GUJARATI’S EBE SOLUTIONS MANUAL
CHAPTER 1 EXERCISES
1.1. Consider the regression results given in Table 1.2.
a. Suppose you want to test the hypothesis that the true or population regression coefficient of
the education variable is 1. How would you test this hypothesis? Show the necessary
calculations.
The equation we are looking at is:
wagei = b1 + b2*(femalei) + b3*(nonwhitei) + b4*(unioni) + b5*(educationi) + b6*(experi) + ei
Here we are testing:
H0: β5 = 1
H1: β5 ≠ 1
From Table 1.2, we have: t = (1.370301 - 1)/0.065904 = 5.618794.
From the t table, the critical t statistic for α = 1% is 2.576 (df = 1289 – 6 = 1283, so we can use df =
∞). Since 5.619 > 2.576, we can easily reject the null hypothesis at the 1% level.
b. Would you reject or not reject the hypothesis that the true union regression coefficient is
1?
Here we are testing:
H0: β4 = 1
H1: β4 ≠ 1
From Table 1.2, we have: t = (1.095976 - 1)/0.506078 = 0.189647.
From the t table, the critical t statistic for α = 10% is 1.645 (using df = ∞). Since 0.190 < 1.645, we
cannot even reject the null hypothesis at the 10% level. (Note that from the output, if we were testing H0:
β4 = 0 vs. H1: β4 ≠ 0, we could reject the null hypothesis at the 5% level.)
c. Can you take the logs of the nominal variables, such as gender, race and union status?
Why or why not?
No, because these are categorical variables that often take values of 0 or 1. The natural log of 1 is
0, and the natural log of 0 is undefined. Moreover, taking the natural log would not be helpful as
the values of the nominal variables to not have a specific meaning.
d. What other variables are missing from the model?
We could have included control variables for region, marital status, and number of children on the
right-hand side. Instead of including a continuous variable for education, we could have controlled
for degrees (high school graduate, college graduate, etc). An indicator for the business cycle (such
as the unemployment rate) may be helpful. Moreover, we could include state-level policies on the
minimum wage and right-to-work laws.
e. Would you run separate wage regressions for white and nonwhite workers, male and
female workers, and union and non-union workers? And how would you compare them?
We would if we felt the two groups were systematically different from one another. We can run
the models separately and conduct an F test to see if the two regressions are significantly different.
If they are, we should run them separately. The F statistic may be obtained by running the two
,together – the restricted model – then running the two separately – jointly, the unrestricted model.
We then obtain the residual sum of squares for the restricted model (RSSR) and the residual sum of
squares for the unrestricted model (RSSUR, equal to RSS1 + RSS2 from two separate models). F =
[(RSSR – RSSUR)/k] / [RSSUR/(n-2k)] ~ Fk,n-2k. I would then see which model was a better predictor
of the outcome variable, wage.
f. Some states have right-to-work laws (i.e., union membership is not mandatory) and some
do not have such laws (i.e, union membership is permitted). Is it worth adding a dummy
variable taking the value of 1 if the right-to-work laws are present and 0 otherwise? A priori,
what would you expect if this variable is added to the model?
Since we would expect these laws to have an effect on wage, it may be worth adding this variable.
A priori, we would expect this variable to have a negative effect on wage, as union wages are
generally higher than nonunion wages.
h. Would you add the age of the worker as an explanatory variable to the model? Why or
why not?
No, we would not add this variable to the model. This is because the variable Exper is defined as
(age – education – 6), so it would be perfectly collinear and not add any new information to the
model.
, CHAPTER 2 EXERCISES
2.1. Consider the following production function, known in the literature as the transcendental
production function (TPF).
Qi = B1 LiB2 K iB3 e B4 Li + B5 Ki
where Q, L and K represent output, labor and capital, respectively.
(a) How would you linearize this function? (Hint: logarithms.)
