Homework III Economics 515 Fall 2016

1. Firm B

Low Price Med Price High Price FirmA Low Price 400,400 320,720 560,600

Med Price 500,300 450,525 540,500 High Price 375,420 300,378 525, 750

Is there a dominant strategy for firm A and B? If so what are the strategies? What payoff will each firm receive?

2. Firm B

Small Cars Large Cars FirmA Small Cars 1, 1 3,21

Large Cars 21,3 4,2 Is the pricing decision a prisoner's dilemma? Why or why not What is the cooperative outcome? What is the non-cooperative outcome? If they cooperate is there an incentive to cheat? Explain.

3. A monopolistic competitive firm faces the following revenue and cost schedules

TC = Q3 _8Q2 + 57Q + 20 TR = 55Q - 0.5Q2

Determine the profit maximizing output, price and profit level. Calculate and graph TFC, TVC, TC, MC, MR, P, AVC, ATC, PROFIT and Marginal Profit

4. Using E-Views estimate the regression models for the following data sets and explain your results. Explain the t-values, F-values, and R2

1. Wrestling Model 2. Dow Jones Industrial Average 3. Sports Car Production Costs 4. Price Elasticity of Demand for Compact Discs 5. Retail Sales Model 6. DVD Demand 7. Demand for Roses 7.20 8. Gold Prices, CPI and NYSE 3.22 9. Smoking and Lung Cancer 5.20 10. Demand for Chicken 7.19 11. Cobb-Douglas Production Function Greek Economy 12. Real Consumption Function 7.24 13. Read Qualcom Stock Price Example 7.25 and use your Stock Data to Estimate

86 Chapter, Additional Hodel.1ng Techniques

Intercept Dummies Dummy variables are often defined in this manner:

if perwn is male

otherwise

Usually, one of the possible values for the dummy variable is zero. This dummy var'i­able can change the intercept for males in the sample as can be seen from the follow­ing example. (Note that defining "XI = l if female. 0 otherwise" would work fine also.)

Professional Wrestling Needs Dummies Advertisers need to know who will be watching before committing money for TV com­mercials. Here. we wanl to see who spends time watching professional wreslling. It is likely that men and women have different viewing habits for this spon. so a dummy variable for gender is needed.

HOURS = Bo + BIMALE + B~INCOME + BjAGE + e (5-1)

where HOURS == hours spent watching professional wrestling on television, per month

MALE = I if person is male, 0 otherwise INCOME = individual's annual income

AGE == individual's age

Suppose a survey gives us the data shown in Table 5-A. Table 5-B shows the results from using this data to estimate the professional wrestling model.

Both MALE and INCOME are statistically significant at a 5% error level, and both have coeffiCients with the signs we would expect The coefJicient for AGE is statist;­caJJy insignificant. Allhough ilS coefficient is positive. the high p-value of 0.27 indi­cates a 27% chance that the true value of the coefficient IS actually zero. These results tell us that we can't be confident that older people tend 10 watch more wrestling. We can say. though. that people with higher incollH:S lypically watch less wrestling. The INCOME coefficIent is negative and significant at a 5%: enor level. Although at first glance the coefficient value of -0.000065 seems small, it really is not when you con­sider how INCOME and HOURS are measured. Income is annual income in dollars, so the -0.000065 value represents the average decrease in viewing hours for a trivial $1 increase in annual salary. keeping other variables constanr. If 50meone's income increases by 55000, on average. keeping other variables constant. our re~ulrs indicate that ht: or 5he will watch 0.3255 fewer hours of wrestling a month, or about 20 min­utes less I [$5000 x -0.000065'-= -0.33 (indicating fewer hours of television): -033 hours x 60 minutes in an hour = -19.8 minutes)

I Thl; may mcan Ihal people whe- dCl Ilor "'~Ich much wrc>tlJng Icnd to m~~e more monel'. I Ullfortllrl~lely.

Ihi, doe,n't me<J1l Ihol if y"llJ ':UI b"c~ On "all'hrng \\'re>lling by 20 minule~ a monlh. yoor <Jnnu~1 income

will go up by £5000:) ThJ~ que,l,on of whelher Ihe re/Jlion~hip belween "n Independent ~nd dependent vari­

able i, Ihe olher way "round I' called Ihe que;tion of c;Jll~alil\' Te\ling for c:Ju;~)iIY will be di-clI\si'd In Chapler 10.

5-1 Dummy Variables Aren't Stupid: When a Variable Is Not a Number 87

Table 5-A

Data for Professional Wrestling Model

Person HOURS MALE INCOME AGE

I 0 0 $58,600 41 I

2

3

4

5

0.5

I

2.5

0

i

-

0

I

0

0

I I

15.000

76.700

20.500

72,100

I

j

19

54

22

48

!

I I

6

7

/

/0

I

I

84,200

12.700

i

I

52

24

8 175 0 I

42.250 60

9

10

4

0

/ I I'-~

32,750

142,300 I

28

58

,

II

12

I

U

I -

/

I I

I-

98,250

24.200 I

35

67 i 13

14

15

16

17

175

4

0.25

I

16

I

I

/

I

0

0

I ,

l8.800

3/.000

22.000

30.200

27,260

I 30 I

I 21

I 32 I I

I 33

~~ 18 2 I 42,250 . 56 I

! 19 6 I I 27.850 I 31

i

! 20 0.25 0 37,500 57

MALE has a posilive coefficient, 4.47, indicating that on average men watch wrestling 4.47 nlOre hours a month than women. k.eeping TNCOME and AGE constant. MALE is an intercept dummy variable, so the significant coefficient means thaI men and women have different intercepts. To see this, consider the resulls wriuen Iik.e this:

HOURS := 0.86 + 4.47 MALE - 0.000065 INCOME + 0.073 AGE (5-2)

5-3 Designing Your Oym F-Te,! 97

Suppose we test the nu II hypothesis 8., = B4 =°for a typical ordinary least squares regression. If the null hypothesis is true. the unrestricted model is likely to give esti­mates for B) and B. thaI are close to zero and insignificant. By removing X3 and X4

from the restricted model, 8.) and B4 are forced to be zero. The restricted and unre­stricted models will perform in a similar manner: RSS"",.,oricl<d will be close in value to that of RSSr"'''Cl<d' Looking al Equation (5-1 S). if RSSr<;trimd and RSS"nre"r1clcd are close in value. the F-statistic will tend to be small and statistically insignificant. The null hypothesis will not be rejected.

Now suppose thai the null hypothesis is still B) = B. = 0, but the null hypothesis is aClUally false, so that at least one of the coefficients is not equal to zero. The unre­stricted model will give unbiased estimates of Bj and B. if the classical assumptions hold. In the restricted model. B) and B. are automatically forced to be zero, since X3 and XJ are not in the model. The restricted model will not fit the data as well as the unrestricted model did, since at least one of Bj and B-l is not really zero. This Ciluses the difference between RSSre,orKl<d and RSS"n«suiet<d to be a lot larger than if B3 and B4

actually are equal to zero. Looking at Equation (5-1 S), lhe F-statistic will tend 10 be larger and statistically significant. The null hypothesis can be rejected. The next exam­ple demonstrates the F·test.

