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Consumer Expenditure and Money Stock data, table 8.1, p. 203
data p203; input Year Quarter Expend Stock ; cards; 1952 1 214.6 159.3 1952 2 217.7 161.2 1952 3 219.6 162.8 1952 4 227.2 164.6 1953 1 230.9 165.9 1953 2 233.3 167.9 1953 3 234.1 168.3 1953 4 232.3 169.7 1954 1 233.7 170.5 1954 2 236.5 171.6 1954 3 238.7 173.9 1954 4 243.2 176.1 1955 1 249.4 178.0 1955 2 254.3 179.1 1955 3 260.9 180.2 1955 4 263.3 181.2 1956 1 265.6 181.6 1956 2 268.2 182.5 1956 3 270.4 183.3 1956 4 275.6 184.3 ; run;
Table 8.2 and fig. 8.1, p. 203.
goptions reset = all; symbol v=dot h=.8 c=blue; proc reg data = p203; model expend = stock; output out=temp student.=student; run; data temp; set temp; id = _n_; run; symbol v=dot h=.8 i=join c=blue; axis1 label=(angel=90 'Standardized Residuals'); proc gplot data = temp; plot student*id / vaxis=axis1 vref=0; run; quit;
The REG Procedure
Model: MODEL1
Dependent Variable: Expend
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 6395.76657 6395.76657 403.22 <.0001
Error 18 285.51093 15.86172
Corrected Total 19 6681.27750
Root MSE 3.98268 R-Square 0.9573
Dependent Mean 243.47500 Adj R-Sq 0.9549
Coeff Var 1.63576
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 -154.71916 19.85003 -7.79 <.0001
Stock 1 2.30037 0.11456 20.08 <.0001
Code for calculating the Durbin-Watson statistic for autocorrelation, p. 205.
proc autoreg data = p203; model expend = stock/ dw = 2 dwprob; run; quit;
The AUTOREG Procedure
Dependent Variable Expend
Ordinary Least Squares Estimates
SSE 285.510934 DFE 18
MSE 15.86172 Root MSE 3.98268
SBC 115.919967 AIC 113.928502
Regress R-Square 0.9573 Total R-Square 0.9573
Durbin-Watson Statistics
Order DW Pr < DW Pr > DW
1 0.3282 <.0001 1.0000
2 0.7356 0.0007 0.9993
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for
testing negative autocorrelation.
Standard Approx
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 -154.7192 19.8500 -7.79 <.0001
Stock 1 2.3004 0.1146 20.08 <.0001
Since Pr < DW was less than 0.05 the conclusion is that positive autocorrelation is present. One solution is to use the Cochrane-Orcutt procedure which can be done by utilizing the cocr macro.
%cocr(temp, expendit, stock index)
Housing Starts data, table 8.4, p. 211.
data p211;
input H P D ;
label H = 'Housing Starts'
P = 'Population'
D = 'Mortgage Money Index';
cards;
0.09090 2.200 0.03635
0.08942 2.222 0.03345
0.09755 2.244 0.03870
0.09550 2.267 0.03745
0.09678 2.280 0.04063
0.10327 2.289 0.04237
0.10513 2.289 0.04715
0.10840 2.290 0.04883
0.10822 2.299 0.04836
0.10741 2.300 0.05160
0.10751 2.300 0.04879
0.11429 2.340 0.05523
0.11048 2.386 0.04770
0.11604 2.433 0.05282
0.11688 2.482 0.05473
0.12044 2.532 0.05531
0.12125 2.580 0.05898
0.12080 2.605 0.06267
0.12368 2.631 0.05462
0.12679 2.658 0.05672
0.12996 2.684 0.06674
0.13445 2.711 0.06451
0.13325 2.738 0.06313
0.13863 2.766 0.06573
0.13964 2.793 0.07229
;
run;
Table 8.5 and Fig. 8.3, p. 212.
proc autoreg data = p211; model H = p/dw=1 dwprob; run; symbol v=dot h=.8 c=blue; proc reg data = p211 noprint; model h = p d; output out=temp student.=student; run; data temp; set temp; id = _n_; run; symbol v=dot h=.8 i=join c=blue; axis1 label=(angel=90 'Standardized Residuals'); proc gplot data = temp; plot student*id / vaxis=axis1 vref=0; run; quit;
The AUTOREG Procedure
Dependent Variable H
Housing Starts
Ordinary Least Squares Estimates
SSE 0.00038291 DFE 23
MSE 0.0000166 Root MSE 0.00408
SBC -199.77982 AIC -202.21757
Regress R-Square 0.9252 Total R-Square 0.9252
Durbin-Watson 0.6208 Pr < DW <.0001
Pr > DW 1.0000
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for
testing negative autocorrelation.