Taking the natural log of both sides, the transcendental production function above can be written
in linear form as:
ln Qi = ln B1 + B2 ln Li + B3 ln Ki + B4 Li + B5 K i + ui
(b) What is the interpretation of the various coefficients in the TPF?
The coefficients may be interpreted as follows:
ln B1 is the y-intercept, which may not have any viable economic interpretation, although B1 may
be interpreted as a technology constant in the Cobb-Douglas production function.
The elasticity of output with respect to labor may be interpreted as (B2 + B4*L). This is because
ln Qi B ln Qi ln Qi
= B2 + 4 = B2 B4 L . Recall that =
( )
.
ln Li 1 + ln Li 1 Li
L L
Similarly, the elasticity of output with respect to capital can be expressed as (B3 + B5*K).
(c) Given the data in Table 2.1, estimate the parameters of the TPF.
The parameters of the transcendental production function are given in the following Stata
output:
. reg lnoutput lnlabor lncapital labor capital
Source | SS df MS Number of obs = 51
-------------+------------------------------ F( 4, 46) = 312.65
Model | 91.95773 4 22.9894325 Prob > F = 0.0000
Residual | 3.38240102 46 .073530457 R-squared = 0.9645
-------------+------------------------------ Adj R-squared = 0.9614
Total | 95.340131 50 1.90680262 Root MSE = .27116
lnoutput | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lnlabor | .5208141 .1347469 3.87 0.000 .2495826 .7920456
lncapital | .4717828 .1231899 3.83 0.000 .2238144 .7197511
labor | -2.52e-07 4.20e-07 -0.60 0.552 -1.10e-06 5.94e-07
capital | 3.55e-08 5.30e-08 0.67 0.506 -7.11e-08 1.42e-07
_cons | 3.949841 .5660371 6.98 0.000 2.810468 5.089215
B1 = e3.949841 = 51.9271.
B2 = 0.5208141
B3 = 0.4717828
B4 = -2.52e-07
, B5 = 3.55e-08
Evaluated at the mean value of labor (373,914.5), the elasticity of output with respect to labor is 0.4266.
Evaluated at the mean value of capital (2,516,181), the elasticity of output with respect to capital is
0.5612.
(d) Suppose you want to test the hypothesis that B4 = B5 = 0. How would you test these
hypotheses? Show the necessary calculations. (Hint: restricted least squares.)
I would conduct an F test for the coefficients on labor and capital. The output in Stata for this test
gives the following:
. test labor capital
( 1) labor = 0
( 2) capital = 0
F( 2, 46) = 0.23
Prob > F = 0.7992
This shows that the null hypothesis of B4 = B5 = 0 cannot be rejected in favor of the alternative
hypothesis of B4 ≠ B5 ≠ 0. We may thus question the choice of using a transcendental production
function over a standard Cobb-Douglas production function.
We can also use restricted least squares and perform this calculation “by hand” by conducting an F
test as follows:
(RSSR − RSSUR ) /(n − k + 2 − n + k )
F= ~ F 2,46
RSSUR /(n − k )
The restricted regression is:
ln Qi = ln B1 + B2 ln Li + B3 ln Ki + ui ,
which gives the following Stata output:
. reg lnoutput lnlabor lncapital;
Source | SS df MS Number of obs = 51
-------------+------------------------------ F( 2, 48) = 645.93
Model | 91.9246133 2 45.9623067 Prob > F = 0.0000
Residual | 3.41551772 48 .071156619 R-squared = 0.9642
-------------+------------------------------ Adj R-squared = 0.9627
Total | 95.340131 50 1.90680262 Root MSE = .26675
lnoutput | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lnlabor | .4683318 .0989259 4.73 0.000 .269428 .6672357
lncapital | .5212795 .096887 5.38 0.000 .326475 .7160839
_cons | 3.887599 .3962281 9.81 0.000 3.090929 4.684269
The unrestricted regression is the original one shown in 2(c). This gives the following:
(3.4155177 − 3.382401) /(51 − 5 + 2 − 51 + 5)
F= = 0.22519 ~ F2,46
3.382401/(51 − 5)
CHAPTER 1 EXERCISES
1.1. Consider the regression results given in Table 1.2.
a. Suppose you want to test the hypothesis that the true or population regression coefficient of
the education variable is 1. How would you test this hypothesis? Show the necessary
calculations.