F-Test Your Way to Riches

Here, dummy variables for different seasons are used to look for a pattern in the Dow Jones Industrial Average (DllA). For example, perhaps stock prices tend to be lower in the winter and higher in the summer. We could use such a pattern. if it cOJltinues, to make money. Those of you who are familiar with the efficient market hypothesis know that most economists are skeptical that such a pattern exists. 4 Quarterly data are used for this model.

DIlA = Bo + B,GDP + B2UNEMPLOY

+ B)SUMMER + B4 FALL + BsWlNTER + e (5-16)

where DJIA = Dow Jones Industrial Average. averaged quarterly GOP = real Gross Domestic Product. in billions of 1996 dollars. per

quarter UNEMPLOY = average unemployment rate (%), cach qUaJ1cr

SUMMER = I during summer. 0 otherwise FALL = J during fall, 0 otherwise

WJNTER = I during winter. 0 otherwise

The data is given in Table S-G, the results in Table SoH.

, For a discussion of the efficienl market hypothesis_ see Roger Miller. Daniel B('njamin. anJ Dougl",' North. "Pure Compt'li!ion on Wall Slree'" in The Economics of Public (Hues Illh eJ. (New York: AJdison-We,ley, 1998), pp. 75 -80.

------- ---

-----

--

98 Chapter S Additional Modeling Techniques

Table 5-G

Data for Seasondl DJIA Model

o

o

o

o

o

o)3ni41.9

Years: QUllrtt'r D.JIA GOP I UNEI\IPI.OY SUMMER FA 1.1. WINTEI{

I\.)'!\I 3. C)6 2. L)8"1 I 7-188. "I

I .5 0 0 () I

' ­ I -'

I'N5:2 -1.377 663 I 7.503.3 5 7 I 0 0 f- ­ I

I '!\.).":3 -1.690.2li3 7.';61-1 S 7 0 I 0 I ­I 1')'J.'.l -1,94-1.123

; ,7.6~ /9 56 I) 0 I I

I . ­ - - - - - - . ­ --,-- ­ - 0-1I Il)'!():1 5.'136."1 '0 7.676.4 5.6 (I 0 ~ - ­ --I--­

1L)1.)6·2 I 5.6~2.693 7 SO::'.') 55 I (I (I I

I i

19%:4 (l,250.14 "I 7.93l.3 S3 0

1\)<,)71 (>.8.:11.'1;77 8.016.-1 5.3 ()

I L)97:2 7.160.:187 8131'! 50 -I 19'!7:3 7.935.223 8.2166 -18 0 I 1l)L)J:.1 7.8~ 1000 8.27~.') .:1.7 0

----- ­1L)t)K: I 8.2KO.-177 8.-10-1.9 n 0

1991U S.l)96~23 ::I.-J6S/, -1.-1

19\)::1:.' 8.-195/51) 8.537.C) -1.5 0

1­4.4 0 o19\);;:4 S.T:'9633 8.65.:15

I- I

I 19l)l).1 9.-17.:11-13 8.730.0 4.3 I

19')l): ::' '- 10/'67130 !US:12 -J3I II-- I - -----,- - -1-- -- -I ­199'!:3 1(I.90057() K90S8 ,U

--~--

o

o

()0

Il

---0

1')L)t):-1 10.S 176~O <)OK-1 I --- --L

lIlO() I 10.7688!l0 \) 191 S 4.1:=¢t" 9.3088 40 J~O()Il:2 ~0762.::'.:I1l

All (\:\(:1 :lr~ ;l\ :lll.d.>k al the texl web)il~ ,\r h{lp:I/\\.·\\·\\·.~"::olltllnJ.~i(.\.·<..\m.A,hern()lI\'C'I~. (nr GDP. :,e..: IHlp //\\"\\ \\',l'c-.u.il'\· ,gc.w/hc-,dJli J

.htmll,: L'.S DCr:1rllll,,'l1t of COmJllc:'lu:~. BLllealll.1l· Fen!1I'1l1Il' ..:\n~d.\·,l.'. F"l" L)i\Ef'vIPLO'r. so;:'e 11lIp:li\~\\"\\ ~[I'i.rl"b.nl~/I·rc-I..Ui!l(!e.\.hlI1l1

b~: tile' Sl. Ll)ll:.., Fed. DJ 1.-\ is ;1\·~lil.lbl.::' ;:\1 hllp.l/iJldc:'\t.'~ do\\ jl1llc ..... <.:omJhonlc:' hlllll n~ De'I\' .11'lIe .. iJnd el'

L

5·5 Log Models: Estimating Elasticities 105

5-4 POLYNOMIAL MODELS: CURVES CAN BE LINEAR REGRESSIONS

Most CO$! and production functions are curves with changing slopes, not Slr<light Ijne~

with constant slopes. The polynomial model enables us to estim<lte curves. In the poly­nomial model, independent variables Ciln be squared, cubed, or raised to ilny exponen­tiill power. The coefficients still <lppear in the regression in a linem fashion, allowing the regression to be estimilted as usual (see Section 2·2). Some examples of polyno­mial models are

Y '" Bo + BIX, + B2 (X I? + BJXJ + e (5-22)

Y = Bo + BIX, + B2(X I )2 + BJ(X,)J + e (5-23)

At first glance, the independent vari<lbles in each of these polynomial models seem to be highly correlated, since the same independent variables appear more than once,

in different forms (A ~imilm problem was discussed in Section 5-2, where we added an interactionterrn to the professional wrestling model.) Multicollinearity may not exist here, however. because the variables are not linearly correlated. For example, In Equa­

tion (5-23), XI' (X,)2, and (XI)] share il nonlinear correlation.

Sports Car Production Costs

Suppose <In exotic spons Cilr company wants to estimate its average cost function. Here

is its polynomial model for estimating the average cost function:

(5-24)

where AVECOST = average cost per car, in dollars

CARS = number of cars produced per week

Table 5-M gives data for the aver<lge cost at ditferent levels of weekly prodUCtion,

Table 5-:'J gives the resulls for this model We can use these results to estimate the average cost for different levels of pro­

duction, which gives us the average COSt curve shown in Figure 5-3. Such information is useful in planning future production schedules

5-5 LOG MODELS: ESTIMATING ELASTICITIES

In a log model. the natural logarithm is used to transform one or more of the variables Log models express nonlinear relationships between the dependent and independent variables. As with the polynomial model. even though the rel~lIionship is nonlinear. the coefficient enters the regression in a linear manner. We'll examine two commonly used

log models. the double-log and semi-log models.

