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Econometric Methods (AF6208)

Dr. Steven X.. Wei

Problem Set I [Due at 11:00 pm on 15 March 2022]

Note: The problem set is designed for three purposes: (1) your understanding of the lecture notes; (2) reading research papers and

working on your research projects/dissertation; and (3) preparing for your final exam (which will include some similar types of

questions to those in the two problem sets). It is strongly suggested to work on the two problem sets independently first, and then meet

with your team members to discuss your solutions. Each group submits ONE write-up, while please try to type your solutions as much

as possible.

Questions from your textbook:

Chapter 2:

Q3 (Please use Excel in your computer to do the estimation of part (i) of the

to understand the differences between each pair of the terms below:

question. In addition, please use the question

? and GPA

(a) GPA

(b) u? and u, where GPA = ?0 + ?1 ACT + u. It is noted that u is unobservable!

(c) ?? 1 (the estimate of ?1) and ?1, (similarly, ?? 0 and ?0 )

(d) s2 (=

1 n ?2

? u i ) and ?2 (=Var(u|ACT)) where we have n = 8.

n ? 2 i ?1

In addition, please use the results to numerically demonstrate that

8

? ACT * u?

i ?1

i

i

? 0.

Explain why it should be zero theoretically, and why it is not exactly zero when you calculate it.

Q6 [Pay attention to how to interpret the coefficient of the independent variable!], Q7

Econometric Methods (AF6208)

Dr. Steven X.. Wei

Chapter 3: Q2.

Additional Questions

Q1. Understand Descriptive Statistics:

Taking Table 1 of the article titled Accounting earnings and gross domestic product as an example, you are asked to construct a

similar table of descriptive statistics to the Table 1 (including skewness and excess kurtosis) for the following variables:

CRSP value-weighted, equal-weighted stock market index return [in the markets of NYSE/AMS/NASDAQ/ARCA];

US 30 day T-bill rate; US inflation rate.

Please retrieve the above return data (rather than the level data) for each variable at monthly frequency for the sample period: Jan.,

1926 Dec., 2020 from the Warton database (see the Introduction to the WRDS )?Do not report your data.

To help you get the right data, I put some information below for your data confirmation before you work on the retrieved data:

ValueWeighted

Calendar

Return-incl.

Date

dividends

30/1/1926

0.000561036

27/2/1926

-0.03304553

31/3/1926

-0.06400222

30/10/2020

30/11/2020

31/12/2020

-0.02017795

0.1237065

0.04504809

EqualWeighted

30 Day TReturn-incl.

Bill

dividends

Returns

0.02317378

0.002951

-0.05351035

0.002768

-0.09682397

0.002778

0.000583933

0.1744121

0.07285263

0.000073

0.000066

0.00006

Rate of

Change in

Consumer

Price Index

0

0

-0.005587

0.000415

-0.000611

0.000941

Econometric Methods (AF6208)

Dr. Steven X.. Wei

Hint: You could use Excel to work out all the details while any other statistical or econometric software is OK too.

(1) It is suggested to get your output in a format as in the following table, while please briefly talking about your descriptive

statistics. You may use % for mean, median, standard deviation, range, minimum and maximum.

Whole Sample

Value-Weighted

Index Return

Equal-Weighted

Index Return

T-bill Rate

Inflation

Rate

Mean

Median

Standard Deviation

Kurtosis

Skewness

Range

Minimum

Maximum

Count

(2) Do the following economic analysis. After a high inflation period of 1970s, or since 1982, the Fed has used 3% inflation rate

as one of its monetary policy targets. It generated a steady growth for the US economy. Here, you are asked to roughly work

out the evidence of a better stock market performance over the period 1982 2000 than that in 1970s. For your analysis, you

can use the descriptive statistics for the different sample periods to reach your conclusion [No need to conduct a serious

hypothesis testing at this stage].

Q2. [Understanding the simple linear regression model] Write out clearly the expression of the OLS estimates of ?0 and ?1 for the

simple linear regression model:

y ? ?0 ? ?1×1 ??.

Econometric Methods (AF6208)

Dr. Steven X.. Wei

Answer the following questions:

(1) If the model missed another independent variable x2, i.e., the true model is

y ? ? 0 ? ?1 x1 ? ? 2 x2 ? ?

,

then demonstrate mathematically the OLS estimator of ?1 is usually biased under the first five classical assumptions in lecture note

2.

(2) Discuss in which special case(s), the OLS estimator could be unbiased in (1).

(3) Through the analysis, you should understand that in your thesis, missing a control variable is in fact a serious problem in principle.

Investigate how the missing variable problem is dealt with or mitigated in research (or more clearly in published research papers).

Q3. Consider a special case of simple linear regression model,

yi ? ? ? ui ,

i ? 1, 2, …, n

where ui ? N(0, ?2).

