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 to show 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 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 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 associated with exogenous row and column variables represents covariance between the two exogenous variables. An off-diagonal element in 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 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 defined in the RAM model matrix . Elements in represent either intercepts or means. An element in associated with an exogenous row variable represents the mean of the variable. An element in 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 or effects from (column) exogenous variables to (row) endogenous variables.

As shown in the matrix partitions, there are four submatrices. The two 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 .

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 application, 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 TCALIS.

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 in your model there is a path from V2 to V1 and the associated effect parameter is named parm1 with as the starting value, you can use one of the following list-entries of the RAM statement to specify this parameter location:

 _A_      V1     V2   parm1  (0.5), 

 path     V1 <-  V2   parm1  (0.5), 

 path     V2 ->  V1   parm1  (0.5), 

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

If V2 is an exogenous variable in the RAM model and you want to specify its variance as a parameter named parm2 with as the starting value, you can use one of the following list-entries of the RAM statement:

_P_      V2     V2   parm2  (10.), 

pvar     V2          parm2  (10.), 

If V1 is an endogenous variable in the RAM model and you want to specify its partial variance or error variance as a parameter named parm3 with as the starting value, you can use one of the following list-entries of the RAM statement:

_P_      V1     V1   parm3  (5.), 

pvar     V1          parm3  (5.), 

Notice that this partial variance specification has the same format as that for the variance parameter of an exogenous variable.

If you want to specify (partial) covariance between two variables V3 and V4 as a parameter named parm4 with as the starting value, you can use one of the following list-entries of the RAM statement:

    _P_      V3     V4   parm4  (5.), 

    pcov     V3     V4   parm4  (5.), 

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

Means and intercepts are specified when the mean structures of the model are of interest. If V5 is an exogenous variable and you want to specify its mean as a parameter named parm5 with as the starting value, you can use one of the following list-entries of the RAM statement:

_W_      V5    parm5  (11.), 

mean     V5    parm5  (11.), 

If V6 is an endogenous variable and you want to specify its intercept as a parameter named parm6 with as the starting value, you can use one of the following list-entries of the RAM statement:

_W_           V6    parm6  (7.), 

intercept     V6    parm6  (7.), 

Note that this specification bears the same format as that for the mean parameter of an exogenous variable.

Specifying Parameters without Initial Values

If you do not have any knowledge about the initial value for a parameter, you can omit the initial value specification and let PROC TCALIS to compute it. For example, you can just provide the parameter locations and parameter names as shown in the following specifications:

_A_         V1     V2   parm1, 

intercept   V6          parm6,

Specifying Fixed Parameter Values

If you want to specify a fixed parameter value, you do not need to provide the parameter name. Instead, you just provide the fixed value without parentheses at the end of the list-entry. The following shows examples for specifying fixed parameters:

_PCOV_      V3   V4     5., 

mean        V5          11., 

A Complete RAM Model Specification Example

Essentially, you can specify all parameters and their locations of the RAM model in the RAM statement. No subsidiary model specification statements are needed in the RAM model—that is, all information regarding the model is specified in the RAM statement. Each parameter location specification is a list-entry in the RAM statement, and these list-entries are separated by commas. For example,

   ram
      path     V1 <-  V2   parm1  (.5), 
      path     V1 <-  V3   parm2      , 
      pvar     V1          errv   (1.), 
      pvar     V2          parm3  (10.), 
      pvar     V3          parm4  (10.), 
      pcov     V3     V2   parm5  (5.); 

Notice that although the order of the list-entries is not critical to the model specification, it is the same order that is used in output displays. See the RAM statement for more details about the syntax.

Automatic Free Parameters

The set of automatic parameters in the RAM model is essentially derived in the same way as those in the LINEQS model. That is, in the LINEQS model the set of automatic parameters includes the variances of and covariances among manifest exogenous variables, the means of manifest exogenous variables when the mean structures are modeled, and the variances of all exogenous latent variables (factors and errors). All these automatic free parameters are applied similarly to the RAM model.

The only difficulty about this generalization is the definition of error variances in the RAM model. Unlike the LINEQS model, error terms in the RAM model are not named explicitly. The question is what the error variances in the LINEQS model correspond to in the RAM model. As described previously, error variances in the RAM model are recast as the partial variances of the corresponding endogenous variables. As a result, the set of automatic parameters in the RAM model includes the variances or partial variances of all variables—that is, variances when the variables are exogenous, partial variances when the variables are endogenous.

As in the LINEQS model, the reason for automatic parameter generation is to safeguard a proper RAM model specification. See the section Rationale of the Default Parameters in the LINEQS Model for a more detailed discussion.

In terms of the RAM model matrices, the set of automatic parameters in the RAM model includes:

• all matrix diagonal elements, which are either variances of exogenous variables (manifest or latent) or partial variances of endogenous variables (manifest or latent)

• those matrix off-diagonal elements with their rows and columns corresponding to manifest exogenous variables

• those matrix elements with their rows corresponding to manifest exogenous variables, provided that the mean structures are modeled

Each of these automatic mean, intercept, variance, partial variance, covariance, and partial covariance parameters is named with the prefix _Add, and appended with an unique integer.

Default Fixed Zeros

All unspecified parameter locations that are neither model-restricted nor automatic free parameters in the RAM model matrices are fixed zeros by default.