Sets EC220
Problem 3 9,
Michaelmas Term 2011
2011/2012
Note: Problem Sets 1 and 2 were distributed with the initial hand-out.
For the class in the week beginning October 31
PROBLEM SET 3
Note: For Items 1 and 2 you will use Stata and an EAEF data set derived from the US National
Longitudinal Survey of Youth. For a description of the EAEF data sets, see Appendix B at the end
of the text. You should download your data set from the website at
http://econ.lse.ac.uk/courses/ec220/G/iedata/eecs/stata/
To download, double click on the data set, choose save this file to disk, and then decide where you
wish to save the file in your H:space. You will download the data set only once. Thereafter you will
work with it from your H:space.
You may save Stata output by blocking it with the cursor and copying directly to a wordprocessing application such as Word. You should then change the typeface to Courier to ensure that
the output lines up correctly. Alternatively, you may open a Stata output file. If you do this, you
should choose .log file, rather than the default .smcl file, in the save as file type window.
You will probably find it convenient to do Items 1 and 2 in the same Stata session. In principle,
downloading the data set, launching Stata, running the regressions, and saving to a word processor
should take no more than five minutes. Obviously, since this is your first time, it will take longer.
(1)*
Exercise 1.6 in the text
Does educational attainment depend on intellectual ability? In the United States, as in most
countries, there is a positive correlation between educational attainment and cognitive ability. S
(highest grade completed by 2002) is the number of years of schooling of the respondent.
ASVABC is a composite measure of numerical and verbal ability with mean 50 and standard
deviation 10 (both approximately; for further details of the measure, see Appendix B). Perform a
regression of S on ASVABC and interpret the regression results. Comment on the value of R2.
Stata command
reg S ASVABC
(2)*
Exercise 1.7 in the text
Do earnings depend on education? Using your EAEF data set, fit an earnings function parallel to
that in Table 1.2, regressing EARNINGS on S, and give an interpretation of the coefficients.
Comment on the value of R2.
Stata command
reg EARNINGS S
(3)
Exercise 1.16 in the text.
Come to class prepared to present an answer to this exercise. A written answer is not required.
PROBLEM SETS, MICHAELMAS TERM
2
Two individuals fit earnings functions relating EARNINGS to S using EAEF Data Set 21.
first individual does it correctly and obtains the result found in Section 1.6:
The
EARNINGS = 13.93 + 2.46 S
The second individual makes a mistake and regresses S on EARNINGS, obtaining the following
result:
S 12.29 0.070 EARNINGS
From this result the second individual derives
EARNINGS 175.57 14.29S
Explain why this equation is different from that fitted by the first individual.
(4)
Exercise 2.2 in the text.
For the model Yi 2 X i u i , the OLS estimator of 2 is b2
n
i 1
X i Yi
n
j 1
X2 .
j
Demonstrate that b2 may be decomposed as
b2 2
n
d u
ii
i 1
where d i
Xi
and hence demonstrate that it is an unbiased estimator of 2.
n
X
2
j
j 1
(5)*
Exercise 2.4 in the text.
An investigator correctly believes that the relationship between two variables X and Y is given by
Y = 1 + 2X + u
Given a sample of n observations, the investigator estimates 2 by calculating it as the average value
of Y divided by the average value of X. Discuss the properties of this estimator. What difference
would it make if it could be assumed that 1 = 0?
For the class in the week beginning November 7
PROBLEM SET 4
(1)*
Exercise 2.12 in the text
Suppose that the true relationship between Y and X is Yi 1 2 X i u i and that the fitted
model is Yi b1 b2 X i . In Section 1.4, it was shown that if Yi* 1 2Yi , and Y * is regressed
*
*
on X, the slope coefficient b2 2 b2 . How will the standard error of b2 be related to the standard
error of b2?