106 Chapter 5 Additional Modeling Techniques

Table 5-M

Data for Polynomial Sports Car Model

Observation

I

CARS

50

AVECOST

42,250

I ,

2 100 40.825

3 150 36.540

4 200 37.135

5

6

250

300

29.500

32.250 ,

7 350 25.230

8 400 31.842

9 450 2J .568

10

II

12

500

550

600

18.225

2~.759

36.102

I I

13 1>50 33.215

14 I 700 31.460

15 750 42.5 J 2

Hi

17

i ,

800

850 )

42,890

51.230 I I

lB.

19

20

900

950

1.000

I i

I

55.\185

65.320

74,586

I

I

The Double-Log Model

The double-log model consists of taking lhe nalliral logarithm of the dependenl and independent variables.

(5-25)

107 5-5 Log Models Estimating Elasticities

Table S-N

Results for Polynomial Sports Car Model

Dependent variable: AVECOST

f Variable Coefficient Standard Error I t-Stalistic p-val::l

T[ con~ta~; 51.67110 ~.870231 18002 0001I I .

-121.662CARS 12590 ! -9664 0.001! II - ­f (CARS? o 142 00116 I~224 0001i ! _._--_..... ---­

R' ~ 0.9JJ

AdJlIsl<d R' ~ 0.925

Re,iduol Sum vI Square' = 25J.000.000

F-sl:Jllsli<: = 117.929

Figure 5-3

Estimated Average Cost Curve for Polynomial Sports Car Model. The * symbols are data points hom Table S-M.

*

* *

* * *

80,000 ;;;; 70,000 ~

rtl 60,000u ~

OJ 50.000a. +" VI 40,0000 u OJ 30,00001 ~ 20.000OJ > <l: 10,000

0 200 400 600 800 1000

Numbers of cars produced weekly

Except for the intercept, the coefficients of a double-log model are elasticitie5. Elasticity

is the percentage movement in one variable that results when another vnriable ch,mges

1%. The estimate of 8 1

eSllITIates the average percentage change in Y for a 1% ch3nge in X I' keeping the other variables ConSlanl. The most commonly used elasticity is price

elasticity ot' clemand. which measures the percentage change in quantity demanded for a I % change in the good's price. Elasticitie::. of all types can be estimated with double-log

models. Double-log models are appropriate when the theoretical relationship between the independent and dependent variables has a constant elasticity, but not a constant slope.

Chapter ~ Additional ,'1odeling Tecr.nique·;

Estimating the Price Elasticity of Demand for Compact Discs

SUrrll'" J Inrge rel'l)ru .'lore JU,I> J Ilt'\\' ,eClIOIl of rurLlpe~1i1 tech')l) mu,ic di-c,. III order 10 e,tiJ11:lte the prlcc elaqjcit)' of dem,lrlu for the.'<: disc,. the swre conduct' nil experiment. Eoch dal' fur 20 day5, lhe Siore luwer, the price t(lf Ihese CD, and tracl\s how many Me ,old. Tht' store hus ;] lJrge number uf techllo J11L1~ic CD, ill ,tock. so Ihe allHlunt 'old i.'; thi: qU<lntity demJlld<:d. Tuble 5-0 gives data for ,uch all experiment.

5-5 Log Models: Estimating Elasticities 109

The appropriate double-log model for estimating the price elasticity of demand is

In(QUANTlTY) '" 80 + B I In(PRICE) + e (5-26)

Many economics textbooks depict demand curves as linear (so that the slope is con­

stant). but the elasticity changes along the curve. Demand curves need not be linear.

This model gives a demand curve that is nonlinear and has a constant elasticity. Results from this double-log model are given in Table Sop.

The estimate of the price elasticity of demand is 2.937 (Although the slope coef­

ficient estimate is -2.937. the pricc elasticity of demand is stated as <l positive num­beL) For every I % decrease in price. the quantity demanded will typically increase by '2937%.

The Semi-Log Model The semi-log model is an adaptation of the double-log model where only some of the variables are transformed by the natural logarithm. The semi-log model can lake dif­

ferent forms. such as

Y := Bo + B, In(X,) + 82 In(X1 ) + e (5-27)

Y = Bo + RIX 1 + B, In(X 2) + e (5-28)

In(Y) == 8 0 + B,X, + R,X1 + e (5-29)

Independent variables that are. in theory. linearly related to the term on the left-hand side of the eCluation fY or In( Y)J are entered into the regression without a logarithmic

transfornmion. If the theoretical relationship is nonlinear. then a logarithmic transfor­mation may be appropriate. Equations (5-27) and (5-28) model situations where the effect of X on Y becomes smaller as X becomes larger. For example. Equation (5-28) could be used if Y is consumption. and X~ is income. As income increases. con­sumption increases. but consumption increases more slowly when income reaches a cer­

tain level. The effect is represenled properly in the regres!;ion by taking the logarithm

Table 5-P

Results for Techno CD Double-Log Model

Dependent variable: In(QUANTITY)

JO.474

,I --2.9)7 O..~66 ! -8.015 0.001

~---l

Variable Coefficient Standard Error I t-Slatistic p-Value

Constanl 1020 I

10.270 0.001

In(PRICE)

Obse"'"tion~: 20

R' ~ 0781

AJj""etl R' ~ 0.7(,'1

Re"dual Sum of Squ"re~ = 5.1.17

F-q"'islic = 64.248

-------

5-1 Dummy Variables Aren't Stupid: When a Variable Is Not a Number

(- (: tJ E/lr\l)Loj + 62 \Table 5-C --- ._- ----:-=--~--:---------------------

Data for Seasonal Retail Sales Model

II Years: Quarter SALES

11('-'056

I UN Ei\-WLOY i

I 5.5

I SUMMER

I 0

FALL

0

! WINTER 1

I 0

I II 1995: I

I 1995:2 110,~ 13.5 I 5.7 --i

0 !

0 ,

129,957.5

J 3S7N6

121..168.0

13~~018

132.3-10 I

1-12,5839

124.1619 -

135.6.190

136,6569

/45.1358

5.7 o o

56 o

5.6 o

5.5

53 o

53

. -53

5.U

H o

4.7 n I 0 I I

-1.7 o1~6~ 142.5097 <1.-1

4.5 o 1­

152.273.8 1 ~ 04"

114.7909 4.3

J50.:1)46 <1.3

I I

I ':!'}l):4

2000: I I 162,5cl7.2

146.5675 -­ -

<1.1

41 I

0

0

0 i o I ~

I

0

0, 2000:2 ! 158.4470 40 I I

Tlle,~ dat::J .Ire .1V:li!:Jbk ,H lh~ fl'!"\l web:.-ile "If hltp:llwww.~c:onomJg.k.coll1 (S.<.\LES are- llomll1.11 ';J!e'i adju"led

by (he (on")lIm~r pnce lIldex ) Altell13tively. for o0l11ln;',1 :-ak<: :-.e~ U.S. C~n~u~ Otlr~au: Monlhly RetJil Tr:lde

Sur'.. ~y. ;I[ l1t!p:l/w\\l\V.,:ensLl~.go\'/mlb;"'1l,.vw/mrl<,.hlml. (PI J;1I11 3re found ,Il htlp:l/~IJ(~.bl".gov (l)ur~J.u of

Labor SIO(;''''·''. For lJNEMPLOY. 'ce hup:!/ww'W.>lIs.frb.org/fred/inde,\.hrml b~ Ihe SI. LOlli, [oed .

