PCOV Statement |
The PCOV statement is a subsidiary model specification statement for the PATH model. You can use the PCOV statement only with the PATH modeling language. The PCOV statement specifies the covariances of exogenous variables, or the error covariances of endogenous variables in the PATH model. It can also specify the covariance between an exogenous variable and the error term of an endogenous variables, although this usage is rare in practice.
In each assignment of the COV statement, you specify variables in the var_list and the var_list2 lists, followed by the covariance parameter specification in the parameter-spec list. The latter two specifications are optional. The syntax of the PCOV statement is the same as that of the COV statement. See the COV statement for details about specifying within- and between-list (partial) covariances.
The concept behind the PCOV statement is broader than that of the COV statement. The PCOV statement supports the partial covariance parameter specification in addition to the covariance parameter specification, which is the only type of parameter that the COV statement supports. This difference is also reflected from the sets of var_list and var_list2 that you can use in the PCOV statement. In the COV statement, variables on the left-hand side of an assignment must be exogenous. However, in the PCOV statement, you can specify both exogenous and endogenous variables. If a pair of variables are both exogenous in a specification, you are defining a covariance parameter between the variables. If a pair of variables are both endogenous in a specification, you are defining a partial covariance parameter between of the variables. This partial covariance is usually interpreted as the error covariance between the two endogenous variables. If one variable is exogenous while the other is endogenous, you are defining a covariance parameter between the exogenous variable and the error term for the endogenous variable.
You can specify the following five types of the parameters for the partial covariances in the PCOV statement:
an unnamed free parameter
an initial value
a fixed value
a free parameter with a name provided
a free parameter with a name and initial value provided
For example, consider a PATH model with exogenous variables x1, x2, and x3 and endogenous variables y4, y5 and y6. The following PCOV statement shows the five types of specifications in five assignments:
pcov x1 x2 , x1 x3 = (0.5), x2 x3 = 2.0, y4 y5 = psi1, y5 y6 = psi2(0.4);
In this statement, the covariance between x1 and x2 is specified as an unnamed free parameter. For this covariance, PROC CALIS generates a parameter name with the _Parm prefix and appended with a unique integer (for example, _Parm1). The covariance between x1 and x3 is an unnamed free parameter with an initial value of 0.5. PROC CALIS also generates a parameter name for this covariance. The covariance between x2 and x3 is a fixed value of 2.0. This value stays the same during the estimation. The error covariance between endogenous variables y4 and y5 is a free parameter named psi1. The error covariance between endogenous variables y5 and y6 is a free parameter named psi2 with an initial value of 0.4.
The syntax of the PCOV statement is the same as the syntax of the COV statement. See the COV statement for more illustrations about the usage.
Although the PCOV statement specification is conceptually broader than the COV statement specification, their related default set of covariance parameters is the same—that is, all covariances among exogenous manifest and latent variables (excluding error or disturbance variables) are free parameters. Because the PCOV statement applies only to the PATH model, it is easy to understand why the covariances do not apply to the error or disturbance terms. The PATH model, as implemented in PROC CALIS, simply does not use any explicit error or disturbance terms. For the default free covariance parameters, PROC CALIS generate the parameter names with the _Add prefix and appended with unique integer suffixes. You can also use the PCOV statement specification to override these default covariance parameters in situations where you want to set parameter constraints, provide initial or fixed values, or make parameter references.
Another type of default partial covariances are fixed zeros. This default applies to the partial (error) covariances among all endogenous variables, and to the partial covariances between all exogenous variables and all endogenous variables in the path model. Again, you can override the default fixed values by providing explicit specification of these partial or error covariances in the PCOV statement.
If you define a new PATH model by using a reference (old) model in the REFMODEL statement, you might want to modify some parameter specifications from the PCOV statement of the reference model before transferring the specifications to the new model. To change a particular partial covariance specification from the reference model, you can simply respecify the same covariance with the desired parameter specification in the PCOV statement of the new model. To delete a particular partial covariance parameter from the reference model, you can specify the desired partial covariance with a missing value specification in the PCOV statement of the new model.
For example, suppose that you are defining a new PATH model by using the REFMODEL statement and that the covariance between variables F1 and V2 is defined as a fixed or free parameter in the reference model. If you do not want this fixed parameter specification to be copied into your new model, you can use the following specification in the new model:
PCOV F1 V2 = .;
Note that the missing value syntax is valid only when you use it with the REFMODEL statement. See the section Modifying a PATH Model from a Reference Model for a more detailed example of the PATH model respecification.
As discussed in the section Default Covariance Parameters, PROC CALIS generates some default free covariance parameters for the PATH model if you do not specify them explicitly in the PCOV statement. When you use the REFMODEL statement for defining a reference model, these default free covariance parameters in the old (reference) model are not transferred to the new model. Instead, the new model generates its own set of default free covariance parameters after the new model is resolved from the reference model, the REFMODEL statement options, the RENAMEPARM statement, and the PCOV statement specifications in the new model. This also implies that if you want any of the (partial) covariance parameters to be constrained across the models by means of the REFMODEL specification, you must specify them explicitly in the PCOV statement of the reference model so that the same (partial) covariance specification is transferred to the new model.