 Springer Texts in Business and Economics

# . Effect of Additional Regressors on R2

a. Least Squares on the K = K1 + K2 regressors minimizes the sum of squared error and yields SSE2 = min P (Yi - a - "2 X2i -.. - "k Xk і -.. - "kXkD2

i=1

Let us denote the corresponding estimates by (a, b2,..,bKj,..,Ьк). This implies that SSE* = p(Yi - a* - "*X2i - .. - "KXkp - .. - "KXkD2

i= 1

based on arbitrary (a*, "*,.., "K, ., PK) satisfies SSE* > SSE2. In partic­ular, substituting the least squares estimates using only K1 regressors say a,"2,..,"Ki and "Ki+1 = 0,..,"к = 0 satisfy the above inequality. Hence,

n

SSE1 > SSE2. Since P yi2 is fixed, this means that R2 > R*. This is based

i=1

on the solution by Rao and White (1988). 2 2

b. From the definition of R, we get (1 - R ) =  i=1

4.4 This regression suffers from perfect multicollinearity. X2 + X3 is perfectly collinear with X2 and X3. Collecting terms in X2 and X3 we get

Yi = a + ("2 + "4/X2i + ("3 + "4/X3i + "5X2i + "6X2i + ui so

("2 + "4/, ("3 + "4/, "5 and "6 are estimable by OLS.

4.5 a. If we regress et on X2t and X3t and a constant, we get zero regression coef­ficients and therefore zero predicted values, i. e., et = 0. The residuals are

therefore equal to et — et = et and the residual sum of squares is equal to the

total sum of squares. Therefore, R2 = 1-------- = 1 — 1 = 0.

H TSS

. . t t _

b. For Yt = a + bYt + vt, OLS yields Із = Y ytyt/ Y y2 and a = Y — bY.

t=i t=i

. _ — T

Also, Yt = Yt + et which gives Y = Y since Y et = 0. Also, yt = yt + et.

t=i

T T T

Therefore, ytyt = Y y2 since £ ytet = 0. Hence,

 t=1 t=1 TT t=1 b=x y 2/x y 2=1 and a = y — 1 • y = 0

t=i t=i

Yt = 0 + 1 • Yt = Yt and Yt — Yt = et

TT

Y et2 is the same as the original regression, Y yt2 is still the same,

t=i t=i

therefore, R2 is still the same.

TT  ForYt = a + bet + vt, OLS yields Із = ^ etyt / Y

t=i ' t=i

T T T

since e = 0. But yt = yt + et, therefore, Y etyt = Y etyt + Y et2 =

t=i t=i t=i

T T л T, T

Y et2 since £ etyt = 0. Hence, ІЗ = Y et / Y et2 = 1.

t=1 t=1 t=1 ' t=1

Also, the predicted value is now Yt = a + bet = Y+et and the new residual is = Yt — Yt = yt — et = yt. Hence, the new RSS = old regression sum of

T

squares = y2, and t=i

4.6 For the Cobb-Douglas production given by Eq. (4.18) one can test Ho; a + " + " + 8 = 1 using the following t-statistic t = (а+1"+"+^)~1 where the estimates

s. e.(a+|3+y+8)

are obtained from the unrestricted OLS regression given by (4.18). The var(a + O + " + O) = var(a) + var(O) + var(") + var(8) + 2cov(a, O) + 2cov(a, ") + 2cov(a, 8) + 2cov(O,") + 2cov(0, 8) + 2cov(", 8). These variance-covariance estimates are obtained from the unrestricted regression. The observed t-statistic is distributed as tn_5 under Ho.

4.7 a. The restricted regression for Ho; "2 = "4 = "6 is given by Yi = a +

"2(X2i + X4i + X6i) + "зХЗі + "5X5i + "7X7i + "8X8i + .. + "кХКі +

ui obtained by substituting "2 = "4 = "6 in Eq. (4.1). The unrestricted regression is given by (4.1) and the F-statistic in (4.17) has two restrictions and is distributed F2,n_K under Ho.

b. The restricted regression for Ho; "2 = —"3 and "5 — "6 = 1 is given by

Yi +X6i = a + "2(X2i —X3i) C "4X4i + ^5(X5i +X6i) + "7X7i + ..+"кХКі +ui obtained by substituting both restrictions in (4.1). The unrestricted regres­sion is given by (4.1) and the F-statistic in (4.17) has two restrictions and is distributed F2,n_K under Ho.

