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

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

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

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

RootMSE 0.41234 R-square 0.1169

DepMean 6.95074 Adj R-sq 0.1154

C. V. 5.93227

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

Restricted Model (without FEM and FEM*EDUC) Dependent Variable'LWAGE

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

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

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

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

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

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

c. Regression of POIL on SULPHUR Dependent Variable: POIL

Analysis of Variance

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

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

Dependent Variable: POIL

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

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

This indicates the presence of multicollinearity among the regressors.

f. MODEL1 (1950-1987)

Dependent Variable: LNQMG

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

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

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

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

9 10 11 12 13 14

LNCONS

NOTE: 42 obs hidden.

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

Analysis of Variance

Root MSE 0.10041 R-square 0.9999

Dep Mean 11.89979 Adj R - sq 0.9999

C. V. 0.84382

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