Midterm Exam 1: ECON 141 Fall 2021
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Midterm Exam 1: ECON 141
Fall 2021
October 14, 2021
Question 1 (30 points)
Give a brief mathematical derivation to the 5 questions below. All questions have equal weights.
(a) Suppose (X1, ..., Xn) are IID distributed according to N (a/pn, 1) some a 0. Suppose you want to test the null hypothesis of E[X] = 0. (i) What is the distribution of the t-statistic for this null? (ii) Write down the probability of rejecting the null (at a 95% level) as a function of a, in particular, what happens to the probability of rejecting when a ! 1.
(b) Suppose you run the following regression model Y = β1X + U where U is independent of X . Then 0 R2 1. True or False? Explain.
(c) Suppose that X and Y are such that FX(-)1 (τ ) ≥ F1 (τ ) for all τ 2 [0, 1] (i.e., the quantiles of X are
bigger than the quantiles of Y). Then E[X] ≥ E[Y]. True or False? Explain.
(d) Suppose you have a regression Y = β0 + β1X1 + U with U = X1(2)E and E N (0, 1). (i) Is OLS consistent?
(ii) Is OLS BLUE? If not, can you come up with an estimator that it is?
(e) Given an I.I.D. sample X1, ..., Xn, construct an estimator of the population mean that is unbiased but not consistent.
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 = -0.15 and β(ˆ)1 = 0.25. 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 0.23, 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 = 0.23 at 95% - level? What about at a 99%-level? (for the 99% level, use z0.005 2.58). Given these 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 + α2Hi + ei ,
where Hi is the number of parks around gas station i and α2 < 0. Also suppose Hi 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 Traffic and H 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 “incorrect” regression (i.e., equation 1), under the assumption that Traffic and H are negatively correlated?
(iii) Suppose that Cov(Traffic, H) = -0.5 and σTraffic = 1. What would the probability limit
ofβ(ˆ)1 be? Would you be over-estimating, correctly-estimating or under-estimating the efect of tra伍c on sales?
Question 3 (30 points)
Let Yi be the log unemployment rate in city i; let MWi be a “minimum wage indicator” i.e., 1 if city i increased the minimum wage and 0 otherwise. An economist is interesting in studying the efect of an increase in the minimum wage on the unemployment rate. The true model of the world is given by
Yi = β0 + β1MWi - Edi + Ui
where Edi be “high education indicator”, i.e., 1 is city i has a high proportion of highly educated people and 0 otherwise. Moreover,
MWi = 1fπ0 + π1Edi ≥ ig
where i U (0, 1) IID and π1 < 0. Also
Ui = -(Edi - µEd ) + ζi
where ζi N (0, 1) IID and µEd = E[Ed].
An economist decides to run OLS on the following regression:
Yi = γ0 + γ1MWi + Vi
(a) Derive the probability limit of the OLS estimator of the slope (γ1 ).
(b) Based on your answer in (a). Is the OLS estimator of the slope a consistent estimator of β1 ?
Suppose the true efect is zero, i.e., β1 = 0. Will the economist be concluding that the minimum wage increases or decreases unemployment (on average)?
An economics, Mr. Paredes, proposes an alternative estimation technique based on comparing cities that increased the minimum wage to those that did not, but controlling for the education level. That is, E[YjMW = 1, Ed = e] - E[YjMW = 0, Ed = e] where e 2 f0, 1g.
(c) Show that E[YjMW = 1, Ed = e] - E[YjMW = 0, Ed = e] = β1 for any value of e.
Yet another economist, Ms. Romero, proposes to run a multivariate regression model including MW and Ed as regresssors. I.e.,
Y = α0 + α1MWi + α2Edi + Ui
(d) (i) Will the OLS estimator for the slope of Ed be consistent? Explain your answer.
(ii) Will the OLS estimator for the slope of MW be consistent? Explain your answer.
(e) Suppose the true efect of MW on unemployment depends on education, i.e., β1 depends on Ed. Which estimation technique you think will yield better results (i.e., learn the true efects): The one proposed by Mr. Paredes or by Ms. Romero? Please explain. Hint: An heuristic explanation will suffice.
2023-10-18