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Problem Set 6, Econ 120C

Professor Yixiao Sun

In this exercise we will work with a linear demand and supply model and explore the properties

of OLS estimators and IV estimators. The two equations will be

Qdi = 50

Qsi

2(Pi + Ti + Ci ) + Ui

= 5 + Pi + V i

(1)

(2)

where Ui is independent of Vi : To help you understand the equation system, you can think of

the
rst equation as the demand curve where Qdi is the demand and Pi + Ti + Ci is the total

price paid by consumers, where Pi is the sticker price, Ti is the general sales tax, and Ci is the

product-speci
c tax (i.e., cigarettes exercise tax). You can think of the second equation as the

supply curve where Qsi is the supply and Pi is the sticker price. (You may want to ponder upon

the question: why does it make sense to model the demand in terms of the total price while

modeling the supply in term of the sticker price).

1. Solve these two equations to obtain the market price and sales (i.e., let Qdi = Qsi = Qi and

then solve for Qi and Pi in terms of other variables). A student
nds the solution to be

Qi = 20

Pi = 15

2

(Ti + Ci ) +

3

2

(Ti + Ci ) +

3

1

(Ui + 2Vi ) ;

3

1

(Ui Vi ) :

3

(3)

(4)

Do you agree with the above solution?

2. Generate market observations from the above two equations using the following codes:

clear

set obs 1000

set seed 1732

gen T = uniform()*2+1

gen U = 2*uniform() -1

gen V = 2*uniform() -1

gen C = U + uniform() + 1

gen Q = 20-2/3*(T+C)+1/3*(U+2*V)

c 2021 by Yixiao Sun. This document may be reproduced and distributed for non-pro
t educational purposes

only.

1

gen P = 15-2/3*(T+C)+1/3*(U-V)

gen TP = P+T+C

Our observations are Qi ; Pi ; T Pi (total price); Ti and Ci : We do not observe Ui and Vi :

3. Calculate cov(T P; U ) and cov (P; V ) theoretically.

4. Now calculate the sample covariances cov(T

c Pi ; Ui ) and cov

c (Pi ; Vi ) using the sample you

just generated (Note that this is not feasible in practice as Ui and Vi are not observed.

However, for this part and the next two parts of the question we assume that Ui and Vi are

available to us).

5. Compare cov(T P; U ) with cov(T

c Pi ; Ui ): Are they close to each other? Are they both positive?

6. Compare cov (P; V ) and cov

c (Pi ; Vi ). Are they close to each other? Are they both negative?

7. Graph the scatterplot of Qi against T Pi with the true demand line (Q = 50 2 T P ) and

tted OLS line superimposed on the scatterplot. Which line has a higher slope coe¢ cient?

Is this expected? Explain.

8. Graph the scatterplot of Qi against Pi with the true supply line (Q = 5 + P ) and
tted

OLS line superimposed on the scatterplot. Which line has a higher slope coe¢ cient? Is

this expected? Explain.

9. Estimate equation (1) by OLS [reg Q TP,r]. Is the estimated parameter for T P signi
cantly di¤erent from 2? [Stata command: test TP = -2]. (If yes, then we make a wrong

statistical decision.)

10. Estimate equation (2) by OLS [reg Q P,r]. Is the estimated parameter for P signi
cantly

di¤erent from 1? [Stata command: test P = 1]. (If yes, then we make a wrong statistical

decision.)

11. Now we will see whether the IV method does a better job of estimating the structural

parameters. Type ivregress 2sls Q (TP=T),rto estimate the demand equation by IV.

Is the estimated parameter for T P signi
cantly di¤erent from its true value -2? Explain.

12. Now we try a di¤erent IV regression by using ivregress 2sls Q (TP=C),rto estimate

the demand equation by IV. Is the estimated parameter for T P signi
cantly di¤erent from

its true value -2? Explain.

13. Suppose we want to estimate the supply curve. Is C a valid IV? That is, will the IV

regression ivregress 2sls Q (P=C),r gives us a reliable estimator of the true slope of

the supply curve. Explain.

14. Suppose we want to estimate the supply curve. Is T a valid IV? That is, will the IV

regression ivregress 2sls Q (P=T),r gives us a reliable estimator of the true slope of

the supply curve. Explain.

2

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Explanation & Answer:

6 Pages

Tags:

business

Econometrics

problem

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