4.10 a. For the data underlying Table 4.1, the following computer output gives the mean of log(wage) for females and for males. Out of 595 individuals observed, there were 528 Males and 67 Females. The corresponding means of log(wage) for Males and Females being YM = 7.004 and YF = 6.530, respectively. The regression of log(wage) on FEMALE and MALE without a constant yields coefficient estimates aF = YF = 6.530 and aM = YM = 7.004, as expected.

Dependent Variable: LWAGE

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 2 28759.48792 14379.74396 84576.019 0.0001 Error 593 100.82277 0.17002 U Total 595 28860.31068 Root MSE 0.41234 R-square 0.9965 Dep Mean 6.95074 Adj R-sq 0.9965 C. V. 5.93227

Parameter Estimates

 Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| FEM 1 6.530366 0.05037494 129.635 0.0001 M 1 7.004088 0.01794465 390.316 0.0001

FEM=0

 Variable N Mean Std Dev Minimum Maximum LWAGE 528 7.0040880 0.4160069 5.6767500 8.5370000 FEM=1 Variable N Mean Std Dev Minimum Maximum LWAGE 67 6.5303664 0.3817668 5.7493900 7.2793200

b. Running log(wage) on a constant and the FEMALE dummy variable yields a = 7.004 = YM = aM and " = —0.474 = (арТ—~ам). But aM = 7.004.

Therefore, aF = " + aM = 7.004 — 0.474 = 6.530 = YF = aF.

Dependent Variable: LWAGE

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 1 13.34252 13.34252 78.476 0.0001 Error 593 100.82277 0.17002 C Total 594 114.16529

RootMSE 0.41234 R-square 0.1169

C. V. 5.93227

Parameter Estimates

 Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| INTERCEP 1 7.004088 0.01794465 390.316 0.0001 FEM 1 -0.473722 0.05347565 -8.859 0.0001

4.12 a. The unrestricted regression is given by (4.28). This regression runs EARN on a constant, FEMALE, EDUCATION and (FEMALE x EDUCATION). The URSS = 76.63525. The restricted regression for equality of slopes and intercepts for Males and Females, tests the restriction Ho; aF = " = O. This regression runs EARN on a constant and EDUC. The RRSS = 90.36713. The SAS regression output is given below. There are two restrictions and the F-test given by (4.17) yields

(90.36713 - 76.63525//2 „ „ ,

F = ------------------------------- — = 52.94941.

76.63525/591

This is distributed as F2,591 under Ho. The null hypothesis is rejected.

Unrestricted Model (with FEM and FEM*EDUC) Dependent Variable: LWAGE

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 3 Error 591 C Total 594 37.53004 76.63525 114.16529 12.51001 0.12967 96.475 0.0001 Root MSE 0.36010 R-square 0.3287 Dep Mean C. V. 6.95074 5.18071 Adj R-sq 0.3253 Restricted Model (without FEM and FEM*EDUC) Dependent Variable'LWAGE

 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 1 23.79816 23.79816 156.166 0.0001 Error 593 90.36713 0.15239 C Total 594 114.16529 Root MSE 0.39037 R-square 0.2085 Dep Mean 6.95074 Adj R-sq 0.2071 C. V. 5.61625 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 6.029192 0.07546051 79.899 0.0001 ED 1 0.071742 0.00574089 12.497 0.0001  b. The unrestricted regression is given by (4.27). This regression runs EARN on a constant, FEMALE and EDUCATION. The URSS = 77.06808. The restricted regression for the equality of intercepts given the same slopes for Males and Females, tests the restriction Ho; aF = 0 given that у = 0. This is the same restricted regression given in part (a), running EARN on a constant and EDUC. The RRSS = 90.36713. The F-test given by (4.17) tests one restriction and yields (90.36713 - 77.06808//1

77.06808/592

This is distributed as FF592 under Ho. Note that this observed F-statistic is the square of the observed t-statistic of —10.107 for aF = 0 in the unrestricted regression. The SAS regression output is given below.