. "

38 Chapter 2 Ordinary Least Squares, Part Z

c. [f Quinn's predicted DVD e.xpendirures increase b; S30 but RAII':FALL and PR1CE stay tile Silme. hall Inuch does her income innen,e"

9, A, discll~sed in the chapter. each slope coefficielll in a lllultiple legresslOn mea­sures the isol:Hed effeci of a one-unit cilange in it, independent variable whiJe the other independent variable., are kepi constant. Sholl' that rhis i, tme by using tht' following steps: 8. Find the predicted v81ue for DV DEXP u.sing rile resuli, in Table 2- B 8nd the

data for February shown in Table 2-A. b. Find the predicted value., f'or DVDEXP if PR1CE goe.,> up a dollar to $27.49,

but INCOME and RA1NFALL Slay the same 8S in part ,1. c. Subtract part a's 8nswer tram part b's, What do you notice abOutlhe difference

between tilese IWO answers? JO, Consider Equiltion (2·7), which estimate, tile .slope e,tilll<lle 1'8rian-:e:

- ' - - 'T'VAR(B ) '" .,- , - ­

I slim oj' (X - X f added over all obserl';l\ ion,

8. How does the devj,lIion of' X from ii, rnenn affeCt, tile pren,ioll or' BJ ~

b. How does the somplt' size atleci the precision ofB,~

c. How Ciln the answers to parts il and b help yOll design a model'> I J. Using the data in Tilble 2-A. find the ~Ulll of ,yuared re,idunl, Jar both the single

and Illultiple regression model. btirnate

and

DVDEXP = 13 0 + B,lNCOMF. + B~PRICE+ 13,RAINr'ALL+ t'

(Nole: Some 50ftlvare pmgr8ms give th" sum of sq1l8red residuilJs autorllolically: ifyoul' soflware doesn'( do lhi~, use iliO gil"lhe residu'Jis. then sqll:Jre e<Jch resid­UJI and add lhem across all obsen·alions.) il, Which regression has the smillieI' sum of sCjumed resid1l81,;'? b Will your illl,wer ro pall 8 be Ihe S£ll11e .1' differelll data ,amplcs :-Ire u,ed.)

E>:plain. c. Will yOlll' an)\\'er 10 part a change jf you l'ompal'e single ilnd mult,ple regres­

sion" for a different subjecl. one rhat has nothing to do with DVDs" E>:plain. J2. The tilbte belo\\ giles closs-section c1iltiJ 1'01 30 students II'ho buy DVDs from 30

different place.s.

~er\'alion

I 1

J)\,DEXP

970

INCOME

600

I I

PHICE

2~99

RAINfALL

6.0

2 1990 Dl I 2-1.95 ' ,-.­

3

4

49.~<l

926

1120

455

1

I 1999

24.50

I I

1.0

2.7

- ----

39

i

Exercises

Observation DVDEXP INCOME PRICE RAINFALL

5

6

7

8

9

10

II

12

13

14

15

16

17

III

19

20

2J

22

,23 !

24 15.30

25 38.19

26 3266

fJ.-

2190 12 I -----t-----t---- ----I

I 1800 i 2.5 I

475

685

365

98327 50.93

28 27.24 465 I .20.10 1___ 2.2 -jl' 29 3024 783

i

----I ;;Jol ~4.1

:10 30.67 854 22.30 i 28 !

Jel~ leI"hapter 6.) \ -,,.. ,

. 'ii' lr Input. and real capita)

Real capital input (millions of NT $), X,

120,753 122.242 125.263 128.539 131,427 134,267 139,038 146.450 153,714 164,783 176.864 188.146 205.841 .. ~

221.748 239.715

, on Taiwan-1952-1971. A

nieduate Center, City ,~:"'­'. ~

.. , ....".

he output elasticities -: He similar elasticities '~.

MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF ESTIMATION 225

(a) lnteJ1lret your results. (b) What is the rationale for introducing the square of the unemployment rate in

the model? A priori. would you expect 13J to be positive or negative? (c) Is this model really a multiple regression model since onJy one explanatory

variable. the unemployment rate. enters the model? (d) How do your results compare with those obtained in (6.6.2) and in exercise

6.20? (e) Can you compare the R2 terms of the various models? Why or why not? (j) Which model would you choose: the quadratic model given here, the recip·

roca] model given in (6.6.2), or the linear model given in exercise 6.20? What

criteria do you use? 7.20. The demand (o/' wses.· The following table gives quarterly data on these variables:

y = quantity of roses sold, dozens

Xz = average wholesale price of roses. $/dozen XJ = average wholesale price of carnations, $/dozen X. = average weekly family disposable income. $/week Xs = the trend variable taking values of I, 2. and so on, for the period 1971-III to

1975-II in the Detroit metropolitan area

Year and quarter y X, X, X. X,

1971-11l -IV

1972-1 -II -01 -IV

1973-1 -II -Ill -IV

1974-1 -II -UI -IV

1975-1 -II

\ \,484 9,348 8,429

10.079 9.240 8.862 6.216 8.253 8.Q38 7,476 5,911 7,950 6.134 5.868 3.160 5.872

2.26 2.54 3.07 2.91 2.73 2.77 3.59 3.23 2.60 2.89 3.77 3.64 2.82 2.96 4.24 3.69

3.49 2.85 4.06 3.64 3.21 3.66 3.76 3.49 3.13 3.20 3.65 3.60 2.94 3.12 3.58 3.53

\58.1\ 173.36 165.26 172.92 178.46 198.62 186.28 188.98 180.49 183.33 181.87 185.00 184.00 188.20 175.67 \88.00

I 2 3 4 5 6 7 8 9

10 II 12 13 14 15 16

You are asked to consider the following demand functions:

Y, = a\ + a2X2I + aye)/ + ~4' + asXs, + u,

In Y, = 13\ + 132InX2I + 133InXJ' + 13. InX., + 13s lnXs t + u,

(a) Estimate the parameters of the linear model and inteJ1lret the results. (b) ESlimate the parameters of the log-linear model and interpret the results, (c) 132,133, and (3. give. respectively, the own-price, aoss·price, and income elastiC­

ities of demand. What are their a priori signs? Do the results concur with the

a priori expectations?

'\ am indebted to Joe Walsh for collecting these data from a major wholesaler in the Detroit

metropolitan area and subsequently processing them.

