Midterm Exam 1: ECON 141 Fall 2022
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Midterm Exam 1: ECON 141
Fall 2022
October 3, 2022
Question 1 (30 points)
Give a brief mathematical derivation to the 5 questions below. All questions have equal weights.
(a) Suppose you run the following regression model Y(ˆ) = β(ˆ)1X +U(ˆ) . Then 0 R2 1. True or False?
Explain.
(b) Construct an example of two random variables Y and X that are uncorrelated but not independent.
(c) Suppose you have a regression Y = β1X1 + U with U = eX1 E and E N (0, 1). (i) Is OLS consistent?
Why?
(ii) Is OLS BLUE? If not, can you come up with an estimator that it is?
(d) Suppose you have a regression Y = β1X1 + U with U = e2X1 E and E N (0, 1). You run a regression
imposing Homoskedasticity. (i) Will the OLS estimator still be consistent? Why? (ii) Will the standard errors be correct? Why?
(e) Suppose you have a regression Y = β0 + β1X1 + U with U = e2X1 + E and E N (0, 1). (i) Is OLS consistent? Why?
Question 2 (40 points)
Suppose you are consulting for Macy’s online shopping division and they want to learn the efect of a discount (a decrease in the unit price) on money spent in the online site. The data science team ran the following regression:
MSi =β(ˆ)0 + β(ˆ)1Discount i +U(ˆ)i (1)
where i 2 f1, ..., 100g are costumers; MSi is the amout of money spent in the online site (measured in US dollars) by customer i; Discount i takes value 1 if the customer i received the discount and 0 otherwise.
Let β(ˆ)0 = 10.15 and β(ˆ)1 = 20.25. The standard errors are 1 and 5 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 money spent from the discount is larger than 15, then it is proitable to scale the discount and implement it worldwide. Given the value of the OLS estimators, the CEO of Macy’s 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 = 15 at 95% - level? Given these results, what would you tell the CEO? (8 points)
(f) From 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
MSi = α0 + α1Discount i + α2Income i + ei ,
where Income i is the income of customer i and α2 > 0. Also suppose Income i and ei to be indepen- dent. (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 Discount was randomly assigned 2 ? Hint: An heuristic discussion will su伍ce. There is no need to do the algebra.
(ii) As it turns out, Discount was not randomly assigned, in fact Cov(Discount, Income) = 0.95,
σIncome = 1 and σDiscount = 0.25. What would the probability limit of β(ˆ)1 be? Would you be
over-estimating, correctly-estimating or under-estimating the efect of discount on money spent?
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, 0 < π0 < 1 and 0 π 1 + π0 < 1. 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 economist, 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) 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 su伍ce; you don’t have to compute the multivariate regression.
2023-10-18