318

TheestimatedparametersinaIineargressionmodeIminimizethesumofthesquared

regssionesiduaIs

Thestandarderrorofestimatemeasureshowwel>theregressionmode>fitsthedata.IftheSEEissmaIlthemode>fitswell

ThecoefficientofdeterminationmeasuresthefiactionofthetotaIvaiationinthe

dependentvariablethatisexp!ainedbytheindependentvariab>eInalinearregressionwithoneindependentvariabIe,thesimp>estwaytocomputethecoefficientofdetenninationistosquarethecorrelationofthedependentandindependentvaIiables.

Tocalcu>ateaconfidenceinteIaIfbranestimatedregressioncoefficient,wemustknowthestandardeToroftheestimatedcoefficientandthecriticalvauefbthedistribution

atthechosenleveofsignificance,.

TotestwhetherthepopulationvaIueofaregressioncoefficientbl,isequaltoapaIticular

hypothesizedvalue,Bl,wemustknowtheestimatedcoefficient,b,thestandardeorof

theestimatedcoefficient,Iandthecriticalvaluefbrthedistributionatthechosen

!-lleveofsignificance,.Theteststatisticfbrthishypothesisisabsoutevalueofthisstatisticisgreaterthan,thenwereectthenullhypothesisthatbl=Bl.

Intheregssionmodeb0blX8j,ifweknowtheestimatedparameters,

bI,fbranyvalueoftheindependentvaIiable,KthenthepredictedvalueoftheA

dependentvariableyisb0b!..

A

b0and

ThepredictioninteIa>fbraregressionequationfbrapaIticu>arpredictedvalueofthe

dependentvariableis.VwheeVisthesquaerootoftheestlmatedvarianceofthepi.cdictionerrorand/cisthecriticaIleve>fbrthe/-statisticatthechosensignificanceevelThiscomputatlonspecifiesalpeIcentconfidenceintervalForexampIe,if=0.05,thenthiscomputationyieldsag5percentconfidenceinteIval.

PacticePobems

PncticePIoblemsandSolutions:l-l4takenfiomOJmMboves!eSecondEdition,byRichmdA.DcFuSco,CFADcnnisW.McLcavcy,CFAJcraldE.Pinto,CFAandDavid.ERlmklcCFA.CoIIighl2001byCFAInSUlu

|

XyLLL

l245

777772242

120244l0l4l245

I

2.Usethedatasamp>ebelowtoanswerthefbIowingquestions

)(}p

!

/(,

00424>

>--

7=

7

-}

/(/\

,=7=0

8] 5

zzH

-7) 568

ACalcu!atethesampIemean,variance,andstandarddeviationfbrX

BCacuIatethesampIemeanvanance,andstandarddeviationfbrI

CCaIcuIatethesampIecovanancebetweenXandy

DCaIcuIatethesamplecorrelationbetweenXandXI

3StatisticsfbrthreevaIiablesaregivenbelow.Xisthemonthlyretumfbralarge-stockindex,isthemonth!yreturnfbrasmaIstockindex,andZisthemonthlyreturnfbracoporatebondindex.Theeare60observations.

7

z.. z()(y)76,.08i.i-.-720\=0760I j=l-

,

)z-I24.309.i.Zz()=2J}=I0]7j=I /=l

Z(Z)-Z)-=l83.073}}- zZ-l7Iz=0.686jl /=I

< >

ACaIcuIatethesamplevananceandstandarddeviationfbrKKandZ

B.CaIculatethesampIecovanancebetweenXandKXandZandandZ

CCaIcuIatethesampIecoeationbetweenXandy,XandZandyandZ

1.HomesaIesandinterestratesshouIdbenegativelyre>ated.Thefbl>owingtablegivesthenumberofannua>unitsalesfbrPackardHomesandmortgageratesfbrfburrecentyearsCa>culatethesamplecon.e>ationbetweensaIesandmoItgagerates.

31g

YeUnitS!eSInteeStRte

50

70

80

60

8.0

7.0

6.0

7.0

2000

200l

2002

2003

5ThefbllowingtabeshowsthesampIecorreIationsbetweenthemonthlyretumsfbrfburdiffbrentmumalflmdsandtheS&P500.Thecorrelationsarebasedon36monthlyobseIvations.Theflmdsareasfbllows:

Fundl

Fund2

Fund3

Fund4

S&P500

Large-capfimd

Mid-capfhnd

LaIge-capvaluefimd

Emergingmarketsfimd

USdomestlcstockindex

Fnd1FndZFnd3FundSP500

Fundl

Fund2

Fund3.

Fund4

S&P500

I

0g231

0.477l

0.7lll

0.8277

l

0.4l56

0.7238

08223

l

0.3l02

0.57gl

l

0.75l5l

TestthenuI>hypothesisthateachofthesecorrelations,individua>>y,isequa>tozeroagainstthealtemativehypothesisthatitisnotequaltozerUsea5percentsignificancelevel

6BouvierCo.isaCanadiancompanythatselPacificRimcustomersBouvier,ssaIesareveIysensitivetoexchangerates.Thefbllowingtab>eshowsrecentannualsales(inmillionsofCanadiandollars)andtheaverageexchangeIatefbrtheyeaexpssedastheunitsoffbreigncurIcncyneededtobuyoneCanadiandoIlar)

YeExChngeRateSaeSl2345

0.10

036

042

03l

0.33

20

25

l6

30

35

3Z0

6 03 30

A.CalculatethesamplemeanandstadarddeviationfbrXtheexchangerateand(sa>es).

BCalculatethesamplecovaIiancebetweentheexchangerateandsales

CCalcuIatethesamplecocationbetweentheexchangeateandsaes

DCalculatetheinterceptandcoefficientfbranestimatedlineaIregressionwiththeexchangerateastheindependentvaIiableandsalesasthedependentvariable.

7JUIieMoonisanenergyanalystexaminingelectricity,oil,andnaturalgasconsumptionindiffbrentregionsoverdiffbrentseasons.SheIHnaregressionexplainingthevaIiationinenergyconsumptionasafilnctionoftemperatureThetotalvaIiationofthedependentvariab>ewasl40.58,theexplainedvariationwas60.l6,andtheunexp>ainedvariationwas80.42Shehad60monthlyObservations.

AComputethecoefficientofdetennination.

BWhatwasthesamplecorrelationbetweenenergyconsumptionandtemperature?

C.ComputethestandardeIToroftheestlmateofMoon,sregressionmodel

DComputethesamplestandarddeviationofmonthlyenergyconsumption.

8Youareexaminingtheresu!tsofaregressionestimationthatattemptstoexplaintheunitsaesgIowthofabusinessyouaresearching.TheanalysisofvaIianceoutputfbrtheregressionisgiveninthetablebelow.Theregressionwasbasedonfiveobservations5).

ANOVA

-`dfSSMSSF.SigIifknnceF

RegressionResidIIal

Total

88.036.6670.00g04

2.4

I3

88.0

72

95.2

AHowmanyindePendentvaIiablesaintheregressiontowhichtheANOVAefbrs?|

BDefineTotalSS

CCalcuIatethesamplevarianceofthedependentvariableusinginfbnnationintheabovetable.

DDefineRegressionSSandexplainhowitsvalueof88isobtainedintennsofotherquantitiesreportedintheabovetable

EWhathypothesisdoestheFLstatistictest?

321

FExpIainhowthevaIueofthestatisticof36.667isobtainedinteImsofother0

quantitiesreportedintheabovetable

GIsthetestsignificantatthe5percentsignificancelevel?

9Thefirsttab!ebeIowcontainstheregressionresultsfbraregressionwithmonthlyretumsonalarge-capmutuaIfUndasthedependentvariab>eandmonth>yretumsonamarketindexastheindependentvariable.TheanalysisispeIfbrmedusingon>yl2monthlyretumsinpeentThesecondtableprovidessummarystatisticsfbrthedependentandindependentvariables.

AWhatisthepredictedretumonthe>arge-capmutualfimdfbrama%

38.5l

l2

l.56%

6.4l%

4l13

l2

I0IndusnyautomobiIesaIesshouIdbereIatedtoconsumersentlment.ThefbIIowingtabIeprovidesaregressionanalysisinwhichsalesofautomobilesand>ighttmcks(inmil>ionsofvehicles)areestimatedasafimctionofaconsumersent>mentindex.

Regessi0Stathstics

Mu20

C0ecientsSmndadEPr0rStatiSticpVfMue

322

Intercept

SIopecoefficient

607>

00g25>

0.58432

0.00636

I0.38g

>4.54>

00

Portheindependentvaesandag5percentpredictionintervaesifthesentimentindexhasavalueofg0

BFindtheexpectedsaIesanda95percentpredictionintervalfbrsalesifthesentimentindexhasavalueofl00.

|

11Usethefbllowinginfbnnationtocreatearegressionmodel:

46

4

8

--

7

)

)c

Ib

#)(

0

b

-

-

h

/(

z

z

6 0

4

--!7=)

I

-y

D

XI(

(

zzH

0

I8487

4

.

