Exam 2 – Very (Excessively) Brief Answers
FE331 Spring 2013 – Economic Statistics
A. S. Rahman
Name:_______________________________
Thursday, April 4th, 2013
Mark all your answers in the space provided. Complete all the problems. You must
show all your work to receive full credit. 100 points possible. You have up to 110
minutes to complete this exam.
Short Problems:
2) Suppose you wish to analyze the relationship between variables Y and X. Name and
describe a few things you could do to determine whether you should estimate a linear
regression or a log-linear regression. (5 points)
Plot data, plot residuals and look for patterns, conduct MWD test, consider whether you
want elasticity or slope estimates.
3) In a few sentences, explain what kinds of issues arise with multicollinearity. (5 points)
High R2, low t-stats, “strange” results, wider confidence intervals, greater probability of
committing type-II errors.
4) In a few sentences, explain what you can/should do in the presence of multicollinearity.
(5 points)
Drop variables, transform variables, rethink model, get more data
5) In a few sentences, explain what a restricted F-test is. (5 points)
Tests joint significance of only a few coefficients.
Page 1 of 6
Exam 2 – Very (Excessively) Brief Answers
FE331 Spring 2013 – Economic Statistics
A. S. Rahman
Name:_______________________________
6) Explain what happens to regression estimates when you omit relevant variables. Explain
what happens to regression estimates when you include unnecessary variables. (5 points)
Omitted variables – create biased results.
Excessive variables – create multicollinearity
Multiple Variable Regression (25 points). The following regression results were obtained
from a sample of 50 pitchers on major league baseball teams from a single season. The
variables are as follows:
Winsi the number of wins recorded by pitcher
WPcti wins divided by total games for the pitcher’s team
ERAi earned-run-average: the number of earned runs given up per 9 innings pitched
Hi
pitcher’s number of hits given up per 9 innings pitched
HRi
pitcher’s number of home runs given up per 9 innings pitched
BBi
pitcher’s number of walks given up per 9 innings pitched
The regression results are as follows (standard errors for each estimate in parenthesis):
⏞ 𝑖 = 3.069 + 25.8𝑊𝑃𝑐𝑡𝑖 − 1.55𝐸𝑅𝐴𝑖 − 0.586𝐻𝑖 − 2.66𝐻𝑅𝑖 − 0.898𝐵𝐵𝑖
𝑊𝑖𝑛𝑠
(4.37) (2.41)
(0.884)
(0.525)
(1.23)
(0.40)
a) Provide a precise interpretation for the coefficient on the variable HR.
For every home-run given up by the pitcher per 9 innings, the pitcher on average loses
2.66 games, controlling for blah blah blah…
b) How many degrees of freedom do you have if you performed a t-test?
44
Page 2 of 6
Exam 2 – Very (Excessively) Brief Answers
FE331 Spring 2013 – Economic Statistics
A. S. Rahman
Name:_______________________________
c) A Money-Ball enthusiast is convinced that when a pitcher’s ERA rises by one run, the
pitcher on average loses one game per season, controlling for all the other factors listed
above. Test this hypothesis using the results above (show all calculations)? Is she right?
Test hypothesis that B3 = 1. Means you construct a new t-stat. Can’t reject the null.
d) The R2 for this model is 0.82. Perform an F-test for joint significance of the explanatory
variables at significance level α = 0.01 (show all calculations, and be sure to note the
critical value that you use).
Plug in values for F = (R2/(k-1)) / ((1-R)2/(n-k)) and test against critical F. Reject null.
e) Your knowledge of baseball suggests that the number of injuries a team experiences will
negatively affect both Wins and WPct. Unfortunately, you don't have this information
available and thus it was not included in the regression above. Do you believe this would
affect your regression results? If so, explain how. If not, then argue why not.
Negative correlation between WPct and injuries suggest that estimate of B2 is biased
upward. Why?
Page 3 of 6
Exam 2 – Very (Excessively) Brief Answers
FE331 Spring 2013 – Economic Statistics
A. S. Rahman
Name:_______________________________
8) Interpreting regression results (30 points). The dataset “CPS Data” (available on class
website) contains a sample of 1289 with the following characteristics
Wage
Age
Female
Nonwhite
Union
Education
Exper
hourly wage in dollars
age in years
1 if female worker, 0 if not
1 if a nonwhite worker, 0 if white
1 if a union member, 0 if not
years of schooling
potential labor market experience in years
Using this data answer the following questions:
a) First, estimate the following regression, and report results in equation form:
ln Wagei = B1 + B2Educationi + B3Experi + B4Femalei + B5 Nonwhitei
. reg lnwage EDUCATION EXPER FEMALE NONWHITE
Source
SS
df
MS
Model
Residual
147.694398
295.136681
4
1284
36.9235995
.229857228
Total
442.831079
1288
.343812949
lnwage
Coef.
EDUCATION
EXPER
FEMALE
NONWHITE
_cons
.1007782
.0136814
-.2607557
-.1162918
.9080501
Std. Err.