(2)
Exercises 2.17 and 2.22 in the text
Perform a t test on the slope coefficient and the intercept of the earnings function fitted using your
EAEF data set in Exercise 1.7, and state your conclusions. Explain whether it would be possible to
PROBLEM SETS, MICHAELMAS TERM
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perform one-sided tests instead of two-sided tests. If you think that one-sided tests are justified,
perform them, giving your justification, and state whether the use of a one-sided test makes any
difference.
(3)
Exercise 2.27 in the text.
Calculate the 95% confidence interval for the slope coefficient of the earnings function fitted with
your EAEF data set in Exercise 1.7. Check that it agrees with the confidence interval printed as part
of the Stata output.
(4)
Exercise 2.30 in the text.
Calculate the F statistic from ESS and RSS obtained in the earnings function fitted using your
EAEF data set and check that it is equal to the value printed in the output. Check that the F
statistic derived from R2 is the same. Perform an appropriate F test.
(5)
Exercise 2.31 in the text.
Verify that the F statistic in the earnings function regression run by you using your EAEF data set
in Exercise 1.7 is equal to the square of the t statistic for the slope coefficient, and that the critical
value of F at the 1 percent significance level is equal to the square of the critical value of t.
(6)*
Exercise 2.32 in the text.
In Exercise 1.16 both researchers obtained a t statistic of 10.59 for the slope coefficient in their
regressions. Was this a coincidence?
For the class in the week beginning November 14
PROBLEM SET 5
(1)
Write down the name of your class teacher and his or her office hour (location, time).
(The purpose of this is to enable your class teacher to check that you know.)
(2)
Exercise 3.2 in the text.
Does educational attainment depend on parents education? Using your EAEF data set, first regress
S on ASVABC and SM and interpret the regression results. Repeat the regression using SF instead
of SM, and then again including both SM and SF as regressors. There is a saying that if you
educate a male, you educate an individual, while if you educate a female, you educate a nation.
The premise is that the education of a future mother has a future beneficial knock-on effect on the
educational attainment of her children. Do your regression results support this view?
Stata commands
reg S ASVABC SM
reg S ASVABC SF
reg S ASVABC SM SF
(3)*
Exercise 3.7 in the text
Two researchers are investigating the effects of time spent studying on the examination marks
earned by students on a certain course. For a sample of 100 students, they have the examination
mark, M, total hours spent studying, H, hours on primary study, P, and hours spent on revision, R
PROBLEM SETS, MICHAELMAS TERM
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By definition, H = P + R. Researcher A decides to regress M on P and R and fits the following
regression:
M=
45.6 + 0.15 P + 0.21 R
Researcher B decides to regress M on H and P, with regression output
M=
45.6 + 0.21 H 0.05 P
Give an interpretation of the coefficients of both regressions.
negative coefficient of P is implausible. Is this correct?
(4)*
Researcher B thinks that his
Exercise 3.13 in the text.
In Exercise 3.7, the sample means of H, P, and R are 100 hours, 95 hours, and 5 hours,
respectively and the standard deviations of the distributions of H, P, and R are 10.1, 10.1, and 2.1,
respectively. The standard errors of the coefficients of the regression of Researcher A are shown
in parentheses under the coefficients.
M=
45.6 + 0.15 P + 0.21 R
(2.8) (0.03) (0.14)
Perform t tests of the significance of the coefficients of P and R. The researcher says that the
insignificant coefficient of R is to be expected because the students, on average, spent much less
time on revision than on primary study. Explain whether this assertion is correct.
(5)
Exercise 3.10 in the text.
The following earnings functions were fitted separately for males and females, using EAEF Data Set
21 (standard errors in parentheses):
males
EARNINGS = 31.5168 + 3.1408 S + 0.6453 EXP
(7.8708) (0.3693)
(0.2382)
females
EARNINGS = 17.2028 + 2.0772 S + 0.3179 EXP
(4.5797) (0.2805)
(0.1388)
Using equation (3.36) in the text, explain why the standard errors of the coefficients of S and EXP
are greater for the male subsample than for the female subsample, and why the difference in the
standard errors is relatively large for EXP.
