Download 1 - UCSB Economics

Dec. 9, 2003
ECON 240A-1
Final
L. Phillips
Answer all five questions. Each question is worth 30 points.
1. A survey of economics (econometrics) students was conducted about the monthly
rent per person that they paid along with the information about the number of persons
living together, the number of rooms in the unit, the number of blocks from the center
of campus, and the gender of the person responding, female = 1, for yes, 0 for no. The
32 observations of data is appended to the exam. Figure 1.1 shows the histogram and
descriptive statistics for rent per person. Figure 1.2 depicts a regression of rent per
person against the number of persons living together in the unit.
a. Is this cross-section or time series data? Cros-section
Figure 1.1 His togram and Statis tic s for Rent Per Pers on
10
Series: RENTPERPERSON
Sample 1 32
Obs erv ations 32
8
6
4
Mean
Median
Max imum
Minimum
Std. Dev .
Skewnes s
Kurtosis
138.1693
140.0000
285.0000
50.75000
47.11474
1.017101
4.787240
J arque-Bera
Probability
9.776269
0.007535
2
0
50
75 100 125 150 175 200 225 250 275 300
b. Is the distribution more or less symmetric? Yes, but a little skewed and
kurtotic.
c. Is it normally distributed? Explain. No. Based on the Jarque-Bera statistic, it is
skewed enough and kurtotic enough to be significantly non-normal, but only
at the 1% level. It is not wildly non-normal.
d. What is the central tendency of rent per person? Based on the mean about
$138, based in the median about $140 per month.
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L. Phillips
Fig. 1.2 Regression of Monthly Rent Per Person Against Persons
300
y = -11.066x + 165.14
R2 = 0.0959
Rent Per Person
250
200
150
100
50
0
0
1
2
3
4
5
6
7
Persons Living Together
e. Is this bivariate regression significant at the 5% level? Explain. No. Based on
an F statistic of F1, 30 = (R2 /1-R2)(n-2/1) = (0.0959/0.9041) 30 = 3.18 with a critical value
of 4.18 at the 5% level.
2. Another analyst looked at the data and noticed that rent appeared to be higher for
women than for men. There were 10 women in the sample and 22 men. The mean
rent per person for women was $164.17 and the mean rent for men was $126.35. This
analyst was a student of econometrics and calculated an indicator variable for males
as one minus the indicator variable for females and regressed the 32 observations for
rent per person against these two indicator variables. The results are displayed in
Table 2.1. Then the analyst ran another regression, dropping the male indicator
variable and adding a constant term. The results are presented in Table 2.2.
Table 2.1 Regression of Rent Per Person Against Gender Indicator Variables
Dependent Variable: RENTPERPERSON
Method: Least Squares
Sample: 1 32
Included observations: 32
Variable
Coefficient
Std. Error
t-Statistic
Prob.
FEMALE
MALE
164.1667
126.3523
14.02177
9.453475
11.70799
13.36570
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
0.142860
0.114289
44.34073
58983.00
-165.7143
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
138.1693
47.11474
10.48215
10.57375
2.522594
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L. Phillips
Table 2.2: Regression of Rent per Person Against the Female Indicaor Variable and a
Constant Term
Dependent Variable: RENTPERPERSON
Method: Least Squares
Sample: 1 32
Included observations: 32
Variable
Coefficient
Std. Error
t-Statistic
Prob.
FEMALE
C
37.81439
126.3523
16.91089
9.453475
2.236097
13.36570
0.0329
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.142860
0.114289
44.34073
58983.00
-165.7143
2.522594
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
138.1693
47.11474
10.48215
10.57375
5.000131
0.032934
a. Is there a significant difference in monthly rent for men and women? Explain.
Yes. The t-statistic of 2.27 for the coefficient on the female variable, testing
the difference in mean rents, is significant at the 5% level
b. Suppose you were not a student of econometrics, and did not know about
running regressions to answer this question. Could you get a back-of-theenvelope estimate? Hint: Assume the square root of the estimate of the pooled
variance estimator, sp2, is equal to the standard deviation estimate in Figure
1.1. What do you calculate for a t-statistic using this approximation?
t  [( x1  x2 )  (1   2 0]  s (1/ n1  1/ n2 =(37.82-0)/47.11[(1/10)+(1/22)]1/2
=2.10.
3. Someone looking at questions 1 and 2, above, might wonder about combining the two
analyses and regressing rent per person against both the number of persons living
together and gender. In the following multivariate regression, rent per person was
regressed against the number of persons living together, the number of rooms rented,
the number of blocks from the center of campus, and the indicator variable female.
The results follow in Table 3.1.
a. Is the regression in Table 3.1 significant as a whole at the 5% level? Explain.
Yes, the F-stat of 4.27 is significant at the 1% level.
b. Is the indicator variable female significant at the 5 % level? No. the t-stat of
1.51 is not significant at the 5% level.
c. How can the indicator variable female be significant in Table 2.2 but not
significant in Table 3.1? Explain. Female was the only regressor in Table 2.2
Dec. 9, 2003
ECON 240A-4
Final
L. Phillips
but is one of four regressors in Table 3.1, so in the latter case, the question is
whether gender is significant controlling for (conditional on) other factors
such as rooms, persons, and blocks,
Based on the regression results in Table 3.1, two variables, female and blocks were
dropped, although both had t-statistics that were greater than one but not very significant.
The results are reported in Table 3.2
Table 3.1: Multivariat Regression for Rent Per Person
Dependent Variable: RENTPERPERSON
Method: Least Squares
Sample: 1 32
Included observations: 32
Variable
Coefficient
Std. Error
t-Statistic
Prob.
