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).

Подпись: ei2 (n -2 2

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

image171 image172

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

RSS

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

Подпись: et2 and a = Y—be = YПодпись: cForYt = 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

image175

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

DepMean 6.95074 Adj R-sq 0.1154

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

Подпись: Variable DF Parameter Estimate Standard Error T for HO: Parameter=0 Prob>|T| INTERCEP 1 6.122535 0.07304328 83.821 0.0001 FEM 1 -0.905504 0.24132106 -3.752 0.0002 ED 1 0.068622 0.00555341 12.357 0.0001 F_EDC 1 0.033696 0.01844378 1.827 0.0682

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

Подпись: = 102.2.Подпись: Fb. 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

0.2030

OCC

1

-0.161522

0.03690729

-4.376

0.0001

IND

1

0.084663

0.02916370

2.903

0.0038

SOUTH

1

-0.058763

0.03090689

-1.901

0.0578

SMSA

1

0.166191

0.2955099

5.624

0.0001

MS

1

0.095237

0.04892770

1.946

0.0521

FEM

1

-0.324557

0.06072947

-5.344

0.0001

UNION

1

0.106278

0.03167547

3.355

0.0008

ED

1

0.057194

0.00659101

8.678

0.0001

BLK

1

-0.190422

0.05441180

-3.500

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.

image179 image180

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;

Подпись: 4.15 a. MODEL1 (1950-1972)

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;

image182

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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