--

Bq

!

-

9

DH

||

ACaculatethesamplemeanvananceandstandarddeviationfbrXandfbr

BCalculatethesamplecovananceandthecoeationbetweenXandX

C.Calculateb0andbIfbraregressionofthefbrm

ib0bI.

Fortheremainingthepartsofthisquestl9n,assethatthecaculationsshownabovealreadyincoIporatethecoectvaluesfbrb0andbl.

DFindthetota

FFindthestandarderroroftheestimate

1ZThebid-askspreadfbrstocksdependsonthemarketIiquidityfbrstocks.Onemeasureofliquidityisastock,stIHdingvolume.Belowaretheresu!tsofaregressionanaIysisusingthebidaskspreadattheendof2002fbrasampleofl,8lgNASDAQlistedstocksasthedependentvariableandthenaturallogoftradingvo>umeduringDecember2002astheindependentvariable.SeveraIitemsintheregressionoutputhavebeenintentionallyomitted.UsetheIcportedinfbrmationtofiIlinthemissingvaIues.

RegressI0nStthstics

MultipleR

R-squaredStandard

enor

Observanons

X2

X1

X3

l8l9

ShgnifhCnCeFANOVA df SS MSS F

RegressionResidual

Total

l4246

45.8g3

60.l39

X5

X6

X

X7

x8

X9 0

Standard Uppe>C0eCietSErr0FStatiStiCPValUeLOwer9595

Imemept

S>opecoefficient

0.5585l

0075

0.0l870729.85540

0.00l842X10

000.52l82

X11

0.5952

X12

< >

l3AneconomistcollectedthemonthlyretumsfbrKDL,spoItfblioandadiversifiedstockindex.Thedataco>lectedareshownbe>ow:

M0nthP0rtMi0Return(%)IndexRetum(%)l23456

l.ll

72.l0

5.l2

l.0l

l.72

406

-0.5g

64g0

48l

l.68

4.g7

-2.06

Theeconomistcalculatedthecorrelationbetweenthetworetumsandfbundittobe

0.gg6.ThegressionresuItswiththeKDLretumasthedependentvariabeandtheindexretumastheindependentva>iablearegivenasfbllows:

324

RegressI0nStHtbsticS

Mu>tipIeR

R-squaredStandaderror

Observations

0.g96

0.gg2

286l

6

SignificnnceFdf SS MSS FANOVA

RegressionResidu2!

Total

4l0l.62

3Z76

4l34.38

04l0l.62500.79

8.l9

l45

Stnd2d

C0eCientS ErT0r SttisticpVe

1274

0.0477

015l8

0

2.252

l06g

l768

2237g

Intercept

S>ope

|Whenreviewingtheresu!ts,AndreaFusiliersuspectedthattheywereunreliable.ShefbundthattheretumsfbrMonth2shouldhavebeen72lpercentand6.4gpercentinsteadofthelaIgevaluesshowninthefirsttable.Coectingthesevaluesresultedinarevisedcorrelationof0.824andtherevisedregressionresultsshownasfbllows:

|

RegressInStatiSticS

Mu>tip!eR

R-squaredStandarderror

ObseIvations

0.824

0.678

2062

6

SignificanceFSS MSS FdfANOVA

0011RegressionResidual

TotaI

358g

l70l

529l

35.89

425

844l45

Standard

C0eCientS Eror Stt!sticpVa!e

0.060

001q

0.863

0.2l4

2.5g7

2905

2.242

0.623

Intercept

SIope

Exp>ainhowthebaddataaffbctedtheresu>ts.

325

MDietPartneschargesitscIientsasmaIImanagementfbepIusapercentageofgainswheneverportfblioretumsarepositiveC>eoSmithbelievesthatstrongincentivesfbrpotfbiomanagerspoducesupenoretumsfbcIientslnordertodemonstatethisSmithrunsaregressionwiththeDietPartners,poblioremrninpercentasthedependentvariabIeanditsmanagementfbeinpecentastheindependentvariabIe.Theestlmatedregressionfbra60-monthperiodis

RETURN-302l7.062F

-7.28lgS

Thecacuatedvaluesaregiveninparenthesesbelowtheinteceptandslopecoefficients.Thecoefficientofdetenninationfbrtheregressionmode>is0.794

AWhatisthepredictedRETURNifFEEis0percent?IfFEEislpeIcent?

BUsingatwo-tailedtest,istherelationshipbetweenRETURNandFEEsignificantatthe5percentleve>?

C.WouldSmithbejustifiedinconcludingthathighfbesaregoodfbrclients?

ThefowinginfmationeIatestoQuestions1520

KennethMcCoin,CFAisafhirlytoughinteIviewer.Lastyear,hehandedeachjobapplicantasheetofpaperwiththeinfbrmationinthefb>>owingtabIe,andhethenaskedseveralquestionsaboutregressionanalysis.SomeofMcCoin,squestions,aIongwithasamp>eoftheanswershereceivedtoeach,aregivenbeIow.McCointoIdtheappIicantsthattheindependentvaIiab>eistheratioofnetincometosalesfbrrestaurantswithamarketcapofmoIBthan$l00millionandthedependentvariab>eistheratioofcashf>owfiomoperationstosaIesfbrthoserestaurants.WhichofthechoicesprovidedisthebestanswetoeachofMcCoin,squestions?

Regressi0nStdStics

MultipleR

R-squaredStandarderror

ObseIvat>ons

0.8623

07436

0.02l3

24

SigiCnCeFANOVA df SS MSS F

RegressionResidlIaI

TotaI

0.02g

0.0l0

0.040

l

22

23

0.02900063.8l

0000455

0

Standrd

Err0HC0eciets StatisticVaIe

lntercept

SIope

0.077

0.826

0.007

0l03

ll.328

7988

00

326

l5WhatisthevaIueofthecoefficientofdetemination?

A0826l

B07436.

C0.8623

16Supposethatyoude>etedseveraIoftheobservationsthathadsmaIresiduaIva>ues.Ifyoueestimatedtheregressionequationusingthisreducedsample,whatwouldIikeyhappentothestandarderroroftheestimateandtheRsquared?

StandaIdErroroftheEstimate Squared

Decrease

Increase

Decrease

ABCDecrease

Decrease

Increase

17WhatisthecorrelationbetweenXand

A07436.

B0.7436.

C0.8623.

18WheredidthefWalueintheANOVAtablecomefom?

I

AYouookuptheRvalueinatable.TheFdependsonthenumeratoranddenominatordegreesofficedom.

B.DividetheMeanSquare,,fbrtheIgressionbytheMeanSquare,,oftheresiduals.

CTheFWalueisequaItothereciproca>ofthe/-va>uefbrtheslopecoefficient

19Iftheratioofnetincometosalebfbrarestaurantis5percent,whatisthepredictedratioofcashf>owfibmoperationstosales?

A.0007+0.l03(5.0)=0524

B.0.077=0.826(5.0)=-4054.

C0.077+0.826(5.0)=4Z07.

Z0Istherelationshipbetweentheratioofcashfowtooperationsandthemtioofnetincometosalessignificantatthe5pe

328

ThefOowinginfmationeIatestoQuestions2126

HowardGoIub,CFA,isprepanngtowritearesearchreportonSteI>arEneIgyCorp.commonstockOneoftheworld,slaIgestcompanies,StellarisinthebusinessofrefiningandmaIketingoil.Aspartofhisana>ysis,GoIubwantstoevaluatethesensitivityofthestock,sretumstovanouseconomicfactors.ForexampIe,acIientrecentlyaskedGolubwhetherthepnceofSte>IarEneIgyCorporationstockhastendedtoIisefbIowingincreasesinretai>energypnces.GoIubbelievestheassociationbetweenthetwovanabIestobeIegativebuthedoesnotknowthestIngthoftheassociation

GoIubdirectshisassistant,JiI>Batten,tosmdythereIationshipsbetweenStelIarmonth!ycommonstockretumsversusthepreviousmonth,spercentchangeintheUSConsumerPriceIndexfbrEnegyCPNG,andStelarmonthlycommonstockretumsversusthepreviousmonthspercentchangeintheUSProducerPriceIndexfbrCrudeEneIgyMateIia>s(PPICEM).GolubwantsBattentorunbothacoTeationandalinearregressionanalysis.Insponse,Battencompi>esthesummaIystatisticsshowninExhibit1fbrthe248monthsbetweenJanuaryl980andAugust2000.Allofthedataareindecima!fbnn,where0.0lindicatesalpercentretum.BattenalsorunsagressionanalysisusingStelarmonthlyreturnsasthedependentvaabeandthemonthIychangeinCPENGastheindependentvariabIe.Exhibit2displaystheresultsofthisIgressionmodel

