Example 8.2: Comparing Estimates and Models

The AUTOREG Procedure

Example 8.2: Comparing Estimates and Models

In this example, the Grunfeld series are estimated using different estimation methods. Refer to Maddala (1977) for details of the Grunfeld investment data set. For comparison, the Yule-Walker method, the ULS method, and maximum likelihood method estimates are shown. With the DWPROB option, the p-value of the Durbin-Watson statistic is printed. The Durbin-Watson test indicates the positive autocorrelation of the regression residuals.


   title 'Grunfeld''s Investment Models Fit with Autoregressive Errors';
   data grunfeld;
      input year gei gef gec;
      label gei = 'Gross investment GE'
            gec = 'Lagged Capital Stock GE'
            gef = 'Lagged Value of GE shares';
   datalines;
     ... data lines omitted ...
   ;
   
   proc autoreg data=grunfeld;
      model gei = gef gec / nlag=1 dwprob;
      model gei = gef gec / nlag=1 method=uls;
      model gei = gef gec / nlag=1 method=ml;
   run;

The printed output produced by each of the MODEL statements is shown in Output 8.2.1 through Output 8.2.4.

Output 8.2.1: OLS Analysis of Residuals

Grunfeld's Investment Models Fit with Autoregressive Errors

The AUTOREG Procedure

Dependent Variable	gei
	Gross investment GE

Ordinary Least Squares Estimates
SSE	13216.5878	DFE	17
MSE	777.44634	Root MSE	27.88272
SBC	195.614652	AIC	192.627455
Regress R-Square	0.7053	Total R-Square	0.7053
Durbin-Watson	1.0721	Pr < DW	0.0038
Pr > DW	0.9962

Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Variable Label
Intercept	1	-9.9563	31.3742	-0.32	0.7548
gef	1	0.0266	0.0156	1.71	0.1063	Lagged Value of GE shares
gec	1	0.1517	0.0257	5.90	<.0001	Lagged Capital Stock GE

Estimates of Autocorrelations
Lag	Covariance	Correlation	-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
0	660.8	1.000000	\| \|********************\|
1	304.6	0.460867	\| \|********* \|

Preliminary MSE	520.5

Output 8.2.2: Regression Results Using Default Yule-Walker Method

Grunfeld's Investment Models Fit with Autoregressive Errors

The AUTOREG Procedure

Estimates of Autoregressive Parameters
Lag	Coefficient	Standard Error	t Value
1	-0.460867	0.221867	-2.08

Yule-Walker Estimates
SSE	10238.2951	DFE	16
MSE	639.89344	Root MSE	25.29612
SBC	193.742396	AIC	189.759467
Regress R-Square	0.5717	Total R-Square	0.7717
Durbin-Watson	1.3321	Pr < DW	0.0232
Pr > DW	0.9768

Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Variable Label
Intercept	1	-18.2318	33.2511	-0.55	0.5911
gef	1	0.0332	0.0158	2.10	0.0523	Lagged Value of GE shares
gec	1	0.1392	0.0383	3.63	0.0022	Lagged Capital Stock GE

Output 8.2.3: Regression Results Using Unconditional Least Squares Method

Grunfeld's Investment Models Fit with Autoregressive Errors

The AUTOREG Procedure

Estimates of Autoregressive Parameters
Lag	Coefficient	Standard Error	t Value
1	-0.460867	0.221867	-2.08

Algorithm converged.

Unconditional Least Squares Estimates
SSE	10220.8455	DFE	16
MSE	638.80284	Root MSE	25.27455
SBC	193.756692	AIC	189.773763
Regress R-Square	0.5511	Total R-Square	0.7721
Durbin-Watson	1.3523

Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Variable Label
Intercept	1	-18.6582	34.8101	-0.54	0.5993
gef	1	0.0339	0.0179	1.89	0.0769	Lagged Value of GE shares
gec	1	0.1369	0.0449	3.05	0.0076	Lagged Capital Stock GE
AR1	1	-0.4996	0.2592	-1.93	0.0718

Autoregressive parameters assumed given.
Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Variable Label
Intercept	1	-18.6582	33.7567	-0.55	0.5881
gef	1	0.0339	0.0159	2.13	0.0486	Lagged Value of GE shares
gec	1	0.1369	0.0404	3.39	0.0037	Lagged Capital Stock GE

Output 8.2.4: Regression Results Using Maximum Likelihood Method

Grunfeld's Investment Models Fit with Autoregressive Errors

The AUTOREG Procedure

Estimates of Autoregressive Parameters
Lag	Coefficient	Standard Error	t Value
1	-0.460867	0.221867	-2.08

Algorithm converged.

Maximum Likelihood Estimates
SSE	10229.2303	DFE	16
MSE	639.32689	Root MSE	25.28491
SBC	193.738877	AIC	189.755947
Regress R-Square	0.5656	Total R-Square	0.7719
Durbin-Watson	1.3385

Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Variable Label
Intercept	1	-18.3751	34.5941	-0.53	0.6026
gef	1	0.0334	0.0179	1.87	0.0799	Lagged Value of GE shares
gec	1	0.1385	0.0428	3.23	0.0052	Lagged Capital Stock GE
AR1	1	-0.4728	0.2582	-1.83	0.0858

Autoregressive parameters assumed given.
Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Variable Label
Intercept	1	-18.3751	33.3931	-0.55	0.5897
gef	1	0.0334	0.0158	2.11	0.0512	Lagged Value of GE shares
gec	1	0.1385	0.0389	3.56	0.0026	Lagged Capital Stock GE

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