ONOMICS 522-stat代写

ECONOMICS 522
Final Exam
Yulong Wang
Fall 2022
Instructions:
You have 24 hours to do the following 5 problems.
The total number of points is 165.
The problems have not been ordered in terms of di¢ culty. There is huge variability in the di¢ culty of
questions.
Note that most of the questions within each problem can be answered independently of each other.
To obtain full credit, write out numerical answers in decimal form. In the calculations please keep at
least three signicant gures.
The test is open-book.
later than noon Friday 9th. In the title of the email, please write "ECN522-Final-Your Name."
1
Problem 1 (40 points)
A researcher wants to study the e¤ect of income on spending. She uses a dataset that contains information
on spending by those who won a lottery last year. The dataset contains the following variables for each
individual:
S : (annual) spending (in thousands of dollars);
Lott : lottery winnings (in thousands of dollars);
Inc : (annual) income (in thousands of dollars, does not include Lott);
TotInc : = Lott+ Inc.
The researcher estimates the following regressions (assuming homoskedasticity):
Dependent Variable:
S S S S log (S) log (S)
Regressors (1) (2) (3) (4) (5) (6)
TotInc 0.728*** 0.698*** 0.693*** 0.0192***
(0.0580) (0.0618) (0.209) (0.00260)
Lott 0.202 x1
(0.146)
Inc x2
TotInc2 0.00226
(0.0129)
log(TotInc) 1.200***
(0.164)
constant 1.282 -2.096 x3 2.424 2.323*** -1.312*
(4.232) (4.874) (7.777) (0.190) (0.678)
Observations 120 120 120 120 120 120
R2 0.572 0.578 0.572 0.315 0.311
The dependent variable in regressions (1)-(4) is S, and it is log(S) in regressions (5) and (6).
Standard errors in parentheses; *** p<0.01, ** p<0.05, * p<0.1.
1. (6 points) Interpret the coe¢ cients on TotInc in regressions (1) and (5), and the coe¢ cient on
log(TotInc) in regression (6).
2. (4 points) Interpret the coe¢ cient on Lott in regression (2). (Hint: Remember the denition of TotInc.)
3. (4 points) Describe how you test the null hypothesis "neither lottery winnings nor other income has
an e¤ect on spending" using regression (2).
4. (6 points) Consider an individual with TotInc = 50. Compute the e¤ect (in thousands of dollars) of
increasing TotInc by 10% (other things being the same) according to the estimates of regressions (4)
and (6).
5. (10 points) Find x1, x2, x3 for regression (3). (Hint: Use regression (2) and the relation TotInc =
Lott+ Inc:)
6. (10 points) What do you think is the advantage of using the data on individuals who won a lottery to
estimate the e¤ect of income on spending? (Answer in 150 words or less, but please be precise).
Problem 2 (20 points)
The table below presents the results of estimating a Linear Regression (LR) model, a Logit model, and a
Probit model for a binary dependent variable, y, using the explanatory variables x;m;m x, and a constant.
Here m is a dummy (binary) variable, and x is continuously distributed.
2
(1) (2) (3)
Regressors LPM Logit Probit
x 0.468 3.772 2.192
(0.0369) (0.612) (0.325)
m 0.292 2.467 1.436
(0.0393) (0.344) (0.191)
m x -0.0172 0.487 0.258
0.0465 (0.817) (0.434)
constant 0.412 -0.645 -0.375
(0.0322) (0.240) (0.137)
observations 380 380 380
The standard errors are provided in the parentheses.
1. (5 points) Consider an individual with x = 0:5 and m = 0. What is the predicted probability of y = 1
according to the estimated Logit model? According to the Linear Regression model?
2. (10 points) For this question use the estimated Probit model. Suppose Ann has m = 0 and Bob has
m = 1. For both of them, calculate the e¤ect of changing x from 0 to 0:5.
3. (5 points) Can you test the null hypothesis "x has no e¤ect on y" in LPM using the information given
in the table? If yes, perform the test; if not, clearly explain why.
Problem 3 (40 points)
Answer the following questions. The questions are not related to each other and can be answered indepen-
dently of each other. The questions do not require more than 100 words for a complete answer but please
be sure to be precise.
1. (10 points) Alice has a random sample of students test scores (or GPA) and their rst job salary.
Suppose Alice chooses to study the e¤ect of test scores on job salary by regressing the job salary on
the test score. Would her regression analysis su¤er from any bias? Explain.
2. (10 points) You have a dataset with variables Y;X;Z in it. You want to run the instrumental variables
regression of Y on X using Z as an instrument. However, you are concerned that the instruments may
be weak. How would you check if the instruments are weak?
3. (10 points) Consider the problem of estimation of the e¤ect of number of children on womens labor
supply. As we did in class, restrict attention to the subpopulation of women who had at least two
children, and consider variables X ="the woman had more than two children" and Z ="the sex of
her rst two children was the same". Explain why Z is a good (valid) instrument for the endogenous
variable X.
4. (10 points) Suppose that
yi = x