Unrestricted Model (with FEM)

Dependent Variable: LWAGE

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 2 37.09721 18.54861 142.482 0.0001 Error 592 77.06808 0.13018 C Total 594 114.16529 Root MSE 0.36081 R-square 0.3249 Dep Mean 6.95074 Adj R-sq 0.3227 C. V. 5.19093 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 6.083290 0.06995090 86.965 0.0001 FEM 1 -0.472950 0.04679300 -10.107 0.0001 ED 1 0.071676 0.00530614 13.508 0.0001

c. The unrestricted regression is given by (4.28), see part (a). The restricted regression for the equality of intercepts allowing for different slopes for Males and Females, tests the restriction Ho; aF = 0 given that у ф 0. This regression runs EARN on a constant, EDUCATION and (FEMALE x EDUCATION). The RRSS = 78.46096. The SAS regression output is given below. The F-test given by (4.17), tests one restriction and yields:

(78.46096 — 76.63525//1 F = - — = 14.0796.

76.63525/591

This is distributed as F1591 under Ho. The null hypothesis is rejected. Note that this observed F-statistic is the square of the t-statistic (—3.752/ on aF = 0 in the unrestricted regression.

Restricted Model (without FEM)

Dependent Variable: LWAGE

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 2 35.70433 17.85216 134.697 0.0001 Error 592 78.46096 0.13254 C Total 594 114.16529 Root MSE 0.36405 R-square 0.3127 Dep Mean 6.95074 Adj R-sq 0.3104 C. V. 5.23763 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 6.039577 0.07038181 85.812 0.0001 ED 1 0.074782 0.00536347 13.943 0.0001 F_EDC 1 -0.034202 0.00360849 -9.478 0.0001 Dependent Variable: LWAGE Analysis of Variance Sumof Mean Source DF Squares Square F Value Prob>F Model 12 52.48064 4.37339 41.263 0.0001 Error 582 61.68465 0.10599 C Total 594 114.16529 Root MSE 0.32556 R-square 0.4597 Dep Mean 6.95074 Adj R-sq 0.4485 C. V. 4.68377 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 5.590093 0.19011263 29.404 0.0001 EXP 1 0.029380 0.00652410 4.503 0.0001 EXP2 1 -0.000486 0.00012680 -3.833 0.0001
 WKS 1 0.003413 0.00267762 1.275 0.203 OCC 1 -0.161522 0.0369073 -4.376 0.0001 IND 1 0.084663 0.0291637 2.903 0.0038 SOUTH 1 -0.058763 0.0309069 -1.901 0.0578 SMSA 1 0.166191 0.29551 5.624 0.0001 MS 1 0.095237 0.0489277 1.946 0.0521 FEM 1 -0.324557 0.0607295 -5.344 0.0001 UNION 1 0.106278 0.0316755 3.355 0.0008 ED 1 0.057194 0.00659101 8.678 0.0001 BLK 1 -0.190422 0.0544118 -3.5 0.0005

b. Ho : EARN = a + u.

If you run EARN on an intercept only, you would get a = 6.9507 which is average log wage or average earnings = y. The total sum of squares =

n

the residual sum of squares = ^ (yi — y/2 = 114.16529 and this is the

i=i

restricted residual sum of squares (RRSS) needed for the F-test. The unre­stricted model is given in Table 4.1 or part (a) and yields URSS = 61.68465.  Hence, the joint significance for all slopes using (4.20) yields

This F-statistic is distributed as F12 582 under the null hypothesis. It has a p-value of 0.0001 as shown in Table 4.1 and we reject Ho. The Analysis of Variance table in the SAS output given in Table 4.1 always reports this F-statistic for the significance of all slopes for any regression. c. The restricted model excludes FEM and BLACK. The SAS regression out­put is given below. The RRSS = 66.27893. The unrestricted model is given in Table 4.1 with URSS = 61.68465. The F-statistic given in (4.17) tests two restrictions and yields

(66.27893 — 61.68465/ / 2 F = — = 21.6737.

61.68465/582

This is distributed as F2 582 under the null hypothesis. We reject Ho.

Model:Restricted Model (w/o FEMALE & BLACK)

