1. Consider the following equation relating percent share of the vote that the incumbent candidate gets in an election to campaign spending in an election cycle. Pctshare refers to percent of the vote obtained by the incumbent and incexp refers to dollars (in units of hundreds of thousands) spent on advertising by the incumbent. The equation is estimated using hundreds of U.S. Congressional elections in the last ten years.
Pctshare = 51.6 - 0.312 incexp
(16.5) (1.8)
n = 3542 R^{2} = 0.233
a.) Interpret the intercept in this equation. Interpret the slope in this equation.
b.) What is the predicted incumbent share of the vote if he or she spends one million dollars in an election cycle?
c.) What sign would you expect on incexp? Why? Discuss a reason why the coefficient might be biased. Give reasoning for the direction of the bias for what direction you decide.
d.) Is the coefficient on incexp significant at the 5% level? At the 10% level? Use a t-statistic for your answer.
2. 100 schools are given exactly one million dollars each in grant money. They can spend the money on any or all of three programs: math tutoring (math), kickball lessons (kickball), or computer literacy (computer). All the money must be spent on a combination of the three programs at each school. The grant money is intended to improve SAT scores.
a. Why would it not make sense to estimate the following model? describe.
SAT score = β_{0} + β_{1} math + β_{2}kickball + β_{3} computer
b. Indicate how the problem could be fixed so estimation can be possible.
3. A study is done analyzing the average length of stay (in days) for hospital visits. The following equation is estimated for 1,000 patients:
Length = 3.1 + .24 cigarettes - .15 obesity + .86 prescrip - .023 cholesterol
(1.5) (.17) (.82) (1.85) (.062)
n = 1,000 R^{2 }= 0.383
Where cigarettes are the number of packs of cigarettes smoked per day, obesity is the number of pounds a patient is over the obesity threshold for their age and height, prescrip is the number of prescription pills taken each day, and cholesterol is the patient’s blood cholesterol level. Despite standard errors being high and t-stats being low, an F-test reveals the last three variables of obesity; prescript and cholesterol are jointly highly significant.
The equation is re-estimated dropping the variables obesity and cholesterol:
Length = 3.1 + .44 cigarettes + .76 prescrip
(1.1) (.21) (.22)
n = 1,000 R^{2} = 0.343
find out the t-statistic on prescrip in the new and old regression. What is the most likely reason that the standard error for prescrip is much lower in the new regression?
4. The median starting salary for new law school graduates is determined by:
Log (salary) = β_{0} + β_{1}LSAT + β_{2}GPA+ β_{3}log (libvol) + β_{4}log (cost) + β_{5}rank + u
Where LSAT is the median LSAT score for the graduating class, GPA is the median college GPA for the class, libvol is the number of volumes in the law school library, cost is the annual cost of attending law school, and rank is a law school ranking (with rank = 1 being the best.)
a. describe why we expect β_{5} ≤ 0 .
b. What signs do you expect on the other slope parameters? Justify your answers.
c. For a set of law schools, the above estimated equation is:
Log(salary) =8.34 +.0047LSAT + .248GPA+ .095log(libvol) + .038 log(cost) + -.0033rank
n = 136 R^{2} = 0.842
What is the ceteris paribus difference in salary for schools with median GPA different by one point? (Interpret your answer as a percentage.)
d. Interpret the coefficient on the variable log (libvol)
e. Would you say it is better to attend a higher ranked law school? How much is a difference in ranking of 20 worth in terms of starting salary?
5. Consider the following relationship between number of pieces of candy eaten per day and pounds gained per month:
Person Candy Pounds
1 3 6
2 5 7
3 1 2
4 8 10
5 3 5
a. Estimate the relationship between candy and pounds using OLS. That is obtain the intercept and slope intercepts in the equation:
Pounds = β_{0} +β_{1}Candy
b. What is the predicted number of pounds gained per month if someone eats 2 pieces of candy per day?
c. How much of the variation in pounds gained is describeed by pieces of candy eaten? In other words: What is R^{2} for this regression?
6. Consider the following (familiar) equation which estimates the number of hours of sleep per year that someone gets as a function of hours worked per year (totwork), education (in years) and age. Suppose this equation is estimated using a new data set of 64 people and the results are
Sleep = 3434.8 - .118 totwork - 12.05 education +2.35 age
(115.71) (.025) (3.25) (1.01)
n = 64 R^{2} = 0.185
a. Test the null hypothesis that the coefficient on education is equal to 6 at the 5% level.
b. Dropping age and education from the equation yields the following estimate:
Sleep = 3288.0 - .127 totwork
(47.21) (.028)
n = 64 R^{2} = 0.141
Are education and age jointly significant at the 5% level? Justify your answer.