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Economics 375: Introduction to Econometrics
Winter, 2010 Midterm
Student ID #: ___________________________
Please do not place your name anywhere on this exam. I will give partial credit to students who show
work. This is a closed book, closed neighbors, and closed note exam. In some cases, the correct answer
to a question will require a hypothesis test (even if I don’t specifically ask for one). When performing a
hypothesis test, please list your null and alternative hypothesis, your test statistic, the critical value
of the distribution, and your conclusions. Unless otherwise mentioned, all hypothesis tests should
be performed at the 95% level of confidence. Partial credit will be given only if work is shown. The
number of points each problem is worth appears in parenthesis.
1.
In a famous study of economic growth, Robert Barro hypothesized that countries which begin
wealthy, will tend to grow slower over time than countries which begin poor. I have obtained Barro’s
data and reproduced some of it here:
Observation #
Growth Rate
Starting GDP
Per capita
1
5
2
2
3
7
3
4
2
4
7
1
5
2
8
6
3
1
where Starting GDP per capita is the GDP per capita in 1960 measured in tens of thousands of dollars, the
growth rate is the annual percentage growth in GDP between 1961 and 1995. The observations are of six
different nations. From this data, it is true that ∑ 𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒 = 24, ∑ 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝐺𝐷𝑃 = 21,
∑ 𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒 2 = 112, ∑ 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝐺𝐷𝑃2 = 123, ∑ 𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒 × 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝐺𝐷𝑃 = 65,
2
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
∑(𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒𝑖 − 𝐺𝑟𝑜𝑤𝑡ℎ
𝑅𝑎𝑡𝑒) = 16, ∑(𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝐺𝐷𝑃𝑖 − ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝐺𝐷𝑃 )2 = 49.5.
Barro estimated the relationship: 𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒𝑖 = 𝛽0 + 𝛽1 𝑆𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝐺𝐷𝑃𝑖 + 𝜖𝑖
a. Using the above data, estimate β0 and β1. Interpret your estimates. (8)
b. Under Barro’s theory, countries which had a high GDP in 1960 are likely to grow slower than
countries with a low GDP in 1960. Perform a hypothesis test of this conjecture. To make your work
easier (and to help me follow your calculations), I have reproduced the data from page 1 of this exam
along with some blank columns. (15)
Observation #
1
2
3
4
5
6
Growth
Rate
5
3
4
7
2
3
Starting GDP
Per capita
2
7
2
1
8
1
c. The U.S. starting GDP in 1960 was 3.8. Construct a 95% confidence interval of the subsequent annual
US growth rate. (6)
d. What is the R2 from your model? Interpret this number. (6)
2.
In a recent Economic Inquiry article, Kanazawa and Funk explore a potential source of racial
discrimination in professional basketball. Apparently, holding all observable variables constant, white
NBA players are paid a higher salary than their non-white counterparts. One theoretical source for these
wage differences is that fans are more attracted to white basketball players and would therefore be willing
to pay higher ticket prices and create a larger base to which advertisers can target. Each of these would
make the marginal revenue product of a white player higher than a non-white player. To determine if
white players attract a greater fan base, Kanazawa and Funk observed the Nielsen ratings of 210 noncable basketball games in each NBA city. The population regression estimated was:
Nielseni = B0 + B1Win%Loci + B2Win%Visi + B3Primetimei + B4Householdsi + B5Teamsi
+ B6WhiteFansi + B7AllStarsi + B8WhMinLoci + B9WhMinVisi + ui
Where:
Nielsen is the Nielsen rating for the observed basketball game
Win%Loc is the winning percentage of the local (home) team
Win%Vis is the winning percentage of the local team’s opponent
Primetime is a variable that takes a value of 1 if the game started between 6:30 pm and 9:00 pm and a zero otherwise
Households measures the total number of households in the local viewing area (measured in millions of households)
Teams measure the number of other professional and major collegiate teams within the local viewing area
WhiteFans measure the percentage of the population that is white in the local viewing area
AllStars measure the number of NBA All Stars playing in the televised game
WhMinLoc measures the percentage of total minutes played by white players for the local team in the observed game
WhMinVis measures the percentage of total minutes played by white players for the visiting team in the observed game
The results of Kanazawa and Funk’s regression, with standard errors in parenthesis are:
Variables
Model A
Model B
Constant
Win%Loc
Win%Vis
Primetime
Households
Teams
WhiteFans
AllStars
WhMinLoc
WhMinVis
ESS
TSS
N
-13.98
(2.41)
10.6
(1.17)
3.45
(.966)
-.39
(.448)
1.32
(.209)
-.68
(.1161)
9.55
(2.417)
1.81
(.320)
11.79
(2.00)
1.51
(.76)
-13.12
(2.38)
10.2
(1.21)
3.42
(.978)
-.41
(.461)
1.33
(.208)
-.70
(.123)
11.12
(2.491)
1.71
(.341)
7,765
12,345
210
7,317
12,345
210
a. In Model A, the coefficient on the variable Teams has a negative sign. Why might this occur? (3)
b. Using Model A, does an increase use of white players by the visiting team increase local Nielsen
ratings? (12)
c. Using model A, what is this regression’s R2? At the 95% level, did Kanazawa and Funk explain a
statistically significant amount of the variation in Nielsen ratings? (15)
d. General studies of Nielsen ratings indicate that a 1 point increase in Nielsen ratings of a TV show
increase the price of a commercial during that show by $30. Using Model A, how much increased
commercial revenue can a television station expect if the local NBA team increases white playing time by
10%? What is a 95% confidence interval for this estimate? (12)
e. Do the minutes played by white players on the home and visiting team jointly help predict Nielsen
ratings? (12)
3.
I have often considered writing and grading exams in the following “simplified” way: 1) write an
exam with 10 questions of similar difficulty and give it to my class and, after it is turned in, randomly
choose one question of the 10 to grade and assign a percentage score to the entire test based upon the
percent of this one question answered correctly. Relative to the more standard technique of grading all 10
questions and constructing a score based upon all 10, what will happen to the class average and the class
variance of test scores if I were to follow my “simplified” grading process? (6)
4.
In some instances it is likely the case that the true population regression function goes through the
origin (it has a zero constant). Consider the population regression function
Yi = β1Xi + εi. Assuming that the “best fit line” is one that minimizes the sum of n squared residuals,
^
produce an equation for B1 , the OLS estimate of β1. (15)
5.
Consider IQ which is commonly believed to be of mean 100 and variance 100. (3 ea.)
a. What is the probability of observing an individual with an IQ of 118 or greater?
b. What is the probability of observing two random people each with an IQ of 118 or greater?
c. What is the probability of observing two random people who average an IQ of 118 or greater?
d. Is the probability of observing a married couple with an IQ of 118 or greater lower or higher than that
found in part c?
6.
What are the 5 classical assumption in the univariate regression model? If these classical
assumptions are true, what happens? (10)