Example 19.3 Vector AR(1) Estimation
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
... 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;
Output 19.3.1
Model Summary for the Unrestricted Model
gei whi gef gec whf whc |
ge_int ge_f ge_c wh_int wh_f wh_c ar_l1_1_1(0) ar_l1_1_2(0) ar_l1_2_1(0) ar_l1_2_2(0) |
gei whi |
F(ge_int, ge_f, ge_c, wh_int, wh_f, wh_c, ar_l1_1_1, ar_l1_1_2) |
F(ge_int, ge_f, ge_c, wh_int, wh_f, wh_c, ar_l1_2_1, ar_l1_2_2) |
NOTE: At SUR Iteration 9 CONVERGE=0.001 Criteria Met. |
Output 19.3.2
Estimation Summary for the Unrestricted Model
The MODEL Procedure
SUR Estimation Summary
0.000609 |
0.002798 |
0.005411 |
6.243E-7 |
720.2454 |
1.374476 |
Output 19.3.3
Estimation Results for the Unrestricted Model
The MODEL Procedure
5 |
15 |
9374.5 |
625.0 |
24.9993 |
0.7910 |
0.7352 |
Gross Investment GE |
5 |
15 |
1429.2 |
95.2807 |
9.7612 |
0.7940 |
0.7391 |
Gross Investment WH |
-42.2858 |
30.5284 |
-1.39 |
0.1863 |
GE Intercept |
0.049894 |
0.0153 |
3.27 |
0.0051 |
GE Lagged Share Value Coef |
0.123946 |
0.0458 |
2.70 |
0.0163 |
GE Lagged Capital Stock Coef |
-4.68931 |
8.9678 |
-0.52 |
0.6087 |
WH Intercept |
0.068979 |
0.0182 |
3.80 |
0.0018 |
WH Lagged Share Value Coef |
0.019308 |
0.0754 |
0.26 |
0.8015 |
WH Lagged Capital Stock Coef |
0.990902 |
0.3923 |
2.53 |
0.0233 |
AR(ar) gei: LAG1 parameter for gei |
-1.56252 |
1.0882 |
-1.44 |
0.1716 |
AR(ar) gei: LAG1 parameter for whi |
0.244161 |
0.1783 |
1.37 |
0.1910 |
AR(ar) whi: LAG1 parameter for gei |
-0.23864 |
0.4957 |
-0.48 |
0.6372 |
AR(ar) whi: LAG1 parameter for whi |
Output 19.3.4
Model Summary for the Restricted Model
gei whi gef gec whf whc |
ge_int ge_f ge_c wh_int wh_f wh_c gei_l1(0) whi_l1(0) |
gei whi |
Output 19.3.5
Estimation Results for the Restricted Model
The MODEL Procedure
4 |
16 |
10558.8 |
659.9 |
25.6890 |
0.7646 |
0.7204 |
Gross Investment GE |
4 |
16 |
1669.8 |
104.4 |
10.2157 |
0.7594 |
0.7142 |
Gross Investment WH |
-30.1239 |
29.7227 |
-1.01 |
0.3259 |
GE Intercept |
0.043527 |
0.0149 |
2.93 |
0.0099 |
GE Lagged Share Value Coef |
0.119206 |
0.0423 |
2.82 |
0.0124 |
GE Lagged Capital Stock Coef |
3.112671 |
9.2765 |
0.34 |
0.7416 |
WH Intercept |
0.053932 |
0.0154 |
3.50 |
0.0029 |
WH Lagged Share Value Coef |
0.038246 |
0.0805 |
0.48 |
0.6410 |
WH Lagged Capital Stock Coef |
0.482397 |
0.2149 |
2.24 |
0.0393 |
AR(gei) gei lag1 parameter |
0.455711 |
0.2424 |
1.88 |
0.0784 |
AR(whi) whi lag1 parameter |