The equation we are looking at is:
wagei = b1 + b2*(femalei) + b3*(nonwhitei) + b4*(unioni) + b5*(educationi) + b6*(experi) + ei
Here we are testing:
H0: β5 = 1
H1: β5 ≠ 1
From Table 1.2, we have: t = (1.370301 - 1)/0.065904 = 5.618794.
From the t table, the critical t statistic for α = 1% is 2.576 (df = 1289 – 6 = 1283, so we can use df =
∞). Since 5.619 > 2.576, we can easily reject the null hypothesis at the 1% level.
b. Would you reject or not reject the hypothesis that the true union regression coefficient is
1?
Here we are testing:
H0: β4 = 1
H1: β4 ≠ 1
From Table 1.2, we have: t = (1.095976 - 1)/0.506078 = 0.189647.
From the t table, the critical t statistic for α = 10% is 1.645 (using df = ∞). Since 0.190 < 1.645, we
cannot even reject the null hypothesis at the 10% level. (Note that from the output, if we were testing H0:
β4 = 0 vs. H1: β4 ≠ 0, we could reject the null hypothesis at the 5% level.)
c. Can you take the logs of the nominal variables, such as gender, race and union status?
Why or why not?
No, because these are categorical variables that often take values of 0 or 1. The natural log of 1 is
0, and the natural log of 0 is undefined. Moreover, taking the natural log would not be helpful as
the values of the nominal variables to not have a specific meaning.
d. What other variables are missing from the model?
We could have included control variables for region, marital status, and number of children on the
right-hand side. Instead of including a continuous variable for education, we could have controlled
for degrees (high school graduate, college graduate, etc). An indicator for the business cycle (such
as the unemployment rate) may be helpful. Moreover, we could include state-level policies on the
minimum wage and right-to-work laws.
e. Would you run separate wage regressions for white and nonwhite workers, male and
female workers, and union and non-union workers? And how would you compare them?
We would if we felt the two groups were systematically different from one another. We can run
the models separately and conduct an F test to see if the two regressions are significantly different.
If they are, we should run them separately. The F statistic may be obtained by running the two
,together – the restricted model – then running the two separately – jointly, the unrestricted model.
We then obtain the residual sum of squares for the restricted model (RSSR) and the residual sum of
squares for the unrestricted model (RSSUR, equal to RSS1 + RSS2 from two separate models). F =
[(RSSR – RSSUR)/k] / [RSSUR/(n-2k)] ~ Fk,n-2k. I would then see which model was a better predictor
of the outcome variable, wage.
f. Some states have right-to-work laws (i.e., union membership is not mandatory) and some
do not have such laws (i.e, union membership is permitted). Is it worth adding a dummy
variable taking the value of 1 if the right-to-work laws are present and 0 otherwise? A priori,
what would you expect if this variable is added to the model?
Since we would expect these laws to have an effect on wage, it may be worth adding this variable.
A priori, we would expect this variable to have a negative effect on wage, as union wages are
generally higher than nonunion wages.
h. Would you add the age of the worker as an explanatory variable to the model? Why or
why not?
No, we would not add this variable to the model. This is because the variable Exper is defined as
(age – education – 6), so it would be perfectly collinear and not add any new information to the
model.
, CHAPTER 2 EXERCISES
2.1. Consider the following production function, known in the literature as the transcendental
production function (TPF).
Qi = B1 LiB2 K iB3 e B4 Li + B5 Ki
where Q, L and K represent output, labor and capital, respectively.
(a) How would you linearize this function? (Hint: logarithms.)
Taking the natural log of both sides, the transcendental production function above can be written
in linear form as:
ln Qi = ln B1 + B2 ln Li + B3 ln Ki + B4 Li + B5 K i + ui
(b) What is the interpretation of the various coefficients in the TPF?
