This example shows the estimation of a two-variable vector AR(1) error process for the Grunfeld model (Grunfeld and Griliches, 1960) by using the %AR macro. First, the full model is estimated. Second, the model is estimated with the restriction that the errors are univariate AR(1) instead of a vector process. The following statements produce Output 19.3.1 through Output 19.3.5.

data grunfeld;
   input year gei gef gec whi whf whc;
   label gei = 'Gross Investment GE'
         gec = 'Capital Stock Lagged GE'
         gef = 'Value of Outstanding Shares GE Lagged'
         whi = 'Gross Investment WH'
         whc = 'Capital Stock Lagged WH'
         whf = 'Value of Outstanding Shares Lagged WH';
datalines;
1935     33.1      1170.6    97.8      12.93     191.5     1.8
1936     45.0      2015.8    104.4     25.90     516.0     .8
1937     77.2      2803.3    118.0     35.05     729.0     7.4

   ... more lines ...   

title1 'Example of Vector AR(1) Error Process Using Grunfeld''s Model';
/* Note: GE stands for General Electric
         WH stands for Westinghouse     */

proc model outmodel=grunmod;
   var gei whi gef gec whf whc;
   parms ge_int ge_f ge_c wh_int wh_f wh_c;
   label ge_int = 'GE Intercept'
         ge_f   = 'GE Lagged Share Value Coef'
         ge_c   = 'GE Lagged Capital Stock Coef'
         wh_int = 'WH Intercept'
         wh_f   = 'WH Lagged Share Value Coef'
         wh_c   = 'WH Lagged Capital Stock Coef';
   gei = ge_int + ge_f * gef + ge_c * gec;
   whi = wh_int + wh_f * whf + wh_c * whc;
run;

The preceding PROC MODEL step defines the structural model and stores it in the model file named GRUNMOD.

The following PROC MODEL step reads in the model, adds the vector autoregressive terms using %AR, and requests SUR estimation by using the FIT statement.

title2 'With Unrestricted Vector AR(1) Error Process';

proc model data=grunfeld model=grunmod;
   %ar( ar, 1, gei whi )
   fit gei whi / sur;
run;

The final PROC MODEL step estimates the restricted model, as shown in the following statements:

title2 'With restricted AR(1) Error Process';

proc model data=grunfeld model=grunmod;
   %ar( gei, 1 )
   %ar( whi, 1)
   fit gei whi / sur;
run;