,~,19, Th" 1e/'iliIlIlS!,i!) /I"'II'C(-'II //()ll/illll!",I','/'ullge I'U/" lIlId rd",i,',- pl'i,·"". From dllnll;i1

OOSel'l311011S flOll1 I9RS 10 200S, the I'ollo\\'ing rcgfl:ssioll resulb were (,lblailled,

where)' = exchange r~le "fthe C<ln~tlilln dollar lothe U.S. doll;H (CI)/S) and.\' =:

r~li() ol'lhe U.S, Ct11lSUlllt:r pl'lee illde,\ 10 Ihe CllllZl\Ji,lllcOIl,sullier price illd('\; Ih~1 is, .\' rl'p,csenls lhe rclalivc p!'lces ill the two counlries:

I~, = --0.<,112 -t- 2.25()X, r~ = 0.440

so.: = 0096

,/. Interprel IhlS rq;rl'ssioll How 1I'01iid you illterprel I'~')

h. Docs Ihe Iltlsilive value or x, ma"e eC(,lIH)mic sense') Wlt;tl is the underlying ecollomic Iheory"

c. Suppose we were 10 redc/inc .\' <IS lhl' I<llill or lite Canadian CPI 10 lite U.S. CPI. Would Ihat change rhe sign ol'K.' Why')

.1.20. Table .I (, gives data on indexes of ()ulpul per Itour (X) and real compcnsation per

hour ()') 1'01' Ihe bllSlncs" and nonl'al'ln lJusilll:ss scclors or Ihl: US, ecollOl11)' lor

I <j()O··200S Thc base year or tlte indexcs is I <,In = 100 and the inde>..es arc seasonally adjusted.

0. 1'101 )' llgainsl X 1'01' the two seCWrs scparately_

Ii, Wl1;l1 is Ihe eCOl1l11l1ie Ihenry bell1nd Ille lelaliollsllip bclwel:lI Ihe 1\1'0 varidbks'1

Does lhe scallcrgralll support the theory'.'

c. ESlnnJle lhe OI_S regreSSion of)' 011 X Save Ihe rcsulis 1'01' a fllrlhcr look ~ner we study Chapler 5.

3,21, FrOl1l a salllpl~ or I [) oh,c['\'alions, Ihe f"llo\\ ing resullS wen: obtained:

L)'; = 1.110 LX, = 1,700 LXi), = ~().\5(J0

L'\) = L I',' =.122,000 132, JOO

\\'ilh eoelficicnt or cOrrCIJliOIl " = O,97SH. BUI on rechecking Ihcsc calculalions it was roulld Ill'll I \\'0 pairs or obscrvallons were recorded:

y yX X

90 120 80 110instead 01 140 220 150 210

Wllat will be Ihe elkel orlhis error on ,." Obtain the correci 1'.

3.22. Table 3.7 gll'cs dalll on gold prices, Ihc Consulller Price Jndc.\ ((PI). ano Ihe New

York Stock Exchange (NYSl) Index 1'01' Ihe United Siales for the perit>d 1974 --200(, The NYSE Index includes 111051 "rlhe SIOl'ks listed on the NYSE, sOllle 1500-plus_

1/. 1'101 illihe S(lnle seallergr~lll gold pliers. CPI, and Ihe NYSr: Illdex,

Ii. An invesllllclll is supposed 10 be a lIedge again" tlltlall(\n If ils price and/(n mil'

Dr return at kasl kccjls pace \\ ilh Jntl~nion, To ICSI Ihi, hypolllcsls, SlJI1I'0,se you

decide In fil Ihe following 1ll(ldt'1. assuming Ihe sctllCrplOI in (il) sliggesh that Ihis i, apl,ropt iale:

Gold proce, = iii + (11 CPI, + II,

NYSL': index, = iii + Ii: ('PJ, -t- II,

I!:} ParI One Slugh·-rl!tltoj/J/I Regrt'}·.\IlHl J\f"dl'J,

1\:.ll ' ! Gold rric~s, New

Year Gold Price NYSE CPI

York Stock EHhallg~ 1974 159.2600 463.5400 49.30000 Indn, and Consumer 1975 1610200 483.5500 53.80000 Price Jnrlex fOl' U.S. 1976 1248400 5758500 5690000 for 1974-2006 1977 157.7100 567.6600 60.60000

1978 193.2200 567.8100 6520000 1979 306.6800 616.6800 72.60000 1980 6125600 720.1500 8240000 1981 460.0300 7826200 90.90000 1982 375.6700 728.8400 9650000 1983 424.3500 979.5200 99.60000 1984 360.4800 977.3300 1039000 1985 3172600 1142.970 1076000 1986 367.6600 1438.020 109.6000 1987

.. 446.4600 1709.790 1136000

1988 436.9400 1585.140 118.3000 1989 3814400 1903360 124.0000 1990 3835100 1939.470 130.7000 1991 362 ..1.L.0~Q . 2181.720 1362000 1992 343.8200 2421510 140 3000 1993 359.7700 2638960 144.5000 1994 384.0000 2687.020 1482000 1995 384.1700 3078560 1524000 1996 387.7700 3787200 1569000 1997 3310200 4827350 1605000 1998 294.2400 5818.260 163.0000 1999 2788800 6546810 166.6000 2000 279.1100 6805.890 , 722000 2001 274.0400 6397.850 1771000 2002 309.7300 5578890 179.9000 2003 3633800 5447460 184.0000 2004 409.7200 6612.620 1889000 2005 4447400 7349.000 195.3000 2006 6034600 8357990 201.6000

3.23 Table :1.8 gives data on gross domestic product (GOP) for Ihe United States for the years 1959-2005

o. Piol Ihe GOr dala in CUITenl and con51,lnl (ie .. 20(0) dollal's against tll11e.

/J. Letting )' denote GOP and X time (measured chronologically starting 1\'1Ih I for 1959.2 for 1060. through 47 for 2(05). sec if the follov-'\ng 1110del lils Ille (j01' data:

[sti,nate this model for both cUITenl and collslant-dollar GOI'

c How would YlHI interpret {i!'}

d. Jf there IS a difference between fi! c'slimaled for cUlTcnl-dollilr GDI' and Ihal estimated for constJnl-dollili GDI', \\hill explain~ thc Jitli::rcnce?

e. flOn) your results what can you say aboLit the nalLlI'C of intblion ill the Uniled SIJte~ over the silnlple period"

142 Part One Sil/Rle-Eq/latiOI/ Rr.Rressiol/ Models

a. Plot the cpr on the vertical axis and the WPI on the horizonk11 axis. A priori, what kind of relationship do you expect between the two indexes? Why?

b. Suppose you want to predict one of these indexes on the basis of the other index. Which will you use as the regressand and which as the regressor? Why?

e. Run the regression you have decided in (b). Show the standard output. Test the hypothesis that there is a one-to-one relationship between the two indexes.

d. From the residuals obtained from the regression in (e), can you entertain the hypothesis that the true error term is normally distributed? Show the tests you use.

5.20. Table 5.11 provides d<1ta on the lung cancer mortality index (100 = average) and the smoking index (100 = averagc) for 25 occupational groups.

a. Plot the cancer mortality index against the smoking index. What general pattern do you observe?

b. Letting Y = cancer mortality index and X = smoking index, estimate a linear regression model and obtain the usual regression sl<ltistics.

c. Test the hypothesis that smoking has no influence on lung cancer at a = 5%.

d. Which are the risky occupations in terms of lung cancer mortality? Can you give some reasons why this might be so?

e. [s there any way to bring occupation category explicitly into the regression analysis?