Dependent Variable: LWAGE

 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 10 47.88636 4.78864 42.194 0.0001 Error 584 66.27893 0.11349 C Total 594 114.16529 Root MSE 0.33688 R-square 0.4194 Dep Mean 6.95074 Adj R-sq 0.4095 C. V. 4.84674 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 5.316110 0.19153698 27.755 0.0001 EXP 1 0.028108 0.00674771 4.165 0.0001 EXP2 1 -0.000468 0.00013117 -3.570 0.0004 WKS 1 0.004527 0.00276523 1.637 0.1022 OCC 1 -0.162382 0.03816211 -4.255 0.0001 IND 1 0.102697 0.03004143 3.419 0.0007 SOUTH 1 -0.073099 0.03175589 -2.302 0.0217 SMSA 1 0.142285 0.03022571 4.707 0.0001 MS 1 0.298940 0.03667049 8.152 0.0001 UNION 1 0.112941 0.03271187 3.453 0.0006 ED 1 0.059991 0.00680032 8.822 0.0001

d. The restricted model excludes MS and UNION. The SAS regression output is given below. This yields RRSS = 63.37107. The unrestricted model is given in Table 4.1 and yields URSS = 61.68465. The F-test given in (4.17) tests two restrictions and yields

(63.37107 - 61.68465//2 F = — = 7.9558.

61.68465/582

This is distributed as F2,582 under the null hypothesis. We reject Ho.

Restricted Model (without MS & UNION) Dependent Variable: LWAGE

 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 10 50.79422 5.07942 46.810 0.0001 Error 584 63.37107 0.10851 C Total 594 114.16529 Root MSE 0.32941 R-square 0.4449 Dep Mean 6.95074 Adj R-sq 0.4354 C. V. 4.73923 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 5.766243 0.18704262 30.828 0.0001 EXP 1 0.031307 0.00657565 4.761 0.0001 EXP2 1 -0.000520 0.00012799 -4.064 0.0001 WKS 1 0.001782 0.00264789 0.673 0.5013 OCC 1 -0.127261 0.03591988 -3.543 0.0004 IND 1 0.089621 0.02948058 3.040 0.0025 SOUTH 1 -0.077250 0.03079302 -2.509 0.0124 SMSA 1 0.172674 0.02974798 5.805 0.0001 FEM 1 -0.425261 0.04498979 -9.452 0.0001 ED 1 0.056144 0.00664068 8.454 0.0001 BLK 1 -0.197010 0.5474680 -3.599 0.0003 From Table 4.1 , using the coefficient estimate on Union, " u = 0.106278, we obtain gu = = e"u 1 e0.106278 1 0.112131 or (11.2131%). If

the disturbances are log normal, Kennedy’s (1981) suggestion yields gu = e"u-0.5-var("u) _ 1 = e0.106278—0.5(0.001003335) _ j = 0.111573 or (11.1573%).

f. From Table 4.1, using the coefficient estimate on MS, "MS = 0.095237, we obtain gMS = e°MS _ 1 = e0095237 _ 1 = 0.09992 or (9.992%).

4.14 Crude Quality

a. Regression of POIL on GRAVITY and SULPHUR. Dependent Variable: POIL

Analysis of Variance

 Sumof Mean Source DF Squares Square F Value Prob>F Model 2 249.21442 124.60721 532.364 0.0001 Error 96 22.47014 0.23406 C Total 98 271.68456 Root MSE 0.48380 R-square 0.9173 Dep Mean 15.33727 Adj R-sq 0.9156 C. V. 3.15442 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 12.354268 0.23453113 52.676 0.0001 GRAVITY 1 0.146640 0.00759695 19.302 0.0001 SULPHUR 1 -0.414723 0.04462224 -9.294 0.0001 Regression of GRAVITY on SULPHUR. Dependent Variable: GRAVITY Analysis of Variance Sumof Mean Source DF Squares Square F Value Prob>F Model 1 2333.89536 2333.89536 55.821 0.0001 Error 97 4055.61191 41.81043 C Total 98 6389.50727 Root MSE 6.46610 R-square 0.3653 Dep Mean 24.38182 Adj R-sq 0.3587 C. V. 26.52017 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 29.452116 0.93961195 31.345 0.0001 SULPHUR 1 -3.549926 0.47513923 -7.471 0.0001

Regression of POIL on the Residuals from the Previous Regression Dependent Variable: POIL

Analysis of Variance

 Sumof Mean Source DF Squares Square F Value Prob>F Model 1 87.20885 87.20885 45.856 0.0001 Error 97 184.47571 1.90181 C Total 98 271.68456 Root MSE 1.37906 R-square 0.3210 Dep Mean 15.33727 Adj R-sq 0.3140 C. V. 8.99157 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 15.337273 0.13860093 110.658 0.0001 RESID. V 1 0.146640 0.02165487 6.772 0.0001

c. Regression of POIL on SULPHUR Dependent Variable: POIL

Analysis of Variance

 Source DF Sum of Squares Mean Square F Value Prob>F Model 1 Error 97 C Total 98 162.00557 109.67900 271.68456 162.00557 1.13071 143.278 0.0001 Root MSE Dep Mean C. V. 1.06335 15.33727 6.93310 R-square Adj R-sq 0.5963 0.5921 Parameter Estimates Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| INTERCEP SULPHUR 1 1 16.673123 -0.935284 0.15451906 0.07813658 107.903 -11.970 0.0001 0.0001