Further data:
males
su
n
rS,EXP
MSD(S)
MSD(EXP)
(6)*
females
14.278
270
0.4029
6.6080
15.8858
10.548
270
0.0632
5.2573
21.4628
What kind of ability is important for educational attainment? Using your EAEF data set, regress S on
SM, SF, ASVAB02, ASVAB03, and ASVAB04, the three components of the ASVABC composite score.
Calculate correlation coefficients for the three ASVAB components. The two verbal components,
PROBLEM SETS, MICHAELMAS TERM
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ASVAB03 (word knowledge) and ASVAB04 (paragraph comprehension) are likely to be particularly
highly correlated. Generate a composite, VERBAL, as their sum, and regress S on SM, SF, ASVAB02,
and VERBAL. Compare the regression results.
Stata commands
reg
cor
gen
reg
(7)
S SM SF ASVAB02 ASVAB03 ASVAB04
ASVAB02 ASVAB03 ASVAB04
VERBAL = ASVAB03 + ASVAB04
S SM SF ASVAB02 VERBAL
Exercise 3.18 in the text
Using your EAEF data set, fit an educational attainment function, regressing S on ASVABC, SM,
and SF. Calculate the F statistic using the explained sum of squares and the residual sum of
squares in the regression output, verify that it matches the F statistic in the output, and perform a
test of the explanatory power of the equation as a whole. Also calculate the F statistic using R2
and verify that it is the same.
Stata command
reg S ASVABC SM SF
(8)*
Exercise 3.19 in the text.
Fit an educational attainment function using the specification in Exercise 3.18, adding the ASVAB
speed test scores ASVAB05 and ASVAB06. Perform an F test of the joint explanatory power of
ASVAB05 and ASVAB06, using the results of this regression and that in Exercise 3.18.
Stata command
reg S SM SF ASVABC ASVAB05 ASVAB06
For the class in the week beginning November 21
PROBLEM SET 6
(1)* Exercise 4.6 in the text.
What is the relationship between weight and height? Using your EAEF data set, regress the (natural)
logarithm of WEIGHT85 on the logarithm of HEIGHT. Interpret the regression results and perform
appropriate tests.
Stata commands
gen LGWT85 = ln(WEIGHT85)
gen LGHEIGHT = ln(HEIGHT)
reg LGWT85 LGHEIGHT
(2)*
Exercise 4.7 in the text
Suppose that the logarithm of Y is regressed on the logarithm of X, the fitted regression being
^
log Y b1 b2 log X
Suppose Y * Y , where is a constant, and suppose that log Y * is regressed on log X . Determine
how the regression coefficients are related to those of the original regression. Determine also how
the t statistic for b2 and R2 for the equation are related to those in the original regression.
PROBLEM SETS, MICHAELMAS TERM
(3)*
6
Exercise 4.9 in the text.
Using your EAEF data set, regress the logarithm of earnings on S and EXP. Interpret the regression
results and perform appropriate tests.
Stata commands
gen LGEARN = ln(EARNINGS)
reg LGEARN S EXP
Save the data set after generating so LGEARN that you do not have to define it again in the future.
Give the new data set the same name as the old one, overwriting the old one.
(4)
Exercise 4.10 in the text.
Using your EAEF data set, evaluate whether the dependent variable of an earnings function should
be linear or logarithmic. Calculate the geometric mean of EARNINGS by taking the exponential of
the mean of LGEARN. Define EARNSTAR by dividing EARNINGS by this quantity and calculate
LGEARNST as its logarithm. Regress EARNSTAR and LGEARNST on S and EXP and compare the
residual sums of squares.
Stata commands
sum LGEARN
In the next instruction, replace Z with the mean of LGEARN from the sum command:
gen
gen
reg
reg
(5)
EARNSTAR
LGEARNST
EARNSTAR
LGEARNST
=
=
S
S
EARNINGS/exp(Z)
ln(EARNSTAR)
EXP
EXP
Exercise 4.12 in the text
Come to class prepared to discuss this item. A written answer is not required.