ExhbitlDesciptiveSmtistdcs

LaggedM0nthlyChangeM0yRetu>PnSteIar

C0mm0nSt0Ck CPIENGPPICEM

Mean

StandardDeviation

CovaIiance,StellarvsCPIENG

CovaIiance,Stellarvs.PPICEM

CovaIianceCPIENGvsPPICEM

CorrelationSte>>arvs.CPlENG

0.0l23

007l7

0.000l7

0.00048

0.00044

0.l452

EXhibhtZRegeSSi0AnySiSwithCPIENG

RegeSSi0nStm0StiCS

Mu>tipIeR

R-squared

Standarderroroftheestimate

0.M52

0.02ll

0.07l0

0.0023

00l60

00042

00534

Regessi0StatistsObservations 248

C0ecientsStdardEr0Statdstic

Intercept

S>opecoefficient

0.0l38

-0.6486

0.0046

0.28l8

30275

-2.30l4|

Z1BattenwantstodetenninewhetherthesamplecoIrelationbetweentheStel>arandCPIENGvariab>es(-0.l452)isstatistical>ysignificant.ThecIiticaIvaIuefbrtheteststatisticatthe0.05Ieve!ofsignificanceisappIoximate>yl.g6.BattenshouIdconcludethatthestatisticalre>ationshipbetweenStellarandCPINGis:

Asignificant,becausethecalculatedteststatistichasalowerabsolutevaIuethanthecriticaIva>uefbrtheteststatistic.

Bsignificant,becausethecalculatedteststatistichasahigherabsoluteva>uethanthecriticaIvaluefbrtheteststatistic.

Cnotsignificant,becausethecalculatedteststatistichasahigherabsolutevaluethanthecIiticalvaluefbrtheteststatistic.I

ZZDidBatten,sregssionanalyzecrosssectionalortimeseIiesdataandwhatwastheexpectedvaueoftheenortennfomthatgression?

ExpectedValueofErIorTermDataType

Time-senes

Time-senes

Cross-sectional

ABC00

Z3BasedontheregressionwhichuseddataindecimalfbIm,iftheCPIENGcsesbyl0percent,whatistheexpectedIturnonStellarcommonstockdulingthenextperiod?

A0.0073(0.73percent)

B00l38(l38percent).

C0.0203(2.03peain21lpercentofthevaIiabilityinCPIENG

BSte>>ar,sretumsexplainl452percentofthevariabi!ityinCPIENG

CChangesinCPINGexp>ain2llpercentofthevaIiabilityinSte>lar,sretums

25ForBatten,sregressionmode,thestandardeoroftheest1mateshowsthatthestandarddeviationoE

329

330

AtheresidualsfiDmtheegessionis0070.

Bvaluesestimatedomtheregressionis0.070.

CStellar,sobseIvedcommonstockretumsis0.07l0

26FortheanalysismnbyBatten,whichofthefbowingisancoecconclusionfiDmtheregssionoutput?

ATheestimatedintemeptcoefficientfomBattensregressionisstatisticaIIysignificantatthe0.05>eve>.

BInthemonthaffertheCPIENGdeclinesSteIarscommonstockisexpectedtoexhibitapositiveretum.

CViewedincombination,theslopeandinterceptcoefficientsfibmBatten,sregressionarenotstatisticaIlysignificantatthe0.05level

SIutins

1ThethreevariablesbthroughqhaveazerocorreationwithXihaszerocovariancewithbutbecauseYihasnovariation,itsconclationwithXisundefined.Noticethat

althoughLandIMreclearlynon>inearlyrelatedtoX(decreasingandthenincreasingasthevalueofXincreases,theiroverallIinearrelationshipwithXiszero.VaIiablehasaconBIationof10withX

2

AThesamplemean,vanance,andstandarddeviationofXare

.YZ.i220l20/=I

szr2IVI0/=I

s6BThesamplemean,varianceandstandarddeviationofyare

.Z385II5/=I

;Zl-I00l

I0

|

CThesamplecovariancebetweenXandyis=

Cov I)( z.i./=>

{ .Cov 56.8=-0809 -

6.633x>0s83$>s}p

{

3

AThesamp>evaIiancesandstandarddeviationsare

iX!2II7wI,/=I

.3.6l0

;7-lIW2I/=I

4.59I

Z-2I3,,3/=I

Zl.762

BThesampIecovariancesare

331

l

Covz!-.7-l7205J55,Z2l2/=l

CovX.ZZI.ZZ-I2I007,,9Il

Cov.ZZ-7Zi--lI7l.8I05,29I2/=I

< >

CThesampIecolTeIationsare

Cov,22I2

(36l0)(4.59>)=07J7\.).-

.

Cov,jZ3.95

(3.6I0)(I.762)=0.6l5Z/.-

$\.$Z

Cov(>.,Z)29I2(1.59I)(l.762)=0.360/}Z= s>sZ

Samplemeansalesare50708060426065.

Sampemeaninterestrateis807.06.07.0280470.

SamplevaIianceofsalesis5065270652806526065235003l667.

Samp>estandarddeviationofsalesisthesquarerootofthevarianceorl29l

SamplevaIianceofinterestratesis8727726727723230.666667.

Samplestandarddeviationofinterestratesisthesquarerootofthisresu!t,or0.8l65

Sampecovariancebetweensa!esandinterestratesis5065877065778065676065773303l0

Samplecon.elationisthecovariancedividedbytheproductofthestandarddeviations:

Cov,. -l00.987/.=

)56}8

0()>9

7

}(

5Thecriticalvauefbr234dfmsinga5percentsignificancelevelandatwotailedtest,is2032.First,takethesma>>estcoITelationinthetable,thecorrelationbetweenFund3andFund1,andseeifitissignificantIydiffbrentfiomzero.Itscalcu>ated/-vaIueis

332

i.03I02z=l.003/=|

--

71~

7

u

I]

0I

Thiscoelationisnotsignificant!ydifferentfiomzero.Ifwetakethenextlowestcorreation,betweenFund2andFund3thiscoelationof0.4l56hasacalculated

vaIueof2664SothiscoreIationissignificantydiffeentfDmzeroatthe5percentleveIofsignificanceAlIoftheothercorre>ationsinthetabIe(besidesthe0.3l02)aregreaterthan0.4l56,sotheytooasignificantlydiffbrentfiomzero.

|

6Thefbllowingtableprovidessevera>usefhlcalculations:

=

Pp>()-

X/;TTi:;

AThesamplemeanandstandarddeviationoftheexchangerateare

|

m

0

6//6>

7---/

/

V

--

l

and

0.0226s=-

Thesamplemeanandstandarddeviationofsalesare

=0

6

I

/

-}

and

FHy= =7.07>>333

BThesamplecovariancebetweentheexchalgeateandsaIesis

Cov.z.-I-I.3,50278/=l

CThesampIecoTelatiobetweentheexchangeateandsaIesis

...Cov -0278==0927/.=

--

027.07IINr$],

D.WewanttoestimatearegressionequationofthefbrmZ=b0+blX}+ei.Notingthatdivisionbylinthenumeratorcanceswithdivisionbylinthedenominatorintheexpressionfbrtheslopecoefficient,theestimatesofthes>opecoefficientandtheinteeptare

zyiYl #4

45|--

090

0

l

0

b!=

7

zXi.I

D

and

0I.726-I40M265s.68I.6

Sotheregressionequationis816l5444X

7

AThecoefficientofdetenninationis

Explainedvaiation0.279otaIvaiation 058

BForalinearregressionwithoneindependentvariabIe,theabsolutevalueofcorrelationbetweentheindependentvariableandthedependentvariableequalsthe

;l;:::iIp.coefficient.)

CThestandarderooftheestimateis

33

7=0D00

l|X

I/2

.lI--

I78

,

(}}- z-l!

ToaIvaia!io! 72837

--

8l5

0

0

H6-

--

I

VZZ7Thesamplestandarddeviationis !4

8

AThedegreesoffiFeedomfbrtheregrCssionisthenumberofslopeparametersintheregression,whichisthesameasthenumberofindependentvariablesintheregIcssionBecauseregressiondf=l,weconc!udethatthereisoneindependentvariableintheregression

BTota!SSisthesumofthesquaeddeviationsofthedependentvariableYaboutitsmean

|

| CThesamplevananceofthedependentvariableisthetotalSSdividedbyitsdegreesoffeedoml5l4asgiven.ThusthesamplevarlanceofthedependentvariabIeis95.223.8.

|

DTheReqssionSSisthepatoftotalsumofsquaresexplainedbytheregression.RegressionSSequalsthesumofthesquareddiffbncesbetweenpredictedvalues

Z1n!thoftheyandthesamplemeanof}.: Zj=I

valuesinthetable,RegressionSSisequaltoTotalSSminusResidualSS:952-7.2=88.

|

ETheRstatistictestswhetherathesIopecoefficientsinaIinearegssionare

equalto0.

