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, 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. The DATA and PROC steps follow:

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;
1935     33.1      1170.6     97.8

   ... more lines ...

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
MAE	19.9433255	AICC	194.127455
MAPE	23.2047973	HQC	193.210587
Durbin-Watson	1.0721	Regress R-Square	0.7053
		Total R-Square	0.7053

Parameter Estimates
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

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
MAE	18.0715195	AICC	192.426133
MAPE	21.0772644	HQC	190.536976
Durbin-Watson	1.3321	Regress R-Square	0.5717
		Total R-Square	0.7717

Parameter Estimates
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

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
MAE	18.1317764	AICC	192.44043
MAPE	21.149176	HQC	190.551273
Durbin-Watson	1.3523	Regress R-Square	0.5511
		Total R-Square	0.7721

Parameter Estimates
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

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
MAE	18.0892426	AICC	192.422614
MAPE	21.0978407	HQC	190.533457
Log Likelihood	-90.877974	Regress R-Square	0.5656
Durbin-Watson	1.3385	Total R-Square	0.7719
		Observations	20

Parameter Estimates
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