The output shows the result of regression of S on SM and its square, SMSQ. Evaluate the
regression results. In particular, explain why the coefficient of SM is negative.
. gen SMSQ = SM*SM
. reg S SM SMSQ
Source |
SS
df
MS
-------------+-----------------------------Model | 519.131914
2 259.565957
Residual | 2685.85142
537 5.00158551
-------------+-----------------------------Total | 3204.98333
539 5.94616574
Number of obs
F( 2,
537)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
540
51.90
0.0000
0.1620
0.1589
2.2364
-----------------------------------------------------------------------------S|
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------SM | -.2564658
.1318583
-1.95
0.052
-.5154872
.0025556
SMSQ |
.0271172
.0060632
4.47
0.000
.0152068
.0390277
_cons |
12.79121
.7366358
17.36
0.000
11.34416
14.23825
------------------------------------------------------------------------------
PROBLEM SETS, MICHAELMAS TERM
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For the class in the week beginning November 28
PROBLEM SET 7
(1)
Exercise 5.1 in the text.
Does the sex of an individual affect educational attainment? Using your EAEF data set, regress S on
ASVABC, SM, SF and MALE, a dummy variable that is 1 for male respondents and 0 for female ones.
Interpret the coefficients and perform t tests. Is there any evidence that the educational attainment of
males is different from that of females?
Stata command
reg S ASVABC SM SF MALE
(2)*
Exercise 5.6 in the text.
Does ethnicity affect educational attainment? In your EAEF data set you will find the following
ethnic dummy variables:
ETHHISP
ETHBLACK
ETHWHITE
1 if hispanic, 0 otherwise
1 if black, 0 otherwise
1 if not hispanic or black, 0 otherwise.
Regress S on ASVABC, MALE, SM, SF, ETHBLACK, and ETHHISP. (In this specification
ETHWHITE has been chosen as the reference category, and so it is omitted.) Interpret the
regression results and perform t tests on the coefficients.
Stata command
reg S ASVABC MALE SM SF ETHBLACK ETHHISP
(3)*
Exercise 5.10 in the text.
Evaluate whether the ethnicity dummies as a group have significant explanatory power for
educational attainment by comparing the residual sums of squares in the regressions in Exercises 5.1
and 5.6.
(4)
Exercise 5.17 in the text.
Is the effect of education on earnings different for males and females? Using your EAEF data set,
define a slope dummy variable MALES as the product of MALE and S:
MALES = MALE*S
Regress LGEARN on S, EXP, ETHBLACK, ETHHISP, MALE, and MALES, interpret the equation
and perform appropriate statistical tests.
Stata commands
gen MALES = MALE*S
reg LGEARN S EXP ETHBLACK ETHHISP MALE MALES
(5)*
Exercise 5.19 in the text.
Are educational attainment functions are different for males and females? Using your EAEF data
set, regress S on ASVABC, ETHBLACK, ETHHISP, SM, and SF (do not include MALE). Repeat the
regression using only the male respondents. Repeat it again using only the female respondents.
Perform a Chow test.
PROBLEM SETS, MICHAELMAS TERM
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Stata commands
reg S ASVABC ETHBLACK ETHHISP SM SF
reg S ASVABC ETHBLACK ETHHISP SM SF if MALE==1
reg S ASVABC ETHBLACK ETHHISP SM SF if MALE==0
(6)*
Exercise 5.21 in the text.
Are there differences in male and female educational attainment functions? This question has
been answered by Item 5 but nevertheless it is instructive to investigate the issue using the dummy
variable approach. Using your EAEF data set, define the following slope dummies combining
MALE with the parental education, cognitive ability, and ethnicity variables:
MALESM = MALE*SM
MALESF = MALE*SF
MALEASVC = MALE*ASVABC
MALEBLAC = MALE*ETHBLACK
MALEHISP = MALE*ETHHISP
and regress S on ETHBLACK, ETHHISP, ASVABC, SM, SF, MALE, MALEBLAC, MALEHISP,
MALEASVC, MALESM, and MALESF. Next regress S on ETHBLACK, ETHHISP, ASVABC, SM,
and SF only. Examine the correlations among the male intercept and slope dummy variables.
