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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.