TABLE 5.11 Smoking and Lung Cancer

Source. hnp:!/libst31. emu.roul DASlJDJlal1lcsl Smol(lngandCancer hrml.

Occupation

Farmers, foresters, fishermen Miners and quarrymen Gas, coke, and chemical makers Glass and ceramic makers Furnace forge foundry workers Electrical and electronic workers Engineering and allied trades Wood workers Leather workers Textile workers Clothing workers Food, drink, and tobacco workers Paper and printing workers Makers of other products Construction workers Painters ana decorators Drivers of engines, cranes, etc. Laborers not included elsewhere Transportation, and communication workers Warehousemen, store keepers, etc. Clerical workers Sales workers Service, sports, recreation workers Administrators and managers Artists and professional and technical workers

Smoking

77 137 117 94

116 102 111 93 88

102 91

104 107 112 113 110 125 113 115 105 87 91

100 76 66

Cancer.

84 116 123 128 155 101 118 113 104 88

104 129 86 96

144 139 113 146 128 115

79 85

120 60 51

220 Part One Single·Equlltion Regrcrsion Models

TABLE 7.7 DomesticWildcat Activity Output

Source: Energy Informalloll Per Barrel (millions of GNP, Admirwilralion, 197R KepON 10 Thousands Price, barrels ConstantCongrcs)

of Wildcats, Constant S per day) S Billions Time (Y) (X2) (Xl) (X4 ) (X s)

8.01 4.89 5.52 487.67 1948 == 1 9.06 483 5.05 490.59 1949 = 2

10.31 4.68 5.41 533.55 1950=3 11.76 4.42 6.16 576.57 1951 = 4 12.43 4.36 6.26 598.62 1952 = 5 13.31 4.55 634 621.77 1953 == 6 13.10 4.66 6.81 61367 1954 = 7 14.94 4.54 7.15 654.80 1955 = 8 16.17 4.44 7.17 668.84 1956 = 9 14.71 4.75 6.71 681.02 1957 = 10 13.20 4.56 7.05 679.53 1958 = 11 13.19 4.29 7.04 720.53 1959=12 11.70 4.19 7.18 736.86 1960 = 13 10.99 4.17 7.33 755.34 1961 == 14 1080 4.11 7.54 79915 1962 = 15 1066 4.04 7.61 830.70 1963 = 16 10.75 3.96 7.80 874.29 1964 = 17

9.47 3.85 8.30 925.86 1965 = 18 10.31 3.75 8.81 980.98 1966 = 19

8.88 3.69 8.66 1,007.72 1967=20 8.88 3.56 8.78 1,051.83 1968 = 21 9.70 3.56 9.18 1,078.76 1969 = 22 7.69 3.48 9.03 1,075.31 1970 = 23 6.92 3.53 9.00 1,107.48 1971 = 24 7.54 3.39 8.78 1,171.10 1972 = 25 7.47 3.68 8.38 1,234.97 1973 = 26 8.63 5.92 8.01 1,217.81 1974 = 27 9.21 6.03 7.78 1,202.36 1975=28 9.23 6.12 788 1,271.01 1976 = 29 9.96 6.05 7.88 1,332.67 1977 = 30

10.78 5.89 8.67 1,385.10 1978=31

7.19. The demand/or chicken in the Uniled States, /960-/982. To study the per capita consumption of chicken in the United States, you arc given the data in Table 7.9,

where Y = per capita consumption of chickens, Ib

X 2 = real disposable income per capilc1, $

X) = real retail price of chicken per Ib, ¢

X4 = real retail price of pork per Ib, ¢

Xs = real retai I price of beef per lb, ¢

X6 = composite real price of chicken substitutes per Ib, ¢, which is a weighted average of the real retai I prices per Ib of pork and beef, the weights being the relative consumptions of beef and pork in total beef and pork consumption

Chapter 7 M/lltiple Regression Anulysis: The Problem of £stimu/ion 221

TABLE 7.8 U.S. Defense Budget Oullays,1962-1981

Defense Budget Outlays GNP

U.S. Military Sales/

Assistance

Aerospace Industry

Sales Conflicts 100,000+

Source: These d:ll~l were Year (Y) (X2) (Xl) (X4) (Xs) c;Olh:-cICd b~ Alben Luccl'1lno

from v<lfJI.)US govlmmCnl

. publtC<JIlOns

1962 1963

51.1 52.3

560.3 590.5

0.6 0.9

16.0 16.4

0 0

1964 53.6 632.4 1.1 16.7 0

1965 49.6 684.9 1.4 17.0 1 1966 56.8 749.9 1.6 20.2 1 1967 70.1 793.9 1.0 23.4 1

1968 80.5 865.0 0.8 256 1

1969 81.2 931.4 1.5 24.6 1

1970 80.3 992.7 1.0 24.8 1

1971 77.7 1,077.6 1.5 21.7 1

1972 78.3 1,185.9 2.95 21.5 1 1973 74.5 1,326.4 4.8 24.3 0

1974 77.8 1,434.2 10.3 26.8 0

1975 85.6 1,549.2 16.0 . 29.5 0

1976 89.4 1,718.0 14.7 30.4 0

1977 97.5 1,918.3 8.3 33.3 0

1978 105.2 2,163.9 11.0 38.0 0 1979 117.7 2,4178 13.0 46.2 0 1980 1359 2,633:1 15.3 57.6 0

1981 162.1 2,937.7 18.0 689 0

TABLE 7.9 Year Y X2 Xl X4 Xs X6

Demand for Chicken in the U.S., 1960-1982

1960 1961

27.8 29.9

397.5 413.3

42.2 38.1

50.7 52.0

78.3 79.2

65.8 66.9

Source: Dnta on Y IJrc rrom 1962 29.8 439.2 40.3 54.0 79.2 67.8 C;'/haJe and on X11hrough X6

3re (rom the U.S. Ikpallln~n( of

A~eullule. I am indeblcd 10

1963 1964

30.8 31.2

459.7 492.9

395 37.3

55.3 54.7

79.2 77.4

69.6 68.7

Roben 1. FIsher (or col1cCllng 1965 33.3 528.6 38.1 63.7 80.2 73.6 lhc dara and rOI the sialislical analysis

1966 1967

35.6 36.4

560.3 624.6

39.3 37.8

69.8 65.9

80.4 83.9

76.3 77.2

1968 36.7 666.4 38.4 64.5 85.5 78.1

1969 38.4 717.8 40.1 70.0 93.7 84.7

1970 40.4 768.2 38.6 73.2 106.1 93.3 1971 40.3 843.3 39.8 67.8 104.8 89.7 1972 41.8 911.6 39.7 79.1 114.0 100.7