Perform an F test of the joint explanatory power of MALE and the slope dummy variables as a group
(verify that the F statistic is the same as in Exercise 5.19) and perform t tests on the coefficients of
the slope dummy variables in the first regression. What was the point of computing the correlations
between the male intercept and slope dummy variables?
Stata commands
gen
gen
gen
gen
gen
reg
MALESM = MALE*SM
MALESF = MALE*SF
MALEASVC = MALE*ASVABC
MALEBLAC = MALE*ETHBLACK
MALEHISP = MALE*ETHHISP
S ASVABC ETHBLACK ETHHISP SM SF MALE MALESM MALESF MALEASVC
MALEBLAC MALEHISP
reg S ETHBLACK ETHHISP ASVABC SM SF
cor MALE MALESM MALESF MALEASVC MALEBLAC MALEHISP
For the class in the week beginning December 5
PROBLEM SET 8
(1)*
Exercise 6.2 in the text.
Using your EAEF data set, regress LGEARN (1) on S and EXP, (2) on S only, and (3) on EXP only.
Calculate the correlation between S and EXP. Compare the coefficients of S in regressions (1) and
(2). Give both mathematical and intuitive explanations of the direction of the change. Also compare
the coefficients of EXP in regressions (1) and (3) and explain the direction of the change.
Stata commands
reg
reg
reg
cor
LGEARN S EXP
LGEARN S
LGEARN EXP
S EXP
PROBLEM SETS, MICHAELMAS TERM
(2)*
9
Exercise 6.3 in the text
Using your EAEF data set, regress LGEARN (1) on S, EXP, MALE, ETHBLACK, and ETHHISP,
and (2) on S, EXP, MALE, ETHBLACK, ETHHISP, and ASVABC. Calculate the correlation
coefficients for the explanatory variables and discuss the differences in the regression results. (A
detailed mathematical analysis is not expected.)
Stata commands
reg LGEARN S EXP MALE ETHBLACK ETHHISP
reg LGEARN S EXP MALE ETHBLACK ETHHISP ASVABC
cor S EXP MALE ETHBLACK ETHHISP ASVABC
(3)*
Variation on Exercise 6.8 in the text.
Using your EAEF data set, regress LGEARN on S, EXP, ASVABC, MALE, ETHBLACK, and
ETHHISP. Repeat the regression, adding SM and SF. Calculate the correlations between SM and
SF and the other explanatory variables. Compare the results of the two regressions.
Stata commands
reg LGEARN S EXP ASVABC MALE ETHBLACK ETHHISP
reg LGEARN S EXP ASVABC MALE ETHBLACK ETHHISP SM SF
cor SM SF S EXP ASVABC MALE ETHBLACK ETHHISP
(4)
A Monte Carlo experiment uses the following model:
Y = 7,125 + 1,500X2 + 25X3 + u
There is a positive correlation between X2 and X3. u is generated as a multiple of a random variable
drawn from a normal distribution with zero mean and unit variance. In one replication of the
experiment, when Y is regressed against X2, omitting X3, the coefficient of X2 is 1,478. Is this what
you would expect? If not, explain what has happened.
For the class in the week beginning January 9
PROBLEM SET 9
(0)
Write down the name and office hour of your class teacher.
(1)*
Exercise 6.9 in the text.
Is potential work experience a satisfactory proxy for actual work experience? Length of work
experience is generally found to be an important determinant of earnings. Many data sets do not
contain this variable. To avoid the problem of omitted variable bias, a standard practice is to use
PWE, potential years of work experience, as a proxy. PWE is defined as AGE, less age at
completion of full-time education (years of schooling plus 5, assuming that schooling begins at the
age of 6):
PWE = AGE S 5.