FThecalculatedvalueofFinthetableisequaltotheRegressionMSSdividedbytheResidua>MSS:88/2.4=36.667.

GYes.Thesignificanceof000904giveninthetableisthep-valueofthetest(thesmallestlevelatwhichwecanrectthenullhypothesisThisvaueof0.00904isessthanthespecifiedsignificancelevelof0.05sowerectthenulhypothesis.Theregressionequationhassignif>cantexplanatoIypower.

335

A.ForthelaIge-capfUnd,thepredictedrateofretum,>,is

0lX-02870802800620

BJhimadvmanceofhpredMineofMveis

H)

=$=

I..2432

7

ThestandaddeviationofthepredictionerIDristhesquaootofthisnumber457.Forl0degreesoffi.eedom,thecIitical/-valueis2228.A95percent

predictionintervalis.cq,ordl292.2284.57or6.l2gl0l8Z

Pob4053y63l0.g5

I0

AForasentimentindexofg0,pIcdictedautosales,,are

0;!..6.07l00925I90I4.3g7aboutl4miion

veesThlmcdvaancefthcpdMionfKgiveis

|+

+

llllL

7s--

J .v2)

I:

+l-

+7

7

D

!8

0-=

(90-,l0083)2 )860

73I()l07

!(

Thestandarddeviationofthepredictionerroristhesquarerootofthisnumber:0.8l67.Forll8degreesoffi.eedomanda0.05leve!ofsignificance,thecritical/-vaIueisapproximateIy1g8.Theg5percentpredictionintervaIfbrX=g0is

.S,orl4397l9808l67orl439716l7.

Pmb(l2.780

>=b0+bI.\=6.07l+00925l(I00)=I5.322

ThedvnccofthcpcdicionSgivcis

H; 7

v=J-

+I-

+I

IIIIIL

,

=`

7

j

l8

0.-- l=0V

7

\0J

890

000l(

(l20-!)(!373068)

Thestandarddeviationofthepdictionerroristhesquarerootofthisnumber:0.8l85g.Forll8degreesofeedom,thecriticalva!ueisapproximatelyl.98A

g5peentpredictionintervaIwouIdbecorl5.322l.g808l85gorl53221621

l

Prob(l3.70l

ThesamplecoeIationbetweenXandis

Cov./=

p

I05=0.90]7

2.73864.226

CThecoefficientsfbrtheIcgressioIequationare

7i/=l !.and

60

bl=

)z-.2I

o

60y-6I.YI6-I403.

Sotheregressionequationis.3Mk

,

ZI14andtheunexpMnedvadatiomsDThetotalvariationis/=I

2

Z )l-60-2dsxlinI/=>

=ll7.6

.Sotheexplainedvariationisl41-26.4

EThecoefficientofvariation,theR-squaredis

ExplainedvaiationllZ0.8I067TotaIvaiation I4

FThestandarderroroftheestimateis

7=//I|7=

\

=

DR

-

-

0

-

PDP

}

|SE

Z/II4

7.=

62

,

(--

I.92

12.TheR-squared(X1)isExplainedvaIiation/TotaIvariation=I4246/60.l39=02369.TheMultipIeR(X2)isthecorrelationbetweenthetwovariableswhichisthesquarerootof

338

|

theRsquaredo867Thestandarderror3isthesquarerootx ividedby2,whichis

=0.l580

TheTotaIdf;X4,isthesampIesizeminusl,orll,8l9ll,8l8.TheRegssiondfX5isequaItothenumberofindependentvariabes,whichis1TheResiduadfX6isthedifferencebetweentheTotaIdfandRegssiondf;whichisalsoklwheeisthesampIesizel,89andisthenumberofindependentvariablesl.X6isl,8lglll,8l7.MSSisthemeansquare,whichisthesumofsquaresdividedbythedegreesoffiedom.ForX7,theMSSregressionisl1.246ll4.246.ForX8,theMSSresidua!is45.893/l,8l7=0025258.TheF(X,)istestingthehypothesisthattheregressioncoefficientequalszero,anditisequaItoMSSregression/MSSresidua!orl4.216/0.025258=5640ZThisFhasldfinthenumemtorandl,8I7dfinthedenominator.ThisvaIuefbrFisextremeylaIge,andtheprObabiityofanFthislageispractical>yzero.

X10thecalculated/-vaIuefbrthesIopecoefficientisthecoefficientdividedbyitsstandarderror:0.043750.00l8423.75.Thisisanextremelylagen3ganveLwithapIobabilityofpracticaIyzero.NoticethatthesquarerootoftheFisequaltothefbrareg>cssionwithoneindependentvariable.)Finally,X11andX12aIctheupperandlowerboundsfbra95percentconfidenceinteIaIfbrthes>opecoefficient.Thecritical/fbra

twotaiIedtestatthe5pecentsignificanceIevwithl,8l7degeesoffi.eedomis

approximatcIyl.96.TheIoweboundX11isbI-0.04375-196000l842

=-004736.Theupperbound,X1ZisbI+/c$$!=-0.04375+l.96(0.00l842)=-0.040l4.

l3TheMonth2datapointisanoutIier,yingfhrawayfiomtheotherdatavalues.Because

thisoutlierwascausedbyadataentIyerrorcorrectingtheoutlierimpIovestheva>idityandIdiabilityoftheregressionInthiscase,thetruecorIclationisreducedfTom0.gg6to0824TherevisedR-squaIcdissubstantiallylower(0.678versus0.992).Thesignificanceoftheregressionisasolower,ascanbeseeninthedeclineofthevaluefiom500.79to814andthedec>ineinthe/-statisticoftheslopecoeffjcientfiom2237gto2.g05.

M

ThetotalsumofsquaresandregressionsumofsquaresweIcgreatlyexaggeratedintheincorIctanaIysis.WiththecoIection,thesopecoefficientchangesfiPom106gto0.623.Thischangeisimportant.Whentheindexmovesupordown,theoriginalmodelindicatesthattheportfblioreturngoesupordownby106gtimesasmuch,whiletherevisedmode>indicatesthatthepoiableinc>udingorexc

Becausetheca>cuIated/exceedsthecritical/,wemayconcludethatthecoefficientofFEEisnotequaItozeroandthatthere!ationshipbetweenRETURNandFEEissignificant.

CSmith,sanalysisisinadequatetoconcludethathighfbesaregoodClearly,highretumscausehighfbes(becauseofthecompensatloncontractthatDietPaItnershaswithitsc>ients)TheregressionmayberecognizingthisrelationshipUnfbrtunateIy,thereversemaynotbetIue-thatfbescauseretums.AsananaIogy,assumethatincometaxesareaftmctionofincomeAregIssionofincomeasafimctionofincometaxeswouldfindastrongpositivereIationship.Doesthismeanthattaxescauseincome,ortheIBverse?Smith,sexperimentistoosimplistictoaddresstheissueofwhetheraparticuIacompensationcontractisgoodorbadfbrclientretums.

15Biscorrect.ThecoefficientofdeterminationisthesameasR-squared

16.CiscoectDeletingobservationswithsmaIlresiduaIswidegradethestrengthoftheregssion,resuItinginanceinthestandarderrorandaceinRsquared

17Ciscorrect.ForaregressionwithoneindependentvariabIe,thecorIdationisthesameastheMultipleRwiththesignofthes>opecoefficient.BecausethesIopecoefficientispositivethecorre>ationis08623.

I8Biscorrect.ThisanswerdescribesthecaculationoftheRstatistic..

19CiscoIect.Tomakeapredictionusingtheregressionmode,multiplythesopecoefficientbythefbrecastoftheindependentvariabIeandaddtheresulttotheintercept

20Ciscorrect.Thep-valueref>ectsthestrengthoftherelationshipbetweenthetwovariablesInthiscasethep-valueislessthan005,andthustheregressionoftheratioofcashf>owfi.omoperat1onstosa>esontheratioofnetincometosalesissignificantatthe5percentlevel.

21Biscorrectbecausethecalcu>atedteststatisticis

.7--/=

=-2.30>7

JmJrBecausetheabsolutevaIueof2307isgeatethanIg6,thecorrelationcoefEcient

isstatistical>ysignificant.Foraregressionwithoneindependentvariab>ethe/-vaIue(andsignificancefbtbesopecoefficientwhichis230shouIdequaItbevaIueandsignificanceofthecoelationcoefficient.ThesIightdiffbncebetweenthesetwo-valuesiscausedbyroundingerror.

22Aiscon.ectbecausethedataaretlmesenes,andtheexPectedva>ueoftheerrorteIm,E

(e)is0

23.CiscoITect.FromtheregressionequationExpectedretum=00l38+-0.6486(-0.0l)=0.0l38+0.006486=0.0203,or2.03percent.

