Example 14.3: Vector AR(1) Estimation

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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