The coefficients may be interpreted as follows:
ln B1 is the y-intercept, which may not have any viable economic interpretation, although B1 may
be interpreted as a technology constant in the Cobb-Douglas production function.
The elasticity of output with respect to labor may be interpreted as (B2 + B4*L). This is because
ln Qi B ln Qi ln Qi
= B2 + 4 = B2 B4 L . Recall that =
( )
.
ln Li 1 + ln Li 1 Li
L L
Similarly, the elasticity of output with respect to capital can be expressed as (B3 + B5*K).
(c) Given the data in Table 2.1, estimate the parameters of the TPF.
The parameters of the transcendental production function are given in the following Stata
output:
. reg lnoutput lnlabor lncapital labor capital
Source | SS df MS Number of obs = 51
-------------+------------------------------ F( 4, 46) = 312.65
Model | 91.95773 4 22.9894325 Prob > F = 0.0000
Residual | 3.38240102 46 .073530457 R-squared = 0.9645
-------------+------------------------------ Adj R-squared = 0.9614
Total | 95.340131 50 1.90680262 Root MSE = .27116
lnoutput | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lnlabor | .5208141 .1347469 3.87 0.000 .2495826 .7920456
lncapital | .4717828 .1231899 3.83 0.000 .2238144 .7197511
labor | -2.52e-07 4.20e-07 -0.60 0.552 -1.10e-06 5.94e-07
capital | 3.55e-08 5.30e-08 0.67 0.506 -7.11e-08 1.42e-07
_cons | 3.949841 .5660371 6.98 0.000 2.810468 5.089215
B1 = e3.949841 = 51.9271.
B2 = 0.5208141
B3 = 0.4717828
B4 = -2.52e-07
, B5 = 3.55e-08
Evaluated at the mean value of labor (373,914.5), the elasticity of output with respect to labor is 0.4266.
Evaluated at the mean value of capital (2,516,181), the elasticity of output with respect to capital is
0.5612.
(d) Suppose you want to test the hypothesis that B4 = B5 = 0. How would you test these
hypotheses? Show the necessary calculations. (Hint: restricted least squares.)
I would conduct an F test for the coefficients on labor and capital. The output in Stata for this test
gives the following:
. test labor capital
( 1) labor = 0
( 2) capital = 0
F( 2, 46) = 0.23
Prob > F = 0.7992
This shows that the null hypothesis of B4 = B5 = 0 cannot be rejected in favor of the alternative
hypothesis of B4 ≠ B5 ≠ 0. We may thus question the choice of using a transcendental production
function over a standard Cobb-Douglas production function.
We can also use restricted least squares and perform this calculation “by hand” by conducting an F
test as follows:
(RSSR − RSSUR ) /(n − k + 2 − n + k )
F= ~ F 2,46
RSSUR /(n − k )
The restricted regression is:
ln Qi = ln B1 + B2 ln Li + B3 ln Ki + ui ,
which gives the following Stata output:
. reg lnoutput lnlabor lncapital;
Source | SS df MS Number of obs = 51
-------------+------------------------------ F( 2, 48) = 645.93
Model | 91.9246133 2 45.9623067 Prob > F = 0.0000
Residual | 3.41551772 48 .071156619 R-squared = 0.9642
-------------+------------------------------ Adj R-squared = 0.9627
Total | 95.340131 50 1.90680262 Root MSE = .26675
lnoutput | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
lnlabor | .4683318 .0989259 4.73 0.000 .269428 .6672357
lncapital | .5212795 .096887 5.38 0.000 .326475 .7160839
_cons | 3.887599 .3962281 9.81 0.000 3.090929 4.684269
The unrestricted regression is the original one shown in 2(c). This gives the following:
(3.4155177 − 3.382401) /(51 − 5 + 2 − 51 + 5)
F= = 0.22519 ~ F2,46
3.382401/(51 − 5)
