, you can partition into the left and right portions,

where is top-down concatenation of the and matrices and is the top-down concatenation of the and matrices. Second, you can partition into the upper and lower portions,

where is the side-by-side concatenation of the and matrices and is the side-by-side concatenation of the and matrices.

In your initial model, because of the arrangement of the endogenous and exogenous variables is a null matrix. But if you improve your model by applying the LM tests on the entries in , some of these entries might become free paths in your improved model. Hence, some exogenous variables in your initial model now become endogenous variables in your improved model. For this reason, is also designated as a parameter region for new endogenous variables, which is exactly what the NEWENDO region means in the LMTESTS statement.

Partition of the Matrix

The matrix is partitioned as

where is an partial covariance matrix for the endogenous variables, is an covariance matrix for the exogenous variables, and is an covariance matrix between the exogenous variables and the error terms for the endogenous variables. Because is symmetric, and are also symmetric.

There are virtually no model restrictions placed on these submatrices. However, in most statistical applications, errors for endogenous variables represent unsystematic sources of effects and therefore they are not to be correlated with other systematic sources such as the exogenous variables in the RAM model. This means that in most practical applications would be a null matrix, although this is not enforced in PROC CALIS.

Partition of the Vector

The vector is partitioned as

where is an vector for intercepts of the endogenous variables and is an vector for the means of the exogenous variables. There is no model restriction on these subvectors.

(1) Specification of Effects or Paths in Model Matrix

If there is a path from V2 to V1 in your model and the associated effect parameter is named parm1 with as the starting value, you can use the following RAM statement:

RAM 
   var= v1 v2 v3,
   _A_   1   2  parm1(0.5);   

The ram-entry that starts with _A_ means that an element of the ram matrix is being specified. The row number and the column number of this element are 1 and 2, respectively. With reference to the VAR= list, the row number 1 refers to variable v1, and the column number 2 refers to variable v2. Therefore, the effect of V2 on V1 is a parameter named parm1, with an initial value of 0.5.

You can specify fixed values in the ram-entries too. Suppose the effect of v3 on v1 is fixed at 1.0. You can use the following specification:

RAM 
   var= v1 v2 v3,
   _A_   1   2  parm1(0.5),   
   _A_   1   3  1.0;   

(2) Specification of the Latent Factors in the Model

In the RAM model, you specify the list of variables in VAR= list of the RAM statement. The list of variables can include the latent variables in the model. Because observed variables have references in the input data sets, those variables that do not have references in the data sets are treated as latent factors automatically. Unlike the LINEQS model, you do not need to use 'F' or 'f' prefix to denote latent factors in the RAM model. It is recommended that you use meaningful names for the latent factors. See the section Naming Variables and Parameters for the general rules about naming variables and parameters.

For example, suppose that SES_Factor and Education_Factor are names that are not used as variable names in the input data set. These two names represent two latent factors in the model, as shown in the following specification:

RAM 
   var= v1 v2 v3 SES_FACTOR Education_Factor,
   _A_   1    4    b1,
   _A_   2    5    b2,
   _A_   3    5    1.0;

This specification shows that the effect of SES_Factor on v1 is a free parameter named b1, and the effects of Education_Factor on v2 and v3 are a free parameter named b2 and a fixed value of 1.0, respectively.

However, naming latent factors is not compulsory. The preceding specification is equivalent to the following specification:

RAM 
   var= v1 v2 v3,
   _A_   1    4    b1,
   _A_   2    5    b2,
   _A_   3    5    1.0;

Although you do not name the fourth and the fifth variables in the VAR= list, PROC CALIS generates the names for these two latent variables. In this case, the fourth variable is named _Factor1 and the fifth variable is named _Factor2.

(3) Specification of (Partial) Variances and (Partial) Covariances in Model Matrix

Suppose now you want to specify the variance of v2 as a free parameter named parm2. You can add a new ram-entry for this variance parameter, as shown in the following statement:

RAM 
   var= v1 v2 v3,
   _A_   1   2  parm1(0.5),   
   _A_   1   3  1.0,   
   _P_   2   2  parm2;   

The ram-entry that starts with _P_ means that an element of the RAM matrix is being specified. The (2,2) element of , which is the variance of v2, is a parameter named parm2. You do not specify an initial value for this parameter.

You can also specify the error variance of v1 similarly, as shown in the following statement:

RAM 
   var= v1 v2 v3,
   _A_   1   2  parm1(0.5),   
   _A_   1   3  1.0,   
   _P_   2   2  parm2,   
   _P_   1   1;   

In the last ram-entry, the (1,1) element of , which is the error variance of v1, is an unnamed free parameter.

Covariance parameters are specified in the same manner. For example, the following specification adds a ram-entry for the covariance parameter between v2 and v3:

RAM 
   var= v1 v2 v3,
   _A_   1   2  parm1(0.5),   
   _A_   1   3  1.0,   
   _P_   2   2  parm2,   
   _P_   1   1,   
   _P_   2   3  (.5);   

The covariance between v2 and v3 is an unnamed parameter with an initial value of 0.5.

(4) Specification of Means and Intercepts in Model Matrix _W_

To specifying means or intercepts, you need to start the ram-entries with the _W_ keyword. For example, the last two entries of following statement specify the intercept of v1 and the mean of v2, respectively:

RAM 
   var= v1 v2 v3,
   _A_   1   2  parm1(0.5),   
   _A_   1   3  1.0,   
   _P_   2   2  parm2,   
   _P_   1   1 ,   
   _P_   2   3  (.5),   
   _W_   1   1  int_v1,   
   _W_   2   2  mean_v2;   

The intercept of v1 is a free parameter named int_v1, and the mean of v2 is a free parameter named mean_v2.