Answer the following two questions:

(1) Using the FOC to work out the OLS estimate of ?, and then construct the OLS estimate of ?2? [Pay special attention to the

divisor of the estimate to make sure that your estimate of ?2 is unbiased].

? , the OLS estimator of ?, is unbiased, and then calculate its variance and estimated standard

(2) Demonstrate mathematically that ?

? ) and Se( ?? ).

deviation, i.e., Var( ?

(3)What are the exact values of R2 and adjusted R2 in (1)? Explain your answer intuitively.

(4) Demonstrate the following statistic follows a (Student) t-distribution,

?? ? ?

se (?? )

Econometric Methods (AF6208)

Dr. Steven X.. Wei

?,

where the first term of the numerator is the OLS estimator of ? and the denominator is the estimated standard deviation of ?

and ? is the true value. [You can check any reference in your textbook, slides, internet, and other reference books. Comments:

Through the exercise, you should understand the unconditional analysis is actually a special case of conditional analysis! This

exercise sets up a link between a conditional analysis and an unconditional analysis! It is also a good exercise for your

understanding the intuition and mathematics behind a simple linear regression model. To some extent, you understand that there

is a link between the results of your descriptive statistics and those in your regression models when you read reference papers or

write your own thesis.].

Q4. Suppose that we have two estimators,

??

?

and ?

of parameters ? and ?, respectively. Both of the estimators are unbiased.

Answer the following questions:

? + ?? be an unbiased estimator of 2? + ??

(1) Would 2 ?

? – 3 ?? be an unbiased estimator of 5? – 3??

(2) Would 5 ?

? * ?? be an unbiased and/or consistent estimator of ?* ??

(3) Would ?

? / ?? be an unbiased and/or consistent estimator of ?/ ?, where ? ? 0?

(4) Would ?

? + ?? } be an unbiased estimator of exp {? + ?}?

(5) Would exp { ?

(6) Through (1) (5) above, could you summarize under which condition(s), a function of the estimators is an unbiased estimator of

the function of the parameters?

————————– This is the end of the questions.

3 The following table contains the ACT scores and the GPA (grade point average) for eight college stu-

dents. Grade point average is based on a four-point scale and has been rounded to one digit after the

decimal.

GPA

ACT

Student

1

2.8

21

2

3.4

24

3

3.0

26

4

3.5

27

5

3.6

29

6

3.0

25

7

2.7

25

8

3.7

30

(i)

Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and

slope estimates in the equation

GPA = Bo + BACT.

Comment on the direction of the relationship. Does the intercept have a useful interpretation

here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by

five points?

6 Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and McClain (1995),

the following equation relates housing price (price) to the distance from a recently built garbage incin-

erator (dist):

log(price) = 9.40 + 0.312 log(dist)

135, R2 = 0.162.

n =

=

(i) Interpret the coefficient on log(dist). Is the sign of this estimate what you expect it to be?

(ii) Do you think simple regression provides an unbiased estimator of the ceteris paribus

elasticity of price with respect to dist? (Think about the city’s decision on where to put

the incinerator.)

(iii) What other factors about a house affect its price? Might these be correlated with distance from

the incinerator?

sav =

a

=

=

e

=

7 Consider the savings function

: Bo + Binc + u, u =

Vincie,

where e is a random variable with E(e) = 0 and Var(e) = 0 . Assume that e is independent

of inc.

(i) Show that E(ulinc) = 0, so that the key zero conditional mean assumption (Assumption SLR.4)

is satisfied. [Hint: If e is independent of inc, then E(elinc) = E(e).]

(ii) Show that Var(u?inc) = ozinc, so that the homoskedasticity Assumption SLR.5 is violated. In

particular, the variance of sav increases with inc. (Hint: Var(e|inc) = Var(e) if e and inc are

independent.)

(iii) Provide a discussion that supports the assumption that the variance of savings increases with

family income.

=

?

2 The data in WAGE2 on working men was used to estimate the following equation:

educ

=

10.36

: 722, R2

.094 sibs + .131 meduc + .210 feduc

.214,

n =

=

where educ is years of schooling, sibs is number of siblings, meduc is mother’s years of schooling,

and feduc is father’s years of schooling.

(i) Does sibs have the expected effect? Explain. Holding meduc and feduc fixed, by how much does

sibs have to increase to reduce predicted years of education by one year? (A noninteger answer

is acceptable here.)

(ii) Discuss the interpretation of the coefficient on meduc.

CHAPTER 3 Multiple Regression Analysis: Estimation

105

(iii) Suppose that Man A has no siblings, and his mother and father each have 12 years of education,

and Man B has no siblings, and his mother and father each have 16 years of education. What is

the predicted difference in years of education between B and A?

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