Midterm Exam 2: ECON 141 Spring 2018
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Midterm Exam 2: ECON 141
Spring 2018
March 21, 2018
Instructions (PLEASE READ CAREFULLY before starting):
• We will grade only what is written on your exam sheet. You will be provided extra sheets for auxiliary calculations, which you do not want/need to include in your answer.
– Each part of each question has to be answered in the sheet designated for the answer. Anything outside of the designated sheet will not be graded.
– Content in the Auxiliary Calculations part will not be graded.
– If you need extra sheets, we will provide them.
• This test has a total of 3 questions and 100 points. You have 1h 20m to solve it, that is, 80 minutes.
• Show your work, unless you are explicitly told not to! No credit will be given for correct answers if you do not justify your argument.
• Please be sure that your handwriting is legible!
• Be precise but brief. If a correct reply is hidden among wrong, or irrelevant, arguments, you will not get full credit.
• If time is running short, you should try to set up the problem without doing the final calculations.
Exam Questions
Question 1 (40 points)
All parts have equal weight
Give a brief answer, explanation, and/or mathematical derivation to the five questions below.
1.a: Suppose you want to study average earnings for male/females. One expert suggests the following regression “Regress earnings on a variable that takes value 1 for males and -1 for females.” Would this suggestion yield misleading results? True or False? Explain.
1.b: “If two OLS coefficients are not statistically significant, then the F-test will not reject the Null Hypothesis of joint significance. True or False? Explain.
1.c Consider the following population relation Yi = β0 + β1Xi + Ui .
Part 1) A linear regression yields βˆ 1 exactly equal to zero. Is the R2 = 0? True or False?
Explain. (5 points)
Part 2) A linear regression yields R2 = 0. Is βˆ 1 is exactly equal to zero? True or False?
Explain. (5 points)
1.d: “Consider the relation: Yi = β0 + β1X1,i + β2X2,i + Ui . Suppose X1,i = X2 2 ,i. Do you have a multicolinearity problem?” True or False? Explain.
Question 2 (30 points)
Consider the following model
Yi = β0 + β1Di + β2Gi + β3GiDi + Ui, (1)
where Yi is the GDP growth of country i; Gi is the government expenditure of country i; and Di takes value 1 if country i is in a recession and 0 otherwise. Also Ui is independent of Di and Gi .
2.a: (12 points) What is the interpretation of the coefficients β0, β1, β2, β3?
2.b: (3 points) Suppose βˆ 1 = 0.10 and βˆ 3 = 0.010 and the corresponding standard errors are SE(βˆ 1) = 0.001 and SE(βˆ 3) = 0.010. What can you conclude regarding the significance of β1 and β3? Can you conclude anything about the role of recession?
2.c: (10 points) Consider the numbers in 2.b and in addition suppose that the t-statistics corresponding to β1 and β3 are independent of each other.
i. Construct a test statistic that allows you to test for H0 : β1 = β3 = 0.
ii. Would you reject or fail to reject the null hypothesis at 95% confidence? Hint: For a Normal random variable, Z, P(|Z| ≤ 2.24) ≈ 0.975 and for a F2,∞ random variable with 2 degrees of freedom, X, P(X ≥ 3) ≈ 0.95.
iii. Describe the intuition behind the result in (ii).
2.d: (5 points) Suppose that instead of running an OLS regression given by 1, you (incorrectly) run
Yi = α0 + α1Di + α2Gi + Vi .
i. Will ˆα2 be a consistent estimator of the causal effect of government expenditure on GDP growth?
Show your answer.
Question 3 (30 points)
All parts have equal weight
Consider the following model
Y = β0 + β1X1 + U
and
U = X2Z + αX2, X2 ≥ 0
where Z ∼ N(0, 1) and Cov(X1, X2) = 0. Suppose one observe an I.I.D. sample (Yi , X1,i, X2,i)ni=1.
1. Consider regressing Y on X1. Is the OLS estimator of β1 consistent? Please provide a formal answer and be explicit about how your answer depends on α.
For the next two questions assume that α = 0
2. Is the OLS estimator BLUE (Best linear Unbiased)? Please explain your answer.
3. If your answer in item 2 is no. By manipulating the variables Y , X1 and X2 construct one estimator that is BLUE.
2023-11-23