MODEL Procedure

Identities are model equations that express relationships among variables in a model that must be satisfied exactly. In contrast, the model equations that are used for estimation include implicit error terms to account for expected deviations in the model data. Identities are useful in specifying models. For example, one identity equation can define a quantity that is subsequently used to define multiple equations for estimation.

In PROC MODEL, you specify identity equations by using an assignment statement in which the left-hand-side variable of the assignment appears neither in an ENDOGENOUS statement nor in the list of equations to be estimated in the FIT statement.

Identity equations can also be useful in specifying the endogeneity of variables in a model. In the following example from Arie ten Cate, the OLS and FIML estimates are identical because the dependence of Y on CONS is not specified in the PROC MODEL program (Ten Cate 2017):

data a;
   input inv cons;
   y = cons + inv;
   datalines;
   1  10
   2  15
   3  29
   6  51
   7  66
   ;

proc model data=a;
   endogenous cons;
   parameters a 0.9 b 0.1;
   cons = a * y + b;
   fit / ols fiml;
quit;

You can represent the full endogenous character of CONS if you include the dependence of Y on CONS by using the identity equation . The OLS and FIML estimates differ in the following estimation because the FIML method captures the dependence of Y on CONS:

proc model data=a;
   endogenous cons;
   parameters a 0.9 b 0.1;
   y = cons + inv;
   cons = a * y + b;
   fit / ols fiml;
quit;