i + "i
where xi and "i are independent and E ["i] = 0 (also assume that (x

i ; "i) is independent of

xj ; "j

for
all i 6= j). Unfortunately, you do not observe xi . Instead, you observe
xi = x

i + vi
where E [vi] = 0 and the v0is are independent of each other and of everything else. Suppose you regress yi
on xi (that is, you run OLS without a constant). Is the OLS estimator consistent in this case? Show your
work.
3
Problem 4 (15 points)
Suppose that Yt follows the stationary AR(2) model,
Yt = 2 + 0:75Yt1 0:125Yt2 + ut
where ut has E[ut] = 0 and V [ut] = 9 and is independent of (Yt1; Yt2; Yt3; :::) (and of (ut1; ut2; ut3; :::)).
Note that the coe¢ cients -2, 0.75, -0.125 and 9 above are assumed to be known rather than estimated.
1. (5 points) Suppose that you observe Yt = 1 and Yt1 = 1, what are your forecasts of Yt+1 and Yt+2 ?
2. (5 points) Assume that this process is stationary. Compute the mean of Yt. (Hint: consider the mean
on both side of the equation and use stationarity.)
3. (5 points) What is the root mean squared forecast errors of the forecasts in (a)? (Hint: it should use
the value of V [ut+1] and V [ut+2])
Problem 5 (50 points)
During the 1880s, a cartel known as the Joint Executive Committee (JEC) controlled the rail transport of
grain from the Midwest to eastern cities in the United States. The cartel preceded the Sherman Antitrust Act
of 1890, and it legally operated to increase the price of grain above what would have been the competitive
price. From time to time, cheating by members of the cartel brought about a temporary collapse of the
collusive pricesetting agreement. In this exercise, you will use variations in supply associated with the
cartels collapses to estimate the elasticity of demand for rail transport of grain. The data le JEC contains
weekly observations on the rail shipping price and other factors from 1880 to 1886.1 A detailed description
of the data is contained in JEC_Description.
Suppose that the demand curve for rail transport of grain is specied as
ln (Qi) = 0 + 1 ln(Pi) + 2Icei +
12X
j=1
2+jSeasj;i + ui,
where Qi is the total tonnage of grain shipped in week i, Pi is the price of shipping a ton of grain by rail,
Icei is a binary variable that is equal to 1 if the Great Lakes are not navigable because of ice, and Seaj is a
binary variable that captures seasonal (monthly) variation in demand (in total 12 of them). Ice is included
because grain could also be transported by ship when the Great Lakes were navigable.
Answer the following questions by running appropriate regressions and explaining the results. Be sure
to attach your STATA output table in questions 1, 4, 5, which contains the code already. (If
you use some other software, please attach the code.)
1. (10 points) Estimate the demand equation by OLS. What is the estimated value of the demand
elasticity and its standard error?
2. (10 points) Is the OLS estimator of the elasticity biased? If yes, why?
3. (10 points) Consider using the variable cartel as instrumental variable for ln(P ). Use economic
reasoning to argue whether cartel plausibly satises the two conditions for a valid instrument.
4. (10 points) Estimate the rst-stage regression. Is cartel a weak instrument?
5. (10 points) Estimate the demand equation by instrumental variable regression. What is the estimated
demand elasticity and its standard error?
6. (Bonus question, 10 points) Does the evidence suggest that the cartel was charging the prot-
maximizing monopoly price? Explain. (Hint: What should a monopolist do if the price elasticity is less
than 1?)