1973 40.4 931.1 521 95.4 124.1 113.5

1974 40.7 1,021.5 48.9 94.2 127.6 115.3

1975 40.1 1,165.9 58.3 123.5 142.9 136.7 1976 42.7 1,349.6 57.9 129.9 143.6 139.2

1977 44.1 1,449.4 56.5 117.6 139.2 132.0 1978 46.7 1,575.5 63.7 130.9 165.5 132.1

1979 50.6 1,759.1 61.6 129.8 203.3 154.4 1980 50.1 1,994.2 58.9 128.0 219.6 174.9 1981 51.7 2,258.1 66.4 141.0 221.6 180.8 1982 52.9 2,478.7 7004 168.2 232.6 189.4

.""flle: The real pri~s \Veri: obt.ained by dividing lhe nominal pnct:s by the Consumer Price lridex for food

222 Part One Single·Equalion Regression Models

Now consider the following demand functions:

InY,=al+a z lnXz r +a}lnXJr +I1, (1)

In Y, = YI + Y2 In X z, + Y3 III X JI + Y4 In X., + 111 (2)

In Y, = AI + Azln Xz, + A} In Xj , + A4lnX5r +11, (3)

In Y, =81 -I- 8z In Xz, + 8J In XJ, + 1)4 In X., + 1)5 In X5, + 11, (4)

In .>', = fJ, + fJz In Xz, + fJ3 In X3, + fJ4 In X6, + 11, (5)

From microeconomic theory it is known that the demand for a commodity generally depends on the real income of the consumer, the real price of the commodity, and the real prices of competing or complementary commodities. In view of these considerations, answer the following questions.

a. Which demand function among the ones given here would you choose, and why?

b. How would you interpret the coefficients of In X z, and In X J, in these models?

c. What is the difference between specifications (2) and (4)?

d. What problems do you foresee if you adopt specification (4)? (Hint.' Prices of both pork and beef are included along with the price of chicken.)

e. Since specification (5) includes the composite price of beef and pork, would you prefer the demand function (5) to the function (4)? Why?

f Are pork and/or beef competing or substitute products to chicken? How do you know')

g. Assume function (5) is the "correct" demand function. Estimate the parameters of this model, obtain their standard errors, and R2 , ft.z, and modified RZ. Interpret your results.

11. Now suppose you run the "incorrect" model (2). Assess the consequences of this mis-specification by considering the values of yz and Y} in relation to fJz and fh. respectively. (Hinl: Pay allention 10 the discussion in Section 7.7.)

7.20. In a study of turnover in the labor market, James F. Ragan, Jr., obtaincd the follow­ing results for the U.S. economy for the period of 1950-1 to 1979-IV· (Figures in the parentheses are the estimated I statistics.)

Ir;r, = 4.47 - 0.34 In X2, + 1.22 In X}, + 1.22 In X.,

(4.28) (-531) (3.64) (3.10)

+ 0.80 In X5' - 0.0055 X6, ft.z = 0.5370

(1.10) (-3.09)

NOle: We will discuss the' statistics in the next chapter.

where Y = quit rate in manufacturing, defined as number of people leaving jobs voluntarily per 100 employees

Xz = an instrumental or proxy vanabk for adult male unemployment rate XJ = percentage of employees younger than 25 X4 = N,_I / N'-4 = ralio of manufacturing employment in quarter (I - I) 10 that

in quaner (I - 4) X5 = percentage of women employees X6 = time trend (1950-1 = I)

'Source: See Ragan's article, "Turnover in the Labor Market: A Study of Quit and LayoH Rates," Economic Review, Federal Reserve Bank of Kansas City, May 1981, pp. 13-22

224 Part One Sillgle·£quation Regression Models

G. Given the data, estimate the above demand function. What are the income and interest rate elasticities of demand for money?

b. Instead of estimating the above demand function, suppose you were to fit the function (M/ Y), = Ctlr~'e"'. How would you interpret the results? Show the necessary calculations.

c. How wi II you decide which is a better specification? (Note. A formal statistical test will be given in Chapter 8.)

7.22. Table 7.11 gives data for the manufacturing sector of the Greek economy for the period 1961-1987.

G. See if the Cohb-Douglas production function fits the data given in the tahle and interpret the resulls. What general conclusion do you draw?

b. Now consider the following model:

Output/labor = A( K/L){!e"

where the regressand represents lahor productivity and the regressor represents the capital labor ralio. What is the economic significance of such a relationship, if any'? Estimate the parameters of this model and interpret your results.

TABLE 7.11 -'.' ---_. ­

Greek Industrial Srctor Observation Output" Capital Labort

Capital-to-Labor Ratio

Source. I am il",dc:bled (0

(jCOr&~ K 00105 of

1961 1962

35.858 37.504

59.600 64.200

637.0 643.2

0.0936 0.0998

ChrlSlOphl't' N..·wpot1 1963 40.378 68800 651.0 0.1057 UujvC('~ity. VlJ~IOI". for (hc~

dala. 1964 1965

46.147 51.047

75.500 84.400

685.7 710.7

0.1101 0.1188

1966 53.871 91.800 724.3 0.1267 1967 56834 99.900 735.2 0.1359 1968 65.4 39 109.100 760.3 0.1435 1969 74.939 120.700 777.6 0.1552 1970 80.976 132.000 780.8 0.1691 1971 90.802 146.600 825.8 0.1775 1972 101.955 162.700 864.1 0.1883 1973 114.367 180.600 894.2 0.2020 1974 101823 197.100 891.2 0.2212 1975 107572 209.600 887.5 0.2362 1976 117.600 221.900 892.3 0.2487 1977 123.224 232.500 9301 02500 1978 130.971 243500 969.9 0.2511 1979 138.842 257.700 1006.9 0.2559 1980 135.486 274.400 1020.9 02688 1981 133.441 289.500 1017.1 0.2846 1982 130.388 301.900 1016.1 0.2971 1983 130.615 314900 10081 0.3124 1984 132.244 327700 985.1 03327 1985 137.318 339.400 977.1 0.3474 1986 137.468 349.492 1007.2 0.3470 1987 135.750 358231 1000.0 0.3582

-' -- ­ --- ­ --- ­ - ---- ­·8dliOll'lO o(Ora..:hma~ al con.nalll 1910 prices.

'Thou,:~nd'li tlf workcr1l fl('r >'f'3f

TABLE 7.12 Real Consumption Expenditure, Real Income, Real Wealth, and Real Interest Rates for the U.S., 1947-2000

Sources: C. YIi. ond quanerly andannual chaio-lype price indo;xes (1996 ~ IDD): Bmeau ofEwnomicAnalysis. u.s. Ortp3rtrncnl of Commerce (hllp:/lwww.bc3.doe.go_lbell! dnl.hlm). Nominal ::annuJI yield on J..month Treasury seculllles: Economic RetJorl of lhe Presiden., 2002. Nominal weahh = end-of·

nominal net wOrlh of holds and non profits

(from nxkral Reserve ftow or fund. data: http://www. raloroJ=crve.go_).