340

|

|

|

|

24CiscorectRsquaredisthecOefficientofdeterminationInthiscase,itshowsthat2.IpercentofthevadabilityinStel!ar,sretumsisexplainedbychangesinCPNG

25Aiscoect,becausethestandardeoroftheestimateisthestandarddeviationofthe

regIssionresiduals.

26Cisthecorrectresponse,becauseitisafalsestatement.Thes!opeandinteIeptarebothstatistica>lysignificant.

341

o

n

aeR

MutipeRegessionandIssuesinRegessionAna|ysis

byRichardA.DeFusco,CFA,DennisW.McLeavey,CFA,JeraldEPinto,CFAandDavidERunkle,CFA

gmmjMborjhveL4sSecondEdiioILbyRicmIdA.DcFusco,CFADcnsW.McLeav,CFAJemldEPinoCFA,andDavidERunkleCFA.Conghl2004byCFAISiue.

LeaningOutcmes

ThecandidateshouldbeabIeto

fbmuIateamutiperegssionequationtodescribethereationbetweenadependentvariab>eandsevera!independentvaIiables,anddetenninethestatisticalsignificanceofeachindependentvariabIe;

binterpretestimatedregressioncoefficientsandtheirpvalues;

cfbnnulateanullandanaltemativehypothesisaboutthepopulationvalueofaregressioncoefficient,calculatethevaIueoftheteststatistic,anddetenninewhethertorCiectthenullhypotheSisatagiven!eveIofsignificance;

dinterprettheresultsofhypothesistestsofgressioncoefficients;

e.calculateandinterptlaconfidenceintervalfbrthepopuationvalueofaregressioncoefficientand2apdictedvauefbrthedependentvariabIe,givenanestimatedregIssionmodeandassumedvaIuesfbrtheindependemvaIiables;

aintheassumptionsofamultipleregressionmode>;

gcaculateandinterpItthestatistic,anddescribehowitisusedinregressionanalysis;

h.distinguishbetweenandintepettheR2andadiustedR2inmuItipIeregression;

ievaIuatehowweIaregressionmodeIexplainsthedependentvaIiablebyanalyzingtheoutputoftheregressionequationandanANOVAtable;

j.fbImuateamultipleregssionequationbyusingdummyvariablestorepresentquaIitativefhctors,andinteIpretthecoefficientsandregressionresults;

312

PacticePbIems

PcicePIDblemsadSoutions:ll6akenfiumgumljMeoMmse,SecondEdiion

byRichaIdADeFusco,CFA,DenmsW.McLcavey,CFA,JemldE.PitCFA.andDavidE.Runkle,CFACoIright2004byCFAInstitu>arandequitymaIketremmsaffbctanasset,sretums.Usethenotationsbelow.

RretumontheassetinpeIiod

hretumontheSP500inperiod

AX}=changeinperiodiinthelogofatrade-weightedindexofthefbreignexchangeva>ueofUSdollaragainstthecurrenciesofabroadgroupofmajorUStradingpartners.

BYouestimatetheregressionfbrArcherDanie>sMidlandCompany(NYSEADM)Youregressitsmonth!ytumsfbrthepenodJanuaIygg0toDecember2002onS&P500Indexretumsandchangesinthe>ogofthetrade-weightedexchangevalueoftheUSdoIIar.Thetab>ebeIowshowsthecoefEcientestimatesandtheir

standaderrors.

C0ecientEstimmesfi.0mRegeSSingADM,sRetrls:M0nthlyDta,Janunry19g0DeCemberZ002

C0eChentSmndmdEr0

0.0015

0.5373

-0.5768

Intercept 00062

0.l332

0.5l2ll56

2FactSeLFedcmlReseeBankofPhiadelphia

DeterminewhetherSP500ItumsafctADM,sretums.Thendetermine

whetherchangesintheva!ueoftheUSdol>araffbctADM,sretums.Usea005significanceleve>tomakeyourdecisions

398

I

CBasedontheestimatedcoefficienton,isitcorrecttosaythatfbrapercentagepointincreaseintheretunontheSP500inperiodweexpecta0.5373percentagepointincreaseinthereturnonADM,,?

2OneofthemostimportantquestionsinfinancialeconomicsiswhatfhctorsdetenninethecrosssectionaIvaationinanasset,setums.Somehavearguedthatb0oktomarketratioandsize(marketva>ueofequity)p>ayanimportantroIe.

AWriteamultipIeregressionequationtotestwhetherbooktomarketratioandsizeexplainthecrosssectionofassetetums.UsethenotationsbeIow.

(B/M)i=book-to-marketIatiofbrasset/

RreturnonassetiinapaIticularmonth

Sizej=naturallOgofthemarketvalueofequityfbrasset/

BThetablebelowshowsthesultsofthelinearregressionfbracrosssectionof66companiesThesizeandbook-to-marketdatafbreachcompanyarefbrDecember2001TheretumdatafbreachcompanyarefbrJanuary2002.

ResultshonReg.essingReturs0ntheBO0kPt0MketRti0ndSize

C0ecientStadmdEr0r

JFactSe

Intercept

(B/M)!

Size

66

00825

-0.054l

-0.0l64

0l644

00588

0.0350

Detenninewhetherthebook-to-marketratioandsizeareeachusefhlfbrexplainingthecmss-sectionofassetretums.Usea0.05significanceleveltomakeyourdecision.

3TheissubstantialcrosssectionalvariationinthenumberoffinanciaIanalystswhofbllowacompany.Supposeyouhypothesizethatacompany,ssize(marketcap)andfinancialIisk(debt-to-equitymtios)inf>uencethenumberoffinancialanalystswhofbllowacompanyYoufbImuatethefbIowingreessionmodel:

(Analystfbllowing)i=b0+blSizei+b2(D/E)i+ei

where

Analystfblowing,thenamIHllogofl,whereiisthenumberofanalysts

fbIIowingcompanyj

39g

400

SizethenaturaIIogofthemaketcapitaIizationofcompanyinmiIIionsofdoIIas

(D/E)!=thedebt-to-equityratiofbrcompany/

InthedefinitionofAnaIystfblIowing,>isaddedtothenumberofanaIystsfbllowingacompanybecausesomecompaniesarenotfbllowedbyanyanalystsandthenaturaIlOgof0isindeteIminate.ThefbIIowingtabIegivesthecoefficientestimatesoftheaboveregIcssionmodeIfbrarandomIyse!ectedsampIeof500companiesThedataafbrtheyea200Z

C0ecdentEstjmesfh0mRegessingAnyStF00wing0SnzefmdDebtt0EquityR0

C0ecientStandardErr0StntiStic

Intecept

Size

(D/E)i500

-0.2815

0.3>9g

-0l8g5

FirsCalIhomsonFinanciaCopusa

0.l080

0.0l52

0.0620

2.6343

2>.046l

-3.0565

AConsidetwocompanies,bothofwhichhaveadebttoequityratioof0.75.Thefirstcompanyhasamarketcapitalizationof$l00million,andthesecondcompanyhasamarketcapitaIizationof$lbiI>ionBasedontheaboveestimateshowmanymoreanalystswiIfb>Iowthesecondcompanythanthefirstcompany?

BSupposethepvaluerepotedfbrtheestimatedcoefficientonE!is0.00236.Statetheinterpretationof000236.

4Inearly200l,USequitymarketplacesstaItedtradingaI>listedsharesinminimalincrementsticksof0.0ldecimaIizationAfferdecimaizationbidaskspreadsofstockstradedontheNASDAQtendedtodeclineInresponse,spreadsofNASDAQstockscross->istedontheTorontoStockExchange(TSE)tendedtodec>ineaswe>lResearchesOppenheimeandSabhewaI2003hypothesizedthatthepercentagedeclineinTSEspreadsofcross->istedstockswasrelatedtocompanysize,thepredecimalizationratioofspreadsonNASDAQtothoseontheTSEandthepercentagedeclineinNASDAQspreads.Thefbl>owingtab>egivestheregressioncoefficientestlmatesfi.omestimatingthatre>ationshipfbrasampleof74companies.Companysizeismeasuredbythenaturallogarithmofthebookva>ueofcompany,sassetsinthousandsofCanadiandoIars.

CoecientEstimntesfT0mReg.essingPecentgeDecliemTSESp.eds0nC0mpySize,PredecimIizti0nR00fNASDAOtoTSESpreadS,dPecemgeDeClme0nNASDAQSpreadS

C0eciemStmstic

Intercept

Size

atioofspreadsi

ecineinNASDAQspreadsi

74

-045

0.05

=006

0.29

=>.86

2.56

-3.77

2.42

;OppemleimeandSabheIwal2003

TheaveragecompanyinthesamplehasabookvalueofassetsofC$g00miIlionandapredecimalizationratioofspreadsequaltol.3BasedontheabovemodelwhatisthepredicteddeclineinspreadontheTSEfbracompanywiththeseaveragecharacteristics,givenalpercentdeclineinNASDAQspreads?

