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ECONOMICS 522

Sample Final Exam

Yulong Wang

Instructions:

You have 120 minutes to do the following 3 problems.

The total number of points is 100.

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.

1

Problem 1 (30 points)

Consider individual monthly spending on the non-alcoholic beverages. The data is a representative sample

of 18-60 years old working people in the US population. The following variables are available in the dataset:

Y : the amount the person spends on the beverages in dollars (permonth);

X : the persons monthly disposable income (in thousands of dollars);

M : dummy variable equal to 1 formales and 0 for females;

Age : the persons age.

For this problem we will suppose that everyone in this population spends a positive amount on beverages,

i.e., Y > 0. Consider the following regressions:

Variablers

ln (X)

(1)

0.254

(0.0513)

(2)

0.219

(0.0739)

0.308

(0.117)

-0.00119

(0.103)

(3)

0.305

(0.0966)

M

0.295

(0.117)

M ln (X)

-0.00162

(0.103)

Age

0.0381

(0.0249)

Age2

-0.000677

(0.000367)

constant

1.673

1.553

1.024

(0.058)

(0.0776)

(0.395)

observations

2330

2330

2330

R2

0.011

0.021

0.023

The dependent variable is ln(Y ). Standard errors in parentheses.

(a) (6 points) Interpret the coe¢ cients on ln(X) and M ln(X) in regression (2).

(b) (6 points) Fixing X and gender, at what age is the spending on beverages the highest according to

regression (3)?

(c) (6 points) Assuming homoskedasticity, test whether age has a statistically signi
cant e¤ect on individual demand for beverages at the 5% signi
cance level.

(d) (6 points) The coe¢ cient on ln(X) in regression (1) is higher than in regression (2). Does this make

sense? Explain briey.

(e) (6 points) Test whether age has any nonlinear e¤ects on ln(Y ) at the 5% signi
cance level.

Problem 2 (40 points)

Consider the following table with Linear Probability Model (LPM), Probit, and Logit regression outputs. It

is based on a random sample of observations on Y , X, S, and N . Here Y is a binary outcome variable, X

is continuously distributed, and S is a dummy variable for someone living in the Southern part of the US.

Finally, N = 1 S, i.e., N is a dummy variable for someone living in the Northern part of the US.

2

Variables

X

S

(1)

LPM

0.236

(0.0914)

-0.106

(0.0551)

(2)

Logit

1.434

(0.552)

-0.624

(0.301)

(3)

Probit

0.820

(0.318)

-0.354

(0.176)

N

(4)

LPM

q1

(5)

LPM

0.690

(0.283)

0.0825

(0.111)

q2

X S

(6)

LPM

q4

q5

-0.525

(0.299)

q6

X N

constant

Observations

R2

0.186

-1.515

-0.914

q3

0.334

(0.0548) (0.312) (0.184)

(0.0995)

500

500

500

500

500

0.017

0.025

Robust standard errors in parentheses

q7

500

Using the output above answer the following questions:

(a) (8 points) What is the predicted probability of Y = 1 for someone living in the South with X = 0.5

according to the Linear Probability Model (1)? Logit? Probit?

(b) (4 points) Test X = 0:5 at the 10% signi
cance level against the two-sided alternative in the Logit

regression.

(c) (4 points) What is the e¤ect of X changing from 0.5 to 0.8 for someone living in the South in the

Logit regression? In the Linear Probability Model (1)?

(d) (5 points) The coe¢ cient on X in Probit regression (3) is higher than the coe¢ cient on X in the

linear regression (1). Does this mean that regression (3) suggests higher e¤ect of X on the probability of Y

= 1 than regression (1)? Explain briey.

(e) (9 points) Find the coe¢ cients in regression (4), i.e., q1 ; :::; q3 .

(f) (10 points) Find the coe¢ cients in regression (6), i.e., q4 ; :::; q7 .

Problem 3 (30 points)

Consider a very large random sample of (Yi ; Xi ; Zi ; Qi ), where

Yi =

0

+

1 Xi

2

X

+ Ui ;

2

Z

and Zi and Qi are some variables. Suppose

= 4;

= 3; 2Q = 2; XU = 0:9; ZX = 0:7 and ZU = ,

for some number and Qi is independent of (Yi ; Xi ; Zi ) (perhaps it was generated randomly on a computer

without any connection to the data we have). Let us de
ne a new variable

Ri = Zi + Qi :

(a) (4 points) Suppose we run OLS regression of Yi on Xi and a constant. Will this result in a consistent

estimator of 1 ?

(b) (4 points) Suppose = 0. Is Zi a valid instrumental variable?

(c) (6 points) Suppose = 0. Is Ri a valid instrumental variable?

(d) (8 points) Suppose 6= 0. Let ^ Z denote the TSLS estimator of 1 that uses Zi as the instrumental

variable. Let bZ denote the probability limit of , i.e., ^ Z !p bZ as n ! 1. Using the information provided

above,
nd bZ . Hint: bZ could depend on .

(e) (8 points) Suppose 6= 0. Supposewe use Ri (instead of Zi ) as the instrumental variable. Let us

denote this estimator ^ R . As in the previous part, let bR denote the probability limit of ^ R , i.e., ^ R !p bR

as n ! 1. Find bR .

3

The economy contains two individuals, Jack and Jill. Each has a utility of U = C L2/2, where

L is the number of hours worked. Jack and Jill earn potentially different wage rates W;. The

government imposes a tax rate of t on income, so that the tax bill of individual i is tw;L?. The

government divides the revenue it collects in half, and gives it back to Jack and Jill in the form of

a lump-sum transfer.

a. Find the labor supply functions for both Jack and Jill.

b. Use this answer to find government revenue and the size of the transfer.

c. What is the elasticity of taxable income with respect to the net-of-tax rate?

d. What tax rate maximizes government revenue?

e. What tax rate would maximize a utilitarian social welfare function? (for the final two parts,

you may utilize the equations derived in the slides).

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