Questions
1) (12 points) Suppose you are interested in estimating whether Canadians give more to charity than
Americans. You have collected a random sample from 1000 people on how much they give
(measured in Canadian dollars), their nationality (Canadian or American), their gender (male or
female), and their age (in years).
a. (4 points) Write down a linear regression model that would estimate the difference
between Canadians and Americans in terms of how much they give. Describe precisely the
variables you will use and the parameters of your model.
b. (4 points) Explain how you would estimate your model, and write down the OLS estimators
for the slope and intercept.
c. (4 points) Describe how you would test the null hypothesis that Canadians and Americans
give the same amount, versus the alternative that they give different amounts. Be clear
about your null hypotheses, test statistic, and decision rule.
2) (12 points) For each statement below, evaluate whether it is true, false, or uncertain, and fully
explain your answer.
a. (4 points) In the simple linear regression model, if R2 = 0 in an OLS estimation, the
parameter �! equals zero.
b. (4 points) Suppose you regress y on x and find that the slope equals 1. If x and y have the
same sample variance, and you then regress x on y, the slope will also equal 1.
c. (4 points) In the linear regression model � = �" + �!� + �, the assumption E[u|x] = 0
implies that there are no unobserved factors that affect y
3) (12 points) Suppose you estimate the linear relationship between y and x, and the resulting sample
regression function is �' = 5 + 2�. The standard error of the slope is 0.2.
a. (4 points) Assume all the standard assumptions of the linear regression model are true and
as such the OLS estimator of the slope is unbiased. True or False: the sampling distribution
of the OLS slope is centered at 2, with standard deviation 0.2.
b. (4 points) Consider scaling the x variable by multiplying all observations by 2. What are the
new intercept and slope estimates?
c. (4 points) Consider scaling the y variable (from the original sample regression function �' =
5 + 2�) by multiplying all observations by 2. What are the new intercept and slope
estimates?
Sample Solution