5Thenegectedcompanyect,,caimsthatcompaniesthataIfbllowedbyfbweranalystswilleamhigherretumsonaveragethancompaniesthatarefb>>owedbymanyanalysts.Totesttheneg>ected-companyeffbct,youhaveco>lecteddataon66companiesandthenumberofanalystsprovidingeamingseSt>matesfbreachcompany.Youdecidetoalsoincludesizeasanindependevariablemeasunngsizeastheogofthemarketvalueofthecompany,sequity,totrytodistinguishanysmallcompanyeffCctomanegectedcompanyeffbct.ThesmalI-companyeffbctasseItsthatsma>>-companystocksmayeamaveragehigherriskPaustedremmsthanlaigecompanystocks.

Thetablebeowshowstheresultsfromest1matmgthemodelRb0blSizeb2

(Numberofana>ysts)!+8ifbracross-sectionof66companies.ThesizeandnumberofanalystsfbreachcompanyafbrDecember200l.TheremmdataarefbrJanuay2002.

Resutsh0mRegressingRetums0SdzMndNumbe0fAlysts

C0efficientStandflrdEoStistic

024g5

=0.4388

08gg5

0.l556

0.0348

0.00l5

0.0388

-0.0l53

000l4

Intercept.

Sizej

Numberofanalysts

dfSSMSSANOVA

00047

00l07

0.00g4

0.6739

0.6833

Regssion

Residual

Total

2

Residstandarderror

R-squared

ObseIanons

0.l034

0.0>38

66

401

02

Sbu.cFirstCaThomsonFinancial,FacSet

AWhattestwouldyouconducttoseewhetherthetwoindependentvariabIesareoMystatisticalIyreatedtoretumsH0:blb20?

BWhatinfbrmationdoyouneedtoconducttheappropriatetest?

CDeterminewhetherthetwovaIiablesjointIyarestatisticaIyrelatedtoretumsatthe0.05significanceIevel

D.ExpIainthemeaningofaustedR2andstatewhetheraljustedR2fbrtheregressionwouldbesmallerthan,equa>toorlargerthan0.0l38

6Somedeve>opingnationsarehesitanttoopentheirequitymarketstofbreigninvestmentbecausetheyfearthatrapidinf>owsandoutf>owsoffbreignfhndswil>increasevolati>ity.InJu>ylgg3,IndiaimplementedsubstantialequitymarketrefbImsoneofwhichallowedfbreigninstitutionalinvestorsintotheIndianequitymarkets.YouwanttotestwhetherthevolatiityofretumsofstockstradedontheBombayStockExchangeSEincreasedaherJulylg93,whenfbreigninstitutionalinvestorsweIBfirstallowedtoinvestinIndia.YouhavecoIlectedmonthlytumdatafbrtheBSEfomFebIuaIylgg0toDecemberlgg7.YourdependentvaIiableisameasureofIcumvolatilityofstockstradedontheBSE;yourindependentvaIiableisadummyvariablethatis.codedliffbreigninvestmtwasal>owedduringthemonthand0otheIwise

YoubelievethatmarketItumvo!atilityacmallycBswiththeopeningupofequitymarkets.Thetablebe>owshowstheresu>tsfiFomyourregression.

Resultsfr0mDummyRegessi0fbrF0egnIvestmentinIndwhthV0mityMeaseaStheDepedentVnmble

Intecept

Dummy

95

jFHcSe.

C0eCient

0.0l33

-0.0075

StandaErmStatistic-

0.00206.535l

0.0027.2.760

AStatenuIandaltemativehypothesesfbrthes>opecoefficientofthedummyvaiablethataeconsistentwithtestingyourstatedbeiefabouttheeffbctofopeningtheequitymarketsonstockretumvolatility.

BDeterminewhetheryoucanreectthenulhypothesisatthe0.05signifiCancelevel(inaone-sidedtestofsignificance).

CAccordingtotheestimatedregressionequation,whatisthelevelofretumvolatiiWbefbreandaHerthemarket-openingevent?

|

|

7Bothreseachesandthepopulapresshavediscussedthequestionastowhichofthetwo>eadingUSpoliticaIpaIties,RepublicansorDemocrats,isbetterfbrthestockmarket.

AWritearegressionequationtotestwhetherovera>>marketretumsasmeasuredbytheannua>returnsontheS&P500Index,tendtobehigherwhentheRepub>icansortheDemocratscontroltheWhiteHouse.Usethenotationsbe>ow.

MretumontheSP500inperiod

PaithepoIiticaIpaycontoingtheWhiteHousefbraRepubIicanpresident;0fbraDemocraticpresident)inpeIiod!

BThetablebeIowshowstheIcsultsofthe>inearregressionfmmPartAusingannualdatafbrtheS&P500andadummyva>iabIefbrthepaItythatcontro>>edtheWhiteHouse.Thedataareh.omlg26to2002.

Resultsfh0mRegeSSingSP500Retuns0nDmmyVriablefbrthePatyThatC0nt0IledtheWheH0use,1926Z002

C0ecieIltStndrdEroStatistic

Intercept

Pa!ty

l

75

76

Residualstandarderror

R-squaredObservations

u2;FactSet.

00625

31287

31gl2

0.2042

0.0lg6

77

0.0625l4g870.2247

00l7

BasedonthecoefficientandstandarderrorestimatesvefytotwodecimaIpacesthestatisticfbrthecoefficientonthedummyvaiabereportedinthetable.

CDetennineatthe0.05significance>evelwhetheroveral!USequitymaanalystswhofbllowacompany.InthatpIublem,companysizeanddebt-to-equiUratiosweretheindependentvariables.YoureceiveasuggestlonthatmembershipintheS&P500Indexshouldbeaddedtothemode>asathirdindependentvariab!e;thehypothesisisthatthereisgIcaterdemandfbranalystcoveragefbrstocksinc>udedintheS&P500becauseofthewidespreaduseoftheS&P500asabenchmark

103

404

AWriteamutipIeregessionequationtotestwhetheanaIystfblIowingissystematicallyhigherfbrcompaniesincludedintheS&P500Index.Alsoinc>udecompanysizeanddebt-to-equityratiointhisequationUsethenotationsbeIow.

(Analystfb>lowing)i=naturallogof(l+Numberofanalystsfbllowingcompany/)

SizenaturallogofthemarketcapitaIizationofcompanyinmillionsofdoIIars

Eidebt-toequityratiofbrcompany

S&Pj=inclusionofcompanyiintheS&P500Index(lifincluded0ifnotincluded)

IntheabovespecificationfbranalystfbIIowing,lisaddedtothenumberofanalystsfbowingacompanybecausesomecompaniesanotfblowedbyanyanalyst,andthenamra>logof0isindeteIminate.

BStatetheappropriatenullhypothesisandaltemativehypothesisinatwosidedtestofsignificanceofthedummyvariable.

CThefb>lowingtab>egivesestimatesofthecoefficientsoftheaboveregressionmodelfbrarandomlyseectedsamp!eof500companies.Thedataafbrtheyear2002Determinewhetheryoucanectthenuhypothesisatthe0.05significance>evel(inatwo-sidedtestofsignificance).

C0eCientEStmateSh0mRegeSSigAnalyStF0ll0wing0lUSize,DebttoEqtyRtn0,ndSP500Membership,200Z

Intercept

Size

Ei

SP

500

C0ecientStaddEnP0rStmstc

-0.0075

02648

-0.l829

0.42l8

0.l2l8

0.0lgl

0.0608

00glg

-0.06l6

l3.863g

30082

4.58g8

y2JFiIstCalIhomsonFinancial,Compusat.

D.Consideracompanywithadebt-to-equitymtioof2/3andamarketcapitaIizationof$l0bil>ion.Accordingtotheestlmatedregressionequat1on,howmanyana>ystswouldfb!>owthiscompanyifitweudedintheS&P500Index,andhowmanywouldfb>>owifitweIcincludedintheindex?

EInProblem3usingthesample,weestimatedthecoefficientonthesizevaIiabIeas0.3lgg,versus0.2648intheabovegressionDiscusswhetherthereisaninconsistencyintheseresuIts.

9YoubeIievethereisaelationshipbetweenbooktomarketatiosandsubsequentretumsTheoutputfomacrosssectionaIegressionandagraphoftheactualandpredictedreIationshipbetweenthebooktomaketatioandetumareshownbeow.

ReSutSfmmRegessigRetls0ntheB00kt0MrketRati0

C0ecientStnddE0StatiStic

Intercept I20l30

/!