Regression of Residuals in part (c) on those in part (b). Dependent Variable: RESID_W

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 1 87.20885 87.20885 376.467 0.0001 Error 97 22.47014 0.23165 C Total 98 109.67900 Root MSE 0.48130 R-square 0.7951 Dep Mean 0.00000 Adj R-sq 0.7930 C. V. 2.127826E16 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 3.082861E-15 0.04837260 0.000 1.0000 RESID. V 1 0.146640 0.00755769 19.403 0.0001 Regression based on the first 25 crudes. Dependent Variable: POIL Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 2 37.68556 18.84278 169.640 0.0001 Error 22 2.44366 0.11108 C Total 24 40.12922 Root MSE 0.33328 R-square 0.9391 Dep Mean 15.65560 Adj R-sq 0.9336 C. V. 2.12882 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 11.457899 0.34330283 33.375 0.0001 GRAVITY 1 0.166174 0.01048538 15.848 0.0001 SULPHUR 1 0.110178 0.09723998 1.133 0.2694

e. Deleting all crudes with sulphur content outside the range of 1-2%.

Dependent Variable: POIL

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 2 28.99180 14.49590 128.714 0.0001 Error 25 2.81553 0.11262 C Total 27 31.80732 Root MSE 0.33559 R-square 0.9115 Dep Mean 15.05250 Adj R-sq 0.9044 C. V. 2.22947 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 11.090789 0.37273724 29.755 0.0001 GRAVITY 1 0.180260 0.01123651 16.042 0.0001 SULPHUR 1 0.176138 0.18615220 0.946 0.3531

SAS PROGRAM

Data Crude;

Input POIL GRAVITY SULPHUR; Cards;

Proc reg data=CRUDE;

Model POIL=GRAVITY SULPHUR;

Proc reg data=CRUDE;

Model GRAVITY=SULPHUR; Output out=OUT1 R=RESID_V; run;

Data CRUDE1; set crude; set OUT1(keep=RESID_V);

Proc reg data=CRUDE1;

Model POIL=RESID_V;

Proc reg data=CRUDE1;

Model POIL=SULPHUR;

Output out=OUT2 R=RESID_W;

Proc reg data=OUT2;

Model RESID_W=RESID_V;

Data CRUDE2; set CRUDE(firstobs=1 obs=25);

Proc reg data=CRUDE2;

Model POIL=GRAVITY SULPHUR;

data CRUDE3; set CRUDE;

if SULPHUR < 1 then delete; if SULPHUR > 2 then delete;

Proc reg data=CRUDE3;

Model POIL=GRAVITY SULPHUR; Dependent Variable: LNQMG

Analysis of Variance

 Sumof Mean Source DF Squares Square F Value Prob>F Model 5 Error 17 C Total 22 1.22628 0.00596 1.23224 0.24526 0.00035 699.770 0.0001 Root MSE 0.01872 R-square 0.9952 Dep Mean C. V. 17.96942 0.10418 Adj R-sq 0.9937

 Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 1.680143 2.79355393 0.601 0.5555 LNCAR 1 0.363533 0.51515166 0.706 0.4899 LNPOP 1 1.053931 0.90483097 1.165 0.2602 LNRGNP 1 -0.311388 0.16250458 -1.916 0.0723 LNPGNP 1 0.124957 0.15802894 0.791 0.4400 LNPMG 1 1.048145 0.26824906 3.907 0.0011 MODEL2(1950-1972) Dependent Variable: QMG_CAR Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 3 0.01463 0.00488 4.032 0.0224 Error 19 0.02298 0.00121 C Total 22 0.03762 Root MSE 0.03478 R-square 0.3890 Dep Mean -0.18682 Adj R-sq 0.2925 C. V. -18.61715 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 -0.306528 2.37844176 -0.129 0.8988 RGNP. POP 1 -0.139715 0.23851425 -0.586 0.5649 CAR. POP 1 0.054462 0.28275915 0.193 0.8493 PMGPGNP 1 0.185270 0.27881714 0.664 0.5144

b. From model (2) we have

logQMG — log CAR = yi + y2 logRGNP — y2 log POP + y3 log CAR — y3 log POP + y4 log PMG — y4 log PGNP + v For this to be the same as model (1), the following restrictions must hold:

