Midterm Exam 1: ECON 141 Fall 2019
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Midterm Exam 1: ECON 141
Fall 2019
October 13, 2020
Question 1 (30 points)
Give a brief mathematical derivation to the 4 questions below. (HINT: If you extend too much on the algebra, your are probably going in the wrong direction)
(a) You take a random sample from some population and form a 96% conidence interval for the popu- lation mean, μ. Which quantity is guaranteed to be in the interval you form? (6 points)
(i) 0
(ii) μ
(iii)
(iv) and μ .
(b) “The Law of Large Numbers gives conditions under which the frequency of times that random variable equals a value x is approximately equal to the probability of x.” True or False? Explain. (6 points)
(c) Construct an example of two random variables Y and X that are uncorrelated but not independent. (6 points)
(d) Given an I.I.D. sample X1, ..., Xn, construct an estimator of the population mean that is unbiased but not consistent. (6 points)
Question 2 (40 points)
Suppose you are consulting for Exxon Mobil that want to invest on new gas station. You ran the following regression:
Sales i =β(ˆ)0 + β(ˆ)1Traffic i +U(ˆ)i (1)
where i = 1, ..., 100 are a set of gas stations picked at random within the US; Sales i are the gas sales (in 1000 of Gallons) in station i; and Traffic i is the tra伍c volume (in 1000 of cars) faced by station i.
Let β(ˆ)0 = 0.35 and β(ˆ)1 = 0.21. The standard errors are 1 and 0.01 respectively.
(a) What is the economic interpretation ofβ(ˆ)1 ? (2.5 points)
(b) Does β(ˆ)0 have a realistic economic interpretation? (2.5 points)
(c) Can you reject the hypothesis that the intercept is equal to zero at 95% level? What is the economic interpretation of this test (and of the result)? (5 points)
(d) It turns out that if the increase of (the population) average sales when increasing tra伍c by 1000 cars is larger than 0.19, then it is proitable to run a nationwide marketing campaign to increase tra伍c volume. Given the value of the OLS estimators, the CEO of Exxon Mobil is ready to launch the campaign. Do you agree? What would you tell the CEO? (4 points)
(e) Can you reject the hypothesis that β1 = 0.19 at 95% - level? What about at a 99%-level? (for the 99% level, use z0:005 2.58). Given this results, what would you tell the CEO? (8 points)
(f) For the previous point does it make more sense to perform a one-sided hypothesis test or a two-sided one (there is no need to compute the one-sided test, just discuss it merits and/or drawbacks vis-a-vis the two-sided one)? (8 points).
(g) Suppose you ind out that you have an omitted variable problem, i.e., the true regression is
Sales i = α0 + α1Traffic i + PCi + ei ,
where PCi is the zip code of gas station i. Also suppose PCi and ei to be independent. (10 points)
(i) How will this new information afect the properties of your estimator of the slope using the “incorrect” regression (i.e., equation 1), under the assumption that PC and Traffic are inde- pendent? Hint: An heuristic discussion will su伍ce. There is no need to do the algebra.
(ii) How will this new information afect the properties of your estimator of the slope using the “in- correct” regression (i.e., equation 1), under the assumption that PC and Traffic are positively correlated? Hint: An heuristic discussion will su伍ce. There is no need to do the algebra.
Question 3 (30 points)
All the parts within this question have equal weight.
Consider the following regression model
Yi = β0 + β1Xi + Ui , Ai = 1, ..., n
with Xi 2 f0, 1g and Ui independent of Xi and E[Ui] = 0.
(a) Consider the following estimators for β0 and β1 : β(˜)0 = and β(˜)1 = -β(˜)0 .
Are they consistent estimators of β0 and β1 resp? Explain your answer.
(b) Compute the OLS estimators and denote them as β(ˆ)0 and β(ˆ)1 . Is it true that β(ˆ)0 = β(˜)0 and β(ˆ)1 = β(˜)1 ?
Explain your answer.
(c) Consider the following variation: Yi = β1Xi+Ui , Xi 2 f0, 1gand Ui = (1 -Xi)+ei with e i N (0, 1). (i) Are Ui and Xi independent? (ii) Is the OLS estimator of β1 consistent? Explain your answer.
2023-10-18