{

\

BoI.!I

MkIucI -92209

ANOVA

RegressionResidua>

Total

df

>

32

33

Residua>standarderror

R-squaredObservat1ons

SS

l549866

4l62.l8g5

13l7176l

ll.4048

0.035g

34

3.5464 3.3874

84454 -l09l8

MSSFSignifhCanCeF

l54g866l.lg2l0.283l

l30.0684

405

|

RetHn

(%)

RetunverSusB00kt0MaTketRti0:ACtualaHdPrediCted

8

7DY0

0

!0

8 =

8 =II0!1

)((M$ 0000 (Mb 0N(]()g0M2

B00kt0MaketRat0

AYouareconcemedwithmodeIspecificationproblemsandregressionassumptionviolations.Focusingonassumptionviolationsdiscusssymptomsofconditiona>heteroskedasticitybasedonthegaphoftheactuaIandpedictedreIationship

B.Describeindetailhowyoucou>dfbnnallytestfbrconditionaIheteroskedasticityinthisregression.

CDescribearecommendedmethodfbrcorIctingfbrconditionalheteroskedasticity

10YouaIexaminingtheeffbctsoftheJanuay200lNYSEimplementatlonofthetradingofsharesinminimalincrements(ticks)of$0.0>(decimalization).InpaIticu>aryouareanalyzingasampleof52Canadiancompaniescross-listedonboththeNYSEandtheTorontoStockExchange(TSE).Youfindthatthebid-askspreadsofthesesharesdeclineonbothexchangesaffertheNYSEdecimalizationYoumnaIinearregssionanalyzingthedec>ineinspreadsontheTSE,andfindthatthedeclineontheTSEisre>atedtocompanysize,predecimalizationratioofNYSEtoTSEspreadsanddec>ineintheNYSEspreads.TherelationshipsaIcstatisticallysignificant.Youwanttobesure,however,thattheresultsarenotinf>uencedbyconditionalheteroskedasticity.Therefbre,youregressthesquaIBdresidualsoftheregessionmodelonthetheindependentvariables.TheRzfbrthisgressionisl4.lpercent.PeIfbrmastatisticaltesttodetermineifconditionalheteroskedasticityispIcsent.

l1.YouareanalyzingifinstitutionaIinvestorssuchasmutualfhndsandpensionftmdsprefbrtoholdsharesofcompanieswithlessvo!ati!eretums.YouhavethepercentageofsharesheldbyinstitutionaIinvestorsattheendoflgg8fbrarandomsampleof750companies.Porthesecompanies,youcomputethestandarddeviationofdailyretumsduIingthat

406

.

S

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yea.ThenyouregesstheinstitutionaIhoIdingsonthestandaddeviationofetums.YoufindthattheregIssionissignificantatthe00IeveIandtheRstatisticis298.TheR2fbrthisregressionisl.7percent.AsexpectedtheregressioncoefficientofthestandarddeviationofremmsisnegativeIts/-statisticis-360whichisa>sosignificantatthe00lIeveI.Befbreconc!udingthatinstitutionsprefbrtohoIdsharesof>essvoIati>estocks,however,youwanttobesurethattheregressionresu!tsanotinfluencedbyconditionalheteroskedasticity.TheIfbreyouregressthesquaredresidualsoftheregressionmodeIonthestandarddeviationofretums.TheR2fbrthisregressionis06percent.

APerfbrmastatistica!testtodetennineifconditionalheteroskedasticityispesent.

BInviewofyouranswertoPartA,whatremedialaction,ifany,isappropriate?

I2InestimatingaIgssionbasedonmonthlyobseIvationsfiomJanuaIylg87toDecember2002inclusive,youfindthatthecoefficientontheindependentvariableispositiveandsignificantatthe0.05level.Youareconcemed,howeveI]thatthe/-statisticontheindependentvariablemaybeinf>atedbecauseofserialcorIclationbetweentheerrortenns.Therefbre,youexaminetheDurbin=Watsonstatistic,whichisl.8g53fbrthisregesson

ABasedonthevalueoftheDubinWatsonstatistic,whatcanyousayabouttheseIialcorIdationbetweenthegIssionsiduals?Aretheypositivelycorrelated,negativelycorreated,ornotcoelatedatall?

BComputethesamplecorrelationbetweentheregressionresidualsfiomoneperiodandthosefiPomthepreviousperiod.

CPefbrmastatistica!testtodetermineifseriaIcorreIationispresent.AssumethatthecIiticalvaluesfbrl92observationswhenthereisasingleindependentvariab>eareabout0.0gabovethecritica>va>uesfbr>00observatlons

13ThebooktomaIketrat1oandthesizeofacompany,sequityaretwofactorsthathavebeenassertedtobeusefUlinexplainingthecross-sectionalvaIiationinsubsequentretums.Basedonthisassertion,youwanttoest1matethefb>lowingregIBssionmode>:

b lBook

Marke( /SizeEi

where

Ri=Retumofcompanyj,sshares(inthefb>lowingperiod)

!-i!Sizej=Marketvalueofcompany/,sequity

AcoIleaguesuggeststhatthisregressionspecificationmaybeerroneous,becausehebelievesthatthebooktomaketratiomaybestrongyre!atedtoconPeatedwithcompanySlze.

407

408

ATowhatproblemisyourcoeaguerefbring,andwhatareitsconsequencesfbrregressionanalysis?

BWithrespecttomulticoIineaIity,critiquethechoiceofvariablesintheregressionmode>above.

Regessi0n0fRetln0B00kt0MmketadSize

C0ecdentStadaFdErr0rStatistic

Intercept

l4.>062

l211l3

4220 3327

g.0406 13430

Size -0.000055020.00005g77-0.g2047

R-sqUaredObservations

C0rreth0Mtix

0.06l56

34

B00kPt0-MarketRatioSize

Book-to-MarketRatio

Size

l0000

-03509 l0000

CStatetheclassicsymptomofmulticoIlineantyandcommentonthatbasiswhethermulticoIinearityappearstobepresent,giventheadditionalfhctthatthRtestfbrtheaboveIgressionisnotsignificant.

14YouareanayzingthevariablesthatexplaintheretumsonthestockoftheBoeingCompanyBecauseovemllmarketretumsare>ikelytoexplainapartoftheretumsonBoeing,youdecidetoincudetheetumsonavaIueweightedindexofalthecompanies>istedontheNYSE,AMEX,andNASDAQasanindependentvalectedthefb>lowingdatafbrthepe

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I

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AX}=changeinmontIMintheIogofatrade-weightediIdexofthefbreignexchangevalueoftheUSdoIaragainstthecurrenciesofabroadgroupofmorUStradingpartners

Thefb!lowingtab>eshowstheoutputfiDmregressingthemonthIyretumonBoeingstockonthetheeindependentvaIiabIes

Regessi0n0fB0ehlgRetus0ThreeExp!nM0yValibleS:MohlyDm,Jnnry1990December2002

C0eCientStandardE0rStntiStiC

Intercept

ALL

R

X}

ANOVA

Regssion

Residual

Total

0.0026

=0l337

0.8875

02005

ResidualstandaIderror

R-squaredObservations

0.0066

0.62l9

06357

0.539g

0393g

0.2l50

13g6l

0.37l4

dfSSMSS

3

l52

l55

0767

0l6l0

l56

0.l720

0.8g47

l0667

00573

0.005g

ucFactSct,FedeIHlRcsccBankofPhiladelphia

Fromthe/-statistics,weseethatnoneoftheexplanatoIyvariablesisstatisticallysignificantatthe5percentleve>orbetter.Youwishtotest,however,ifthethIcevariablesOarestatisticalyreatedtothemmsonBoeing.

AYournullhypothesisisthatalIthepopulationsIopecoefficientsequaI0thatthethreevafiablesbaestatisticalynotelatedtothereturnsonBoeingConducttheappropriatetestofthathypothesis.

BExaminingtheregressionresults,statetheregressionassumptionthatmaybeviolatedinthisexample.Explainyouranswer.

CStateapossibewaytoremedytheviolationoftheregressionassumptionidentifiedinPartB

15Youareanayzingthecrosssectionalvariationinthenumberoffinanciaanalyststhatfbllowacompany(alsothesuhiectofProblems3and8).Youbelievethatthereislessanalystfb>lowingfbrcompanieswithagreaterdebt-to-equityratioandgreateranaIyst

109

410

fbIlowingfbcompaniesincIudedintheSP500Index.ConsistentwiththesebeIiefbyouestimatethefbllowingregressionmodel

Analystsfbllowingb0blDEib2SPi6j

where

(Ana>ystsfbl>owing)i=naturaIlogof(l+NumberofanaIystsfbIlowingcompany/)

(D/E)i=debt-to-equityratiofbrcompanyJ

S&Pj=inclusionofcompany/intheS&P500Index(lifincIuded;0ifnotinc>uded)

Intheprecedingspecification,lisaddedtothenumberofanalystsfbIlowingacompanybecausesomecompaniesarenotfb>lowedbyanyanalysts,andthenatural>ogof0isindetenninate.ThefbIlowingtab>egivesthecoefficientestlmatesoftheaboveregressionmode>fbrarandomlyselectedsampleof500companies.ThedataarefbrtheyearZ002.