"i = yi, "2 = Уз + 1, "3 = (y2 + Уз/ , "4 = y2,

"5 = "4 , and "б = "4.

d. Correlation Analysis

Variables: LNCAR LNPOP LNRGNP LNPGNP LNPMG

Simple Statistics

 Variable N Mean Std Dev Sum Minimum Maximum LNCAR 23 18.15623 0.26033 417.59339 17.71131 18.57813 LNPOP 23 12.10913 0.10056 278.50990 11.93342 12.25437 LNRGNP 23 7.40235 0.24814 170.25415 6.99430 7.81015 LNPGNP 23 3.52739 0.16277 81.12989 3.26194 3.86073 LNPMG 23 -1.15352 0.09203 -26.53096 -1.30195 -0.94675 Pearson Correlation Coefficients/Prob > |R| under Ho: Rho II CD II ГО CO LNCAR LNPOP LNRGNP LNPGNP LNPMG LNCAR 1.00000 0.99588 0.99177 0.97686 0.94374 0.0 0.0001 0.0001 0.0001 0.0001 LNPOP 0.99588 1.00000 0.98092 0.96281 0.91564 0.0001 0.0 0.0001 0.0001 0.0001 LNRGNP 0.99177 0.98092 1.00000 0.97295 0.94983 0.0001 0.0001 0.0 0.0001 0.0001 LNPGNP 0.97686 0.96281 0.97295 1.00000 0.97025 0.0001 0.0001 0.0001 0.0 0.0001 LNPMG 0.94374 0.91564 0.94983 0.97025 1.00000 0.0001 0.0001 0.0001 0.0001 0.0

This indicates the presence of multicollinearity among the regressors.

f. MODEL1 (1950-1987)

Dependent Variable: LNQMG

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 5 Error 32 C Total 37 3.46566 0.02553 3.49119 0.69313 0.00080 868.762 0.0001 Root MSE 0.02825 R-square 0.9927 Dep Mean C. V. 18.16523 0.15550 Adj R-sq 0.9915

 Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 INTERCEP 1 9.986981 2.62061670 3.811 LNCAR 1 2.559918 0.22812676 11.221 LNPOP 1 -2.878083 0.45344340 -6.347 LNRGNP 1 -0.429270 0.14837889 -2.893 LNPGNP 1 -0.178866 0.06336001 -2.823 LNPMG 1 -0.141105 0.04339646 -3.252 MODEL2 (1950-1987) Dependent Variable: QMG. CAR Analysis of Variance Sum of Mean Source DF Squares Square F Value Model 3 0.35693 0.11898 32.508 Error 34 0.12444 0.00366 C Total 37 0.48137 Root MSE 0.06050 R - square 0.7415 Dep Mean -0.25616 Adj R - sq 0.7187 C. V. -23.61671 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 INTERCEP 1 -5.853977 3.10247647 -1.887 RGNP. POP 1 -0.690460 0.29336969 -2.354 CAR. POP 1 0.288735 0.27723429 1.041 PMGPGNP 1 -0.143127 0.07487993 -1.911

 Prob>|T| 0.0006 0.0001 0.0001 0.0068 0.0081 0.0027

 Prob>F 0.0001

 Prob>|T| 0.0677 0.0245 0.3050 0.0644

g. Model 2: SHOCK74 (= 1 if year > 1974) Dependent Variable: QMG. CAR

Analysis of Variance

 Source DF Sum of Squares Mean Square F Value Model 4 0.36718 0.09179 26.527 Error 33 0.11419 0.00346 C Total 37 0.48137

 Prob>F 0.0001

 Root MSE 0.05883 R - square 0.7628 Dep Mean -0.25616 Adj R - sq 0.7340 C. V. -22.96396 Parameter Estimates Parameter Standard for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 -6.633509 3.05055820 -2.175 0.0369 SHOCK74 1 -0.073504 0.04272079 -1.721 0.0947 RGNP. POP 1 -0.733358 0.28634866 -2.561 0.0152 CAR. POP 1 0.441777 0.28386742 1.556 0.1292 PMG. PGNP 1 -0.069656 0.08440816 -0.825 0.4152