Using your EAEF data set, regress LGEARN (1) on S, ASVABC, MALE, ETHBLACK, ETHHISP,
(2) on S, ASVABC, MALE, ETHBLACK, ETHHISP, and PWE and (3) on S, ASVABC, MALE,
ETHBLACK, ETHHISP, and EXP. Compare the results and evaluate whether PWE would have
been a satisfactory proxy for EXP if data for EXP had not been available.
Variation: PWE is not likely to be a satisfactory proxy for work experience for females because it
PROBLEM SETS, MICHAELMAS TERM
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does not take into account time spent out of the labor force rearing children. Investigate this by
running the three regressions for the male and female subsamples separately. You must drop the
MALE dummy from the specification (explain why).
Stata commands
gen
reg
reg
reg
PWE = AGE S
LGEARN S ASVABC
LGEARN S ASVABC
LGEARN S ASVABC
5
MALE ETHBLACK ETHHISP
MALE ETHBLACK ETHHISP PWE
MALE ETHBLACK ETHHISP EXP
Variation:
reg
reg
reg
reg
reg
reg
(2)*
LGEARN
LGEARN
LGEARN
LGEARN
LGEARN
LGEARN
S
S
S
S
S
S
ASVABC
ASVABC
ASVABC
ASVABC
ASVABC
ASVABC
MALE
MALE
MALE
MALE
MALE
MALE
ETHBLACK
ETHBLACK
ETHBLACK
ETHBLACK
ETHBLACK
ETHBLACK
ETHHISP
ETHHISP
ETHHISP
ETHHISP
ETHHISP
ETHHISP
if MALE==1
PWE if MALE==1
EXP if MALE==1
if MALE==0
PWE if MALE==0
EXP if MALE==0
Exercise 6.11 in the text.
Is previous work experience as valuable as experience with the current employer? Using your
EAEF data set, first regress LGEARN on S, EXP, MALE, ETHBLACK, and ETHHISP. Then define
PREVEXP = EXP TENURE.
The variable TENURE in your data set is the number of years spent working with the current
employer. Regress LGEARN on S, PREVEXP, TENURE, MALE, ETHBLACK, and ETHHISP.
The estimates of the coefficients of PREVEXP and TENURE will be different. This raises the
issue of whether the difference is due to random factors or whether the coefficients are
significantly different. Set up the null hypothesis H0: 1 = , where 1 is the coefficient of
PREVEXP and 2 is the coefficient of TENURE. Explain why the regression with EXP is the
correct specification if H0 is true, while the regression with PREVEXP and TENURE should be
used if H0 is false. Perform an F test of the restriction using RSS for the two regressions. Do this
for the combined sample and also for males and females separately.
Stata commands
gen
reg
reg
reg
reg
reg
reg
(3)*
PREVEXP = EXP TENURE
LGEARN EXP S MALE ETHBLACK ETHHISP
LGEARN PREVEXP TENURE S MALE ETHBLACK
LGEARN EXP S MALE ETHBLACK ETHHISP if
LGEARN PREVEXP TENURE S MALE ETHBLACK
LGEARN EXP S MALE ETHBLACK ETHHISP if
LGEARN PREVEXP TENURE S MALE ETHBLACK
ETHHISP
MALE==1
ETHHISP if MALE==1
MALE==0
ETHHISP if MALE==0
Exercise 6.12 in the text.
Using your EAEF data set, regress LGEARN on S, EXP, MALE, ETHBLACK, ETHHISP, and
TENURE. Demonstrate that a t test on the coefficient of TENURE is a test of the restriction
described in Exercise 6.11. Verify that the same result is obtained. Do this for the combined
sample and also for males and females separately.
Stata commands
reg LGEARN S EXP MALE ETHBLACK ETHHISP TENURE
reg LGEARN S EXP MALE ETHBLACK ETHHISP TENURE if MALE==1
reg LGEARN S EXP MALE ETHBLACK ETHHISP TENURE if MALE==0