ROOMS
PERSONS
BLOCKS
FEMALE
C
32.54048
-39.96653
-0.607752
24.87735
165.6059
13.33490
12.90240
0.578149
16.49201
17.72424
2.440249
-3.097605
-1.051202
1.508449
9.343470
0.0215
0.0045
0.3025
0.1431
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.387446
0.296697
39.51192
42152.17
-160.3389
2.300739
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
138.1693
47.11474
10.33368
10.56270
4.269429
0.008336
Table 3.2: Regression for Rent Per Person
Dependent Variable: RENTPERPERSON
Method: Least Squares
Sample: 1 32
Included observations: 32
Variable
Coefficient
Std. Error
t-Statistic
Prob.
ROOMS
PERSONS
C
35.24567
-42.34593
165.3890
13.28083
13.07510
15.63054
2.653875
-3.238670
10.58115
0.0128
0.0030
0.0000
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R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
ECON 240A-5
Final
0.272551
0.222382
41.54703
50058.52
-163.0894
2.638658
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
L. Phillips
138.1693
47.11474
10.38059
10.51800
5.432661
0.009912
d. Did the two variables, female and blocks, add significantly (at the 5% level) to
the explained variance or not? Explain your answer. No. The difference in the
sum of squared residuals between Table 3.2 and Table 3.1 is the sum of
squares explained by the two dropped variables. The F-stat with 2 and 27
degrees of freedom is F2, 27 = [(50,058.52-42152.17)/2]/(42152.17/27) = 2.53.
The critical value at the 5% level is 3.35
e. What distribution statistic did you use for your test in part d, and what was the
critical value of this statistic at the 5% level? F distribution, critical is 3.35.
4. A political science student is investigating voting results on a referendum for a local
school tax. She regresses a variable called yesvote, (one if the individual voted yes, zero
if the vote was no, against property tax paid in $, called proptax, ranging from about $400
to $1800. She also included a household income variable, ranging from about $ 4000 to
$50000, and an indicator variable coded one if the individual was employed as a teacher,
either public or private school, called schooljob. There are 95 observations. The results
are displayed in Table 4.1.
Table 4.1: Regression For Voting On a School Tax Referendum
Dependent Variable: YESVOTE
Method: Least Squares
Sample: 1 95
Included observations: 95
Variable
Coefficient
Std. Error
t-Statistic
Prob.
INCOME
PROPTAX
SCHOOLJOB
C
1.34E-05
-0.000301
0.342434
0.596705
6.34E-06
0.000183
0.154266
0.191382
2.117645
-1.649513
2.219772
3.117882
0.0369
0.1025
0.0289
0.0024
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.095950
0.066146
0.471293
20.21266
-61.28969
2.008855
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.621053
0.487699
1.374520
1.482051
3.219375
0.026396
a. Is this regression significant at the 5 % level? Yes.
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b. Does local school education appear to be a normal good? Explain. Yes.
Income has a positive effect on voting yes, i.e. the demand for local public
school education.
c. Do all three of the explanatory variables have the expected sign? Explain.
Yes, income was discussed in b, teachers would be advocates for education (a
taste variable), as well as self interest for those that are public school teachers,
and people who face paying more property tax to support the schools are less
likely to be enthusiastic, the price effect, which is negative as expected.
The plot of the actual vote, fitted and residual is shown as Figure 4.1. The political
science student is comfortable with the meaning of the dependent variable, which is one
if a person voted yes and zero if a person voted no. She wonders about the meaning of the
fitted dependent variable however, which has values like 0.83 and 0.22
d. If the political science student shows you her results and asks for your
reaction, what kind of advice can you give?
i.
How should she interpret this fitted regression model? The fitted
dependent variable is the estimated probability of voting yes, and the
fitted model is a linear probability model.
ii.
Do the results make economic sense? (see part c. above) Yes, you
should explain the political economy of the results as discussed in part
c, above.
iii.
Is there anything else she might do? Explain. Some of the estimated
probabilities are above one, so she could estimate a logit or probit
model and plot the fitted probabilities against each of the three
explanatory variables, conditional on the average for the other two, to
provide her faculty advisor with a fell for the results.
Figure 4.1: Actual Fitted and Residual from Referendum Regres s ion
1.2
1.0
0.8
0.6
0.4
0.2
1.0
0.0
0.5
0.0
-0.5
-1.0
10
20
30
40
Residual
50
60
Actual
70
80
Fitted
90
Dec. 9, 2003
ECON 240A-7
Final
L. Phillips
5. In a city in America, 50% of the citizens (eligible voters) are Democrats, 30% are
Republicans and 20% are Greens. The fraction of Republicans who voted was 0.65, the
fraction of Democrats who voted was 0.82 and the fraction of Greens who voted was
0.50. An eligible citizen is picked at random and questioned and says they did not vote.
What is the probability this citizen is a Democrat?
P(D/NV) = P(D&NV)/P(NV) = P(D&NV)/[ P(NV/D)*P(D) + P(NV/R)*P(R) +
P(NV/G)*P(G)],
Note: P(NV&D) + P(V&D) = P(D), and P(V&D) = P(V/D)*P(D) = 0.82*0.5,
So P(NV&D) = P(D) – P(V&D) = 0.5 – 0.41 = 0.09
Likewise P(NV&R) = P(R) – P(V&R) = 0.3 – 0.65*0.3 = 0.35*0.3
Likewise P(NV&G) = P(G) –P(G&V) = 0.2 – 0.5*0.2 = 0.5*0.2
P(D/NV) = 0.09/{0.18*0.5 + 0.35*0.3 + 0.5*0.2] = 0.41/[0.09 + 0.105 + 0.1]
P(D/NV) = 0.09/ 0.295 = 0.305