### 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