Chapter 7 Multiple Regression AnI/lysis: The Problem 0/Estimatiult 225

7.23. Monte Carlo experiment: Consider the following model:

Yi = 131 + f32 X2i + f3 JX J, + IIi

You are told thatf3, = 262,132 = -0.006, f3J = -2.4,0 2 =42, and IIi ~ N(0,42). Generate 10 sets of 64 observations on IIi from the given normal distribution and use the 64 observations given in Table 6.4, where Y = CM, X2 = PGNP, and X) = FLR to generate 10 sets of the estimated 13 coefficients (each set will have the three estimated parameters). Take the averages ofeach of the estimated 13 coefficients and relate them to the true values of these coefficients given above. What overall conclusion do you draw?

7.24. Table 7.12 gives data for real consumption expenditure, real income, real wealth, and real interest rates for the U.S. for the years 1947-2000. These dnta will be used again for Exercise 8.35.

a. Given the data in the table, estimate the linear consumption function using income, wealth, and interest rate. What is the fitted equation?

b. What do the estimated coefficients indicate about the variables' relationships to consumption expenditure?

Year

1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976

C

976.4 998.1

1025.3 1090.9 1107.1 1142.4 1197.2 1221.9 1310.4 1348.8 1381.8 1393.0 1470.7 1510.8 1541.2 1617.3 1684.0 1784.8 18976 2006.1 2066.2 2184.2 2264.8 2314.5 2405.2 2550.5 2675.9 2653.7 2710.9 2868.9

Yd

1035.2 1090.0 1095.6 1192.7 1227.0 1266.8 1327.5 1344.0 1433.8 1502.3 1539.5 1553.7 1623.8 1664.8 1720.0 1803.5 1871.5 2006.9 2131.0 2244.6 2340.5 2448.2 2524.3 2630.0 2745.3 2874.3 3072.3 3051.9 3108.5 3243.5

Wealth Interest Rate

5166.8 -10.351 5280.8 -4.720 5607.4 1.044 5759.5 0.407 6086.1 -5.283 6243.9 -0.277 6355.6 0.561 6797.0 -0.138 7172.2 0.262 7375.2 -0.736 7315.3 -0.261 7870.0 -0.575 8188.1 2.296 8351.8 1.511 8971.9 1.296 9091.5 1.396 94361 2.058

10003.4 2.027 10562.8 2.112 10522.0 2.020. 11312.1 1.213 12145.4 1.055 11672.3 1.732 11650.0 1.166 12312.9 -0.712 13499.9 -0.156 13081.0 1.414 11868.8 -1.043 12634.4 -3.534 13456.8 -0.657

Continued

226 Part One Single. Equation Regression Model,

TABLE 7.12 Year C Yd Wealth Interest Rate (Co/ltinued)

1977 2992.1 3360.7 13786.3 -1.190 1978 3124.7 3527.5 14450.5 0.113 1979 3203.2 36286 153400 1.704 1980 3193.0 3658.0 15965.0 2.298 1981 3236.0 3741.1 15965.0 4.704 1982 3275.5 3791. 7 16312.5 4.449 1983 3454.3 3906.9 16944.8 4.691 1984 3640.6 4207.6 17526.7 5.848 1985 38209 4347.8 19068.3 4.331 1986 3981.2 4486.6 20530.0 3.768 1987 4113.4 4582.5 21235.7 2.819 1988 4279.5 4784.1 22332.0 3.287 1989 4393.7 4906.5 23659.8 4.318 1990 4474.5 5014.2 23105.1 3.595 1991 44666 5033.0 24050.2 1.803 1992 4594.5 5189.3 24418.2 1.007 1993 4748.9 5261.3 25092.3 0.625 1994 4928.1 5397.2 25218.6 2.206 1995 5075.6 5539.1 27439.7 3.333 1996 5237.5 5677.7 29448.2 3.083 .1997 5423.9 5854.5 32664.1 3.120 1998 5683.7 6168.6 35587.0 3.584 1999 5968.4 6320.0 39591.3 3.245 2000 6257.8 6539.2 38167.7 3.576

------­ -------­ - -- ----­ - -~--NOIeJ: Year = calendar year

C:::: real consumplion c~pendilures in billions of chained 1996 dolla.r!>. Yd = r~1 ptrwlUl disposable income In billions of d1alncd 1996 dollars.

Weallh = leal we.aJlh in billions orchained 1996 dollars .

Inicrt-sil = nominal annual yield on J·monlh TreJsury ~curitics-inflalion ralC (mc'asured by the' e.Muel % change in annUAl cbained price index).

The nominal real walth vOlriable W3j created tlStnG data from Ihe fedtntl Reserve DOiud's measure of cnd.or.year nel worth fOf households and nonprofits in the Row of funds aceounlS. The price index used 10 con'Vut this nominal weallh variable 10 a reol wcallh

variable was ttl( avtrage of Ihe chained price lndClt from rhe 41h quarter or Ihe currenl yeilr and the Is! quaner or the slIbS(:quent YCilr.

7.25. ESlimaling Qualcomm slOck prices. As an example of the polynomial regression, consider data on the weekly stock prices of Qualcomm, Inc., a digital wireless telecommunications designer and manufacturer over the time period of 1995 to 2000. The full data can be found on the textbook's website in Table 7.13. During the late 1990, teclmological stocks were particularly profitable, but what type of regression model will best fit these data? Figure 7.4 shows a basic plot of the data for those years.

Tills plot does seem to resemble an elongated S Cillve; there seems to be a slight increase in the average stock price, but then the rate increases dramatically toward the far right side of the graph. As the demand for more specialized phones dramatically increased and the technology boom got under way, the stock price followed suit and increased at a much faster rate.

a. Estimate a linear model to predict the closing siock price based on lime. Does this model seem to fit the data well?

h. Now estimate a squared model by using both lime and lime-squared. Is this a bet­ter fit than in (a)?

FIGURE 7.4 Qualcomm stock prices over time.

Chapter 7 Mulliple Regression dnolysis: The PlVblem ofEstimatIon 227

Price

500

450

400

350

300

250

200

ISO

100

50

o __ I! I I '.J I [ I I ! I I I I I I I I J. I I I I I J L-L...L....L.L ~~~~~~~~~~~~~~~~~~~~oooo~oooooooo~~~~~~

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ M~~~~~~~~O~~VOV-~-~-~M~MOONOON~~~7~

~~~~~ON~~~~~~~~~~~ON~~~~~~~~~~ooo~ N~~OO---M~~ -NN7~OO---~~~ -NNV~ -­

Date

c. Finally, fit the following cubic or third-degree polynomial:

Yj = 130 + 13,Xj + 132 X: + 133xl + 1Ij

where Y = stock price and X = time. Which model seems to be the best estimator for the stock prices?

7A.l Derivation of OLS Estimators Given in Equations .(7.4.3) to (7.4.5)

Differentiating the equation

(7.4.2)

partially with respect to the three unknowns and setting the resulting equations to zero, we obtain

Simplifying these, we obtain Eqs. (7.4.3) to (7.4.5).