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ONOMICS 522-stat代写

时间：2022-12-09

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.

Please make sure your answer is readable and send it to Mingzhang by email to miqiao@syr.edu, no

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.

Answer the following questions:

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 womens 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

cartels 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?)

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.

Please make sure your answer is readable and send it to Mingzhang by email to miqiao@syr.edu, no

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.

Answer the following questions:

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 womens 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

cartels 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?)