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 The MODEL Procedure

## Example 14.3: Vector AR(1) Estimation

This example shows the estimation of a two-variable vector AR(1) error process for the Grunfeld model (Grunfeld 1960) 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 produces Output 14.3.1 and Output 14.3.2.

```   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
1938     44.6      2039.7    156.2     22.89     560.4     18.1
1939     48.1      2256.2    172.6     18.84     519.9     23.5
1940     74.4      2132.2    186.6     28.57     628.5     26.5
1941     113.0     1834.1    220.9     48.51     537.1     36.2
1942     91.9      1588.0    287.8     43.34     561.2     60.8
1943     61.3      1749.4    319.9     37.02     617.2     84.4
1944     56.8      1687.2    321.3     37.81     626.7     91.2
1945     93.6      2007.7    319.6     39.27     737.2     92.4
1946     159.9     2208.3    346.0     53.46     760.5     86.0
1947     147.2     1656.7    456.4     55.56     581.4     111.1
1948     146.3     1604.4    543.4     49.56     662.3     130.6
1949     98.3      1431.8    618.3     32.04     583.8     141.8
1950     93.5      1610.5    647.4     32.24     635.2     136.7
1951     135.2     1819.4    671.3     54.38     723.8     129.7
1952     157.3     2079.7    726.1     71.78     864.1     145.5
1953     179.5     2371.6    800.3     90.08     1193.5    174.8
1954     189.6     2759.9    888.9     68.60     1188.9    213.5
;

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

```   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 14.3.1: Results for the Unrestricted Model (Partial Output)

 Example of Vector AR(1) Error Process Using Grunfeld's Model With Unrestricted Vector AR(1) Error Process

 The MODEL Procedure

 Model Summary Model Variables 6 Parameters 10 Equations 2 Number of Statements 6

 Model Variables gei whi gef gec whf whc Parameters 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) Equations gei whi

 Example of Vector AR(1) Error Process Using Grunfeld's Model With Unrestricted Vector AR(1) Error Process

 The MODEL Procedure

 The 2 Equations to Estimate gei = F(ge_int, ge_f, ge_c, wh_int, wh_f, wh_c, ar_l1_1_1, ar_l1_1_2) whi = 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.

 Example of Vector AR(1) Error Process Using Grunfeld's Model With Unrestricted Vector AR(1) Error Process

 The MODEL Procedure

 Nonlinear SUR Summary of Residual Errors Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq Label gei 5 15 9374.5 625.0 24.9993 0.7910 0.7352 Gross Investment GE whi 5 15 1429.2 95.2807 9.7612 0.7940 0.7391 Gross Investment WH

 Nonlinear SUR Parameter Estimates Parameter Estimate Approx Std Err t Value Approx Pr > |t| Label ge_int -42.2858 30.5284 -1.39 0.1863 GE Intercept ge_f 0.049894 0.0153 3.27 0.0051 GE Lagged Share Value Coef ge_c 0.123946 0.0458 2.70 0.0163 GE Lagged Capital Stock Coef wh_int -4.68931 8.9678 -0.52 0.6087 WH Intercept wh_f 0.068979 0.0182 3.80 0.0018 WH Lagged Share Value Coef wh_c 0.019308 0.0754 0.26 0.8015 WH Lagged Capital Stock Coef ar_l1_1_1 0.990902 0.3923 2.53 0.0233 AR(ar) gei: LAG1 parameter for gei ar_l1_1_2 -1.56252 1.0882 -1.44 0.1716 AR(ar) gei: LAG1 parameter for whi ar_l1_2_1 0.244161 0.1783 1.37 0.1910 AR(ar) whi: LAG1 parameter for gei ar_l1_2_2 -0.23864 0.4957 -0.48 0.6372 AR(ar) whi: LAG1 parameter for whi

Output 14.3.2: Results for the Restricted Model (Partial Output)

 Example of Vector AR(1) Error Process Using Grunfeld's Model With Restricted AR(1) Error Process

 The MODEL Procedure

 Model Summary Model Variables 6 Parameters 8 Equations 2 Number of Statements 6

 Model Variables gei whi gef gec whf whc Parameters ge_int ge_f ge_c wh_int wh_f wh_c gei_l1(0) whi_l1(0) Equations gei whi

 Example of Vector AR(1) Error Process Using Grunfeld's Model With Restricted AR(1) Error Process

 The MODEL Procedure

 Nonlinear SUR Summary of Residual Errors Equation DF Model DF Error SSE MSE Root MSE R-Square Adj R-Sq Label gei 4 16 10558.8 659.9 25.6890 0.7646 0.7204 Gross Investment GE whi 4 16 1669.8 104.4 10.2157 0.7594 0.7142 Gross Investment WH

 Nonlinear SUR Parameter Estimates Parameter Estimate Approx Std Err t Value Approx Pr > |t| Label ge_int -30.1239 29.7227 -1.01 0.3259 GE Intercept ge_f 0.043527 0.0149 2.93 0.0099 GE Lagged Share Value Coef ge_c 0.119206 0.0423 2.82 0.0124 GE Lagged Capital Stock Coef wh_int 3.112671 9.2765 0.34 0.7416 WH Intercept wh_f 0.053932 0.0154 3.50 0.0029 WH Lagged Share Value Coef wh_c 0.038246 0.0805 0.48 0.6410 WH Lagged Capital Stock Coef gei_l1 0.482397 0.2149 2.24 0.0393 AR(gei) gei lag1 parameter whi_l1 0.455711 0.2424 1.88 0.0784 AR(whi) whi lag1 parameter

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