Standard Approx Variable
Variable DF Estimate Error t Value Pr > |t| Label
Intercept 1 -0.0609 0.0104 -5.85 <.0001
P 1 0.0714 0.004234 16.87 <.0001 Population
Table 8.6, p. 213.
proc autoreg data = p211; model h = p d/dw=1 dwprob; run;
The AUTOREG Procedure
Dependent Variable H
Housing Starts
Ordinary Least Squares Estimates
SSE 0.00013783 DFE 22
MSE 6.26511E-6 Root MSE 0.00250
SBC -222.10513 AIC -225.76175
Regress R-Square 0.9731 Total R-Square 0.9731
Durbin-Watson 1.8524 Pr < DW 0.2316
Pr > DW 0.7684
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for
testing negative autocorrelation.
Standard Approx
Variable DF Estimate Error t Value Pr > |t| Variable Label
Intercept 1 -0.0104 0.0103 -1.01 0.3220
P 1 0.0347 0.006425 5.39 <.0001 Population
D 1 0.7605 0.1216 6.25 <.0001 Mortgage Money Index
Fig. 8.4, p. 213.
proc reg data = p211 noprint; model h = p d; output out=temp student.=student; run; data temp; set temp; id = _n_; run; symbol v=dot h=.8 i=join c=blue; axis1 label=(angel=90 'Standardized Residuals'); proc gplot data = temp; plot student*id / vaxis=axis1 vref=0; run; quit;
Creating the standardized variables, p. 214.
proc sql;
create table p211std as
select *, (h - mean(h))/std(h) as stdh, (p - mean(p))/std(p) as stdp, (d - mean(d))/std(d)
as stdd
from p211;
quit;
Regressing the standardized h on standardized p and standardized d. Formula at the top of p. 214.
proc reg data = p211std; model stdh = stdp stdd; run; quit;
The REG Procedure
Model: MODEL1
Dependent Variable: stdh
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 23.35385 11.67693 397.58 <.0001
Error 22 0.64615 0.02937
Corrected Total 24 24.00000
Root MSE 0.17138 R-Square 0.9731
Dependent Mean -1.954E-16 Adj R-Sq 0.9706
Coeff Var -8.77065E16
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 7.80117E-17 0.03428 0.00 1.0000
stdp 1 0.46681 0.08654 5.39 <.0001
stdd 1 0.54126 0.08654 6.25 <.0001
The Ski Sales data, table 8.8, p. 217.
data p217; length quarter $ 5; input Quarter Sales PDI Season ; cards; Q1/64 37.0 109 1 Q2/64 33.5 115 0 Q3/64 30.8 113 0 Q4/64 37.9 116 1 Q1/65 37.4 118 1 Q2/65 31.6 120 0 Q3/65 34.0 122 0 Q4/65 38.1 124 1 Q1/66 40.0 126 1 Q2/66 35.0 128 0 Q3/66 34.9 130 0 Q4/66 40.2 132 1 Q1/67 41.9 133 1 Q2/67 34.7 135 0 Q3/67 38.8 138 0 Q4/67 43.7 140 1 Q1/68 44.2 143 1 Q2/68 40.4 147 0 Q3/68 38.4 148 0 Q4/68 45.4 151 1 Q1/69 44.9 153 1 Q2/69 41.6 156 0 Q3/69 44.0 160 0 Q4/69 48.1 163 1 Q1/70 49.7 166 1 Q2/70 43.9 171 0 Q3/70 41.6 174 0 Q4/70 51.0 175 1 Q1/71 52.0 180 1 Q2/71 46.2 184 0 Q3/71 47.1 187 0 Q4/71 52.7 189 1 Q1/72 52.2 191 1 Q2/72 47.0 193 0 Q3/72 47.8 194 0 Q4/72 52.8 196 1 Q1/73 54.1 199 1 Q2/73 49.5 201 0 Q3/73 49.5 202 0 Q4/73 54.3 204 1 ; run;
Table 8.7, p. 215.
proc autoreg data = p217; model sales = pdi/dw = 1; output out = resid r= resid; run; quit;
The AUTOREG Procedure
Dependent Variable Sales
Ordinary Least Squares Estimates
SSE 346.433289 DFE 38
MSE 9.11667 Root MSE 3.01938
SBC 207.245274 AIC 203.867515
Regress R-Square 0.8006 Total R-Square 0.8006
Durbin-Watson 1.9684
Standard Approx
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 12.3921 2.5394 4.88 <.0001
PDI 1 0.1979 0.0160 12.35 <.0001
Fig. 8.5, p. 215. Plotting the residual versus the index with different symbols:
Black Dots = Quarter 1 and 4
Circle = Quarter 2 and 3
data resid; set resid; id + 1; season1 = substr(Quarter, 1, 2); q14 = .; if season1 = 'Q1' then q14 = resid; if season1 = 'Q4' then q14 = resid; q23 = .; if season1 = 'Q2' then q23 = resid; if season1 = 'Q3' then q23 = resid; run; symbol1 v=dot h=.8 c=blue; proc gplot data = resid; plot q14*id='dot' q23*id='circle' / overlay; run; quit;
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