.0048513
.0011667
.0267527
.0373403
.0748281
t
20.77
11.73
-9.75
-3.11
12.14
Number of obs
F( 4, 1284)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
0.002
0.000
=
=
=
=
=
=
1289
160.64
0.0000
0.3335
0.3314
.47943
[95% Conf. Interval]
.0912608
.0113926
-.3132395
-.1895464
.7612514
.1102956
.0159703
-.2082719
-.0430372
1.054849
b) Provide precise interpretations for the coefficients on the variables Exper and Female.
Interpret these as semi-elasticities…
Page 4 of 6
Exam 2 – Very (Excessively) Brief Answers
FE331 Spring 2013 – Economic Statistics
A. S. Rahman
Name:_______________________________
c) A recent study suggests that minorities earn on average 10% less than whites with
comparable education and experience, sparking a great deal of outrage and debate. Can you
confirm or refute this with the results here? Explain, and show all calculations.
Test hypothesis that B4 = -0.1. Fail to reject so confirm the study (at least you cannot refute
it).
d) Now, run the same regression but add Union as an additional explanatory variable.
When you add this variable, what happens to the coefficient on Exper. More importantly,
why does it change?
. reg lnwage EDUCATION EXPER FEMALE NONWHITE UNION
Source
SS
df
MS
Model
Residual
153.064776
289.766303
5
1283
30.6129552
.225850587
Total
442.831079
1288
.343812949
lnwage
Coef.
EDUCATION
EXPER
FEMALE
NONWHITE
UNION
_cons
.0998703
.0127601
-.249154
-.1335351
.1802035
.9055037
Std. Err.
.0048125
.0011718
.026625
.0371819
.0369549
.0741749
t
20.75
10.89
-9.36
-3.59
4.88
12.21
Number of obs
F( 5, 1283)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
0.000
0.000
0.000
=
=
=
=
=
=
1289
135.55
0.0000
0.3457
0.3431
.47524
[95% Conf. Interval]
.0904291
.0104612
-.3013874
-.2064791
.107705
.7599863
.1093115
.015059
-.1969207
-.0605911
.2527021
1.051021
Experience coef goes down because it is positively correlated with Union (why does it
matter?)
e) Now run this regression:
ln Wagei = B1 + B2Educationi + B3Experi + B4Experi2 + B5Femalei + B6Nonwhitei
Interpret your estimates of B3 and B4. What do your results suggest as the general
relationship between work experience and wages?
Page 5 of 6
Exam 2 – Very (Excessively) Brief Answers
FE331 Spring 2013 – Economic Statistics
A. S. Rahman
Name:_______________________________
. reg lnwage EDUCATION EXPER expsq FEMALE NONWHITE
Source
SS
df
MS
Model
Residual
159.123701
283.707378
5
1283
31.8247402
.22112812
Total
442.831079
1288
.343812949
lnwage
Coef.
EDUCATION
EXPER
expsq
FEMALE
NONWHITE
_cons
.0958996
.0404656
-.0006432
-.253479
-.1139374
.779418
Std. Err.
.0048065
.0038973
.0000895
.0262593
.0366258
.0755429
t
19.95
10.38
-7.19
-9.65
-3.11
10.32
Number of obs
F( 5, 1283)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
=
=
=
=
=
=
1289
143.92
0.0000
0.3593
0.3568
.47024
[95% Conf. Interval]
0.000
0.000
0.000
0.000
0.002
0.000
.0864702
.0328197
-.0008187
-.3049949
-.1857905
.6312168
.105329
.0481114
-.0004677
-.2019631
-.0420843
.9276193
Experience positively correlated with wages but at diminishing rates.
f) Finally, you would like to know if female minorities earn more or less than others,
controlling for education and experience. What are your findings? (be sure to specify what
regression you are running, and what the relevant estimates are)
Include cross term female*nonwhite in regression (called femminority):
o
. reg lnwage EDUCATION EXPER FEMALE NONWHITE femminority
Source
SS
df
MS
Model
Residual
148.043031
294.788047
5
1283
29.6086063
.229764651
Total
442.831079
1288
.343812949
lnwage
Coef.
EDUCATION
EXPER
FEMALE
NONWHITE
femminority
_cons
.1007705
.0136267
-.2747165
-.1658789
.0918456
.9160059
Std. Err.
.0048504
.0011673
.0290494
.0549021
.0745616
.0750913
t
20.78
11.67
-9.46
-3.02
1.23
12.20
Number of obs
F( 5, 1283)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
0.003
0.218
0.000
=
=
=
=
=
=
1289
128.86
0.0000
0.3343
0.3317
.47934
[95% Conf. Interval]
.091255
.0113366
-.3317061
-.2735867
-.0544305
.7686907
.110286
.0159167
-.2177269
-.0581711
.2381216
1.063321
No statistically significant influence of female minority status on wages.
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