The t-statistic on SHOCK 74 yields —1.721 with a p-value of 0.0947. This is insignificant. Therefore, we cannot reject that gasoline demand per car did not permanently shift after 1973.

h. Model 2: DUMMY74 (=SCHOCK74 x PMG. PGNP)

Dependent Variable: QMG_CAR

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 4 0.36706 0.09177 26.492 0.0001 Error 33 0.11431 0.00346 C Total 37 0.48137 Root MSE 0.05886 R - square 0.7625 Dep Mean -0.25616 Adj R - sq 0.7337 C. V. -22.97560 Parameter Estimates Parameter Standard T for HO: Variable DF Estimate Error Parameter=0 Prob>|T| INTERCEP 1 -6.606761 3.05019318 -2.166 0.0376 RGNP. POP 1 -0.727422 0.28622322 -2.541 0.0159 CAR. POP 1 0.431492 0.28233413 1.528 0.1360 PMG. PGNP 1 -0.083217 0.08083459 -1.029 0.3107 DUMMY74 1 0.015283 0.00893783 1.710 0.0967

The interaction dummy named DUMMY74 has a t-statistic of 1.71 with a p-value of 0.0967. This is insignificant and we cannot reject that the price elasticity did not change after 1973.

SAS PROGRAM Data RAWDATA;

Input Year CAR QMG PMG POP RGNP PGNP; Cards;

Data USGAS; set RAWDATA;

LNQMG=LOG(QMG);

LNCAR=LOG(CAR);

LNPOP=LOG(POP);

LNRGNP=LOG(RGNP);

LNPGNP=LOG(PGNP);

LNPMG=LOG(PMG);

QMG_CAR=LOG(QMG/CAR);

RGNP_POP=LOG(RGNP/POP);

CAR_POP=LOG(CAR/POP);

PMG_PGNP=LOG(PMG/PGNP);

Data USGAS1; set USGAS; If YEAR>1972 then delete;

Proc reg data=USGAS1;

Model LNQMG=LNCAR LNPOP LNRGNP LNPGNP LNPMG; Model QMG_CAR=RGNP_POP CAR_POP PMGPGNP;

Proc corr data=USGAS1;

VarLNCAR LNPOP LNRGNP LNPGNP LNPMG; run;

Proc reg data=USGAS;

Model LNQMG=LNCAR LNPOP LNRGNP LNPGNP LNPMG; Model QMG_CAR=RGNP_POP CAR. POP PMGPGNP;

run;

data DUMMY1; set USGAS;

If Year>1974 then SHOCK74=0; else SHOCK74=1; DUMMY74=PMG_PGNP*SHOCK74;

Proc reg data=DUMMY1;

Model QMG_CAR=SHOCK74 RGNP. POP CAR. POP PMG_PGNP; Model QMG_CAR=RGNP_POP CAR_POP PMGPGNP DUMMY74; run;

4.16 a. MODEL1

Dependent Variable: LNCONS

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 5 165.96803 33.19361 70.593 0.0001 Error 132 62.06757 0.47021 C Total 137 228.03560 Root MSE 0.68572 R - square 0.7278 Dep Mean 11.89979 Adj R - sq 0.7175 C. V. 5.76243

Parameter Estimates

 Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| INTERCEP 1 -53.577951 4.53057139 -11.826 0.0001 LNPG 1 -1.425259 0.31539170 -4.519 0.0001 LNPE 1 0.131999 0.50531143 0.261 0.7943 LNPO 1 0.237464 0.22438605 1.058 0.2919 LNHDD 1 0.617801 0.10673360 5.788 0.0001 LNPI 1 6.554349 0.48246036 13.585 0.0001

b. The following plot show that states with low level of consumption are over predicted, while states with high level of consumption are under predicted. One can correct for this problem by either running a separate regression for each state, or use dummy variables for each state to allow for a varying intercept.

c. Model 2

Dependent Variable: LNCONS

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 10 226.75508 22.67551 2248.917 0.0001 Error 127 1.28052 0.01008 C Total 137 228.03560 Root MSE 0.10041 R - square 0.9944 Dep Mean 11.89979 Adj R - sq 0.9939 C. V. 0.84382 Plot of PRED*LNCONS. Symbol is value of STATE.

 NN NNN

 CCCC C

9 10 11 12 13 14

LNCONS

NOTE: 42 obs hidden.

 Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| INTERCEP 1 6.260803 1.65972981 3.772 0.0002 LNPG 1 -0.125472 0.05512350 -2.276 0.0245 LNPE 1 -0.121120 0.08715359 -1.390 0.1670 LNPO 1 0.155036 0.03706820 4.182 0.0001 LNHDD 1 0.359612 0.07904527 4.549 0.0001 LNPI 1 0.416600 0.16626018 2.506 0.0135 DUMMY. NY 1 -0.702981 0.07640346 -9.201 0.0001 DUMMY. FL 1 -3.024007 0.11424754 -26.469 0.0001 DUMMY. MI 1 -0.766215 0.08491262 -9.024 0.0001 DUMMY. TX 1 -0.679327 0.04838414 -14.040 0.0001 DUMMY. UT 1 -2.597099 0.09925674 -26.165 0.0001

f. Dummy Variable Regression without an Intercept NOTE: No intercept in model. R-square is redefined. Dependent Variable: LNCONS

Analysis of Variance

 Sum of Mean Source DF Squares Square F Value Prob>F Model 11 19768.26019 1797.11456 178234.700 0.0001 Error 127 1.28052 0.01008 U Total 138 19769.54071

Root MSE 0.10041 R-square 0.9999

Dep Mean 11.89979 Adj R - sq 0.9999

C. V. 0.84382

Parameter Estimates

 Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| LNPG 1 -0.125472 0.05512350 -2.276 0.0245 LNPE 1 -0.121120 0.08715359 -1.390 0.1670 LNPO 1 0.155036 0.03706820 4.182 0.0001 LNHDD 1 0.359612 0.07904527 4.549 0.0001 LNPI 1 0.416600 0.16626018 2.506 0.0135 DUMMY. CA 1 6.260803 1.65972981 3.772 0.0002 DUMMY. NY 1 5.557822 1.67584667 3.316 0.0012 DUMMY. FL 1 3.236796 1.59445076 2.030 0.0444 DUMMY. MI 1 5.494588 1.67372513 3.283 0.0013 DUMMY. TX 1 5.581476 1.62224148 3.441 0.0008 DUMMY. UT 1 3.663703 1.63598515 2.239 0.0269

SAS PROGRAM Data NATURAL;

Input STATE \$ SCODE YEAR Cons Pg Pe Po LPgas HDD Pi; Cards;

Data NATURAL1; SET NATURAL;

LNCONS=LOG(CONS);

LNPG=LOG(PG);

LNPO=LOG(PO) ;

LNPE=LOG(PE) ;

LNHDD=LOG(HDD) ;

LNPI=LOG(PI);

******** PROB16 a **********;

********************************.

Proc reg data=NATURAL1;

Model LNCONS=LNPG LNPE LNPO LNHDD LNPI; Output out=OUT1 R=RESID P=PRED;

Proc plot data=OUT1 vpercent=75 hpercent=100;

plot PRED*LNCONS=STATE;

Data NATURAL2; set NATURAL1;

If STATE=‘NY’ THEN DUMMY_NY=1; ELSE DUMMY_NY=0;

If STATE=‘FL’ THEN DUMMY_FL=1; ELSE DUMMY_FL=0;

If STATE=‘MI’ THEN DUMMY_MI=1; ELSE DUMMY_MI=0;

If STATE=‘TX’ THEN DUMMY_TX=1; ELSE DUMMY_TX=0;

If STATE=‘UT’ THEN DUMMY_UT=1; ELSE DUMMY_UT=0; If STATE=‘CA’ THEN DUMMY_CA=1; ELSE DUMMY_CA=0;

Proc reg data=NATURAL2;

Model LNCONS=LNPG LNPE LNPO LNHDD LNPI

DUMMY_CA DUMMY_NY DUMMY_FL DUMMY. MI DUMMY_TX DUMMY_UT/NOINT; run;

References

Baltagi, B. H. (1987), “Simple versus Multiple Regression Coefficients,” Economet­ric Theory, Problem 87.1.1, 3: 159.

Kennedy, P. E. (1981), “Estimation with Correctly Interpreted Dummy Variables in Semilogarithmic Equations,” American Economic Review, 71: 802.

Rao, U. L.G. and P. M. White (1988), “Effect of an Additional Regressor on R2,” Econometric Theory, Solution 86.3.1,4: 352.

CHAPTER 5

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