C0emcntEStimmes0mRegeSSilgAmhyStF0lI0wing0Debtto

EqityRtioHndSP500Membeship,200Z

C0ecientStadardEroStatistic

Intercept

(D/E)j

S&P/

500

15367

-0.l043

l,222

:FirstCallrhomsonFinaImal,Compusa

00582

0.07l2

0084l

26.4038

-l.161g

l45327

Youdiscussyourresultsw1maco>>eague.ShesuggeststhatthisregressionspecificationmaybeeIoneous,becauseanalystfb!lowingislikelytobealsorelatedtothesizeofthecompany.

AWhatisthisprobemcaIledandwhatareitsconsequencesfbregessionanaIysis?

BToinvestigatetheissueraisedbyyourcol>eague,youdecidetocoI>ectdataoncompanysizealsYouthenestimatethemodeafterincludinganadditionaIvariableSizej,whichisthenatumllogofthemarketcapitalizationofcompanyJinmiIIionsofdoIars.Thefb>IowingtabIegivesthenewcoefEcientestimates.

C0efficientEStimteSfi.0mReglessigA0alyStF0ll0wig0nSe,Debtt0EqdtyRmd0,dSP500MembesMp,200Z

C0ecientStfldadEr0rStatiStic

IItercept-0.0075 0l2l8 =0.06l6

C0emetSmdHdE0Stmisthc

Size

(D/E)i

SP

J00

0.2648

-0.>829

042l8

0.0lg>

0.0608

009lg

>3.863g

3.0082

45898

Sb:FiCaThomsonFiancia.Compusa.

Whatdoyouconcludeabouttheexistenceoftheproblemmentionedbyyourco>>eagueintheoliginaIlchoosetolistonaUSexchange.OneofthefactorsthatyouthinkwillaffbctwhetherornotacompanyIistsintheUnitedStatesisitssizeIc>ativetothesizeofothercompaniesinitshomemarket.

/

AWhatkindofadependentvaIiabledoyouneedtouseinthemodel?

BWhatkindofamodelshouldbeused?

ThefOIIowinginfOmationeIatestoQuestions1722

GaryHansenisasecuritiesanalystfbramutuaIfimdspecializinginsmal>-capita>izationgrowthstocksThehmdregu>arlyinvestsininitialpub>icofferings(IOs)Ifthefhndsubscribestoanoffer,itisallocatedsharesattheoffbrpIice.HansennotesthatIOsfi?equentlyareunderpIiced,andthepnceIiseswhenopenmarkettradingbegins.Theinitia>retumfbranIOiscalculatedasthechangeinpnceonthefirstdayoftradingdividedbytheoffbrprice.HansenisdeveIopingaregressionmodeltopredicttheinitiaIretumfbrIPOs.Basedonpastresearch,hese>ectsthefbllowingindependentvariab>estopredictIPOinitiaIreturns:

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/

Unde>wnterrank

Preoffbrpriceadjustme

Offbrsize($miI>ions)

Fractionretaineda

l-l0,wherel0ishighestrank

(Offbrprice-Initia>fi>ingprice)/Initialfi>ingpnce

SharessoldxOffbrprice

FractionoftotalcompanySharesretainedbyinsiders

--

--

--

aExpressedasadecimal

Hansencollectsasampleofl,725IccentIOsfbrhisregressionmodel.RCgressionresultsappearinExhibitlandANOVAresultsappearinExhibit2.

11

412

ExMbitHaSe,SRegeSSi0lReSultSDePedentVaie:IPOitMRetumExpressedinDecimF0rm,i.e.,10.0l

Vnr!ble C0eCienthSmndadEr0rSttiSthC

Intercept

Unde>

00260

Exhbit2Se!ectedANOVARes!tsfOHse,sRegess0

25.>l

3.06

2l.53

-082

l.g2

DegreeS0fFeCd0mdfSUm0fSqUSSS

RegressionResidual

Total

4

l,720

l,724

MultipleR-squared=0.36

5l.433

gl436

l12.86g

HansenwantstousetheregressionsultstopredicttheinitiaretumfbranupcomingIPO.TheupcomingIOhasthefbllowingchamcteristics:

underwnterrank=6;

preoffbrpriceaustment004;

offbrsize0milion;

fractionretained=0.70

Becausehenotesthatthepre-offerpriceadjustmentappearstohaveanimportanteffbctoninitialretum,Hansenwantstoconstructa95percentconfidenceintervalfbrthecoefficientonthisvariable.HeaIsobe>ievesthatfbreachlpercentincreaseinpre-offbrpriceaqustment,theinitialreturnwillincreasebylessthan0.5percentholdingothervariab>esconstantHansenwishestotestthishypothesisatthe0.05levelofsignificance.

Befbreapplyinghismodel,Hansenasksacoeague,PhilChang,toviewitsspecificationandresuIts.ARerexaminingthemodelChangconcIudesthatthemodeIsuffbrsfiDmtwoprobIems:l)conditionalheteroskedasticity,and2)omittedvariabIebias.ChangmakesthefbI>owingstatements:

Statementl ConditionalheteIDskedasticitywil>resu>tinconsistentcoefficientestimates,butboththestatisticsandstatisticwillbebiased,resultinginfh>seinfbrences.,,

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Statement2 IfanomittedvaIiab>eiscorIdatedwithvariablesalreadyinc>udedinthemode},coefficientestimateswillbebiasedandinconsistentandstandarderrorswilIaIsobeinconsistent.,,

SelectedvaIuesfbrthe/-distributionand/MistributionappearinExhibits3and4,respectively

Exhibit3.SeectedVeMbtheDisbuti0ndf

ArenRightTilValue

0.050

0.025

0.0l0

0005

l.645

l960

2326

2.576

Exhibit4SelectmValueSfOrtheRStrbuti0n001df1dh:

Numerat0rDen0mhMlt0Degees0fFreed0m

dH

4

4I6.00l3.50d

3.32l.00

17BasedonHansen,sregression,thepredictedinitialretUmfbrtheupcomingOiscBstO:

A0.0943.

B0l064

C0.l54l.

I8Theg5percentconfidenceintervalfbrtheIgIssioncoefficientfbrthepIofferpriceaustmentiscsto:

A0.l56to0.74

B0.3g5to0475

C0402to0.468

4I3

l,.TheosappropnatenuhypothesisandthemappropnateconclusionregardingHansen,sbeIiefaboutthemagnitudeoftheinitia!etumrelativetothatofthepIcoffbrpnceadjustmentefectedbythecoecientbae:

ConcIusionabouth/(0.05LevelofSignificance)NullHypothesis

H>:h/=0.5

H0:h0.5

H0:b0.5

ABCRectH0

FailtorectH0

RectH

Z0TheosappropriateinteIpretationofthemuItipeRsquaredfbrHansen,smodelisthat:

Aunexplainedvariationinthedependentvariableis36percentoftotalvariation.

BcorIdationbetweenpredictedandactua>va>uesofthedependentvariableis036.

CcorIdationbetweenpredictedandactua>va>uesofthedependentvariab>eis060.

21IsChang,sStatementlcorrect?

AYes.

BNo,becausethemodel,sFBstatisticwil>notbebiased.

CNo,becausethemodel,s/-statisticswilInotbebiased

22IsChang,sStatement2correct?

AYes.

BNo,becausethemodeI,scoefficientestimateswiIlbeunbiased.

CNo,becausethemode,scoefficientestimateswilIbeconsistent.

ThefowinginfOmationeIatestoQuestiohs2328

AdeleChiesaisamoneymanagerfbrtheBiancoFund.SheisinterestedinrecentfindingsshowingthatceItainbusinessconditionvaIiablespredictexcessUSstockmarketretums(one-monthmarketretumminusonemonthTbiItum.SheisalsofHmiIiarwithevidenceshowinghowUSstockmarketretumsdiffbrbythepo>iticaIpmyaffi>iationoftheUSPresideHt.ChiesaestimatesamuItipleregressionmodeItopredictmonthlyexcessstockmaIketretumsaccountingfbrbusinessconditionsandthepoIiticalpaiationoftheUSPresident:

Exccsssockmakccum

!DeuIspeadI2TemspeadlPespaydummyI

DefaultspreadisequaltotheyieIdonBaabondsminustheyieldonAaabondsTermspreadisequatotheyieldonal0yearconstant-maturityUSTreasuIyindexminustheyieldonalyea

4M

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constantmatUrityUSTreasuindexPrespaydummyisequaItoIiftheUSPesidentisamemberoftheDemocraticPartyand0ifamemberoftheRepublicanPay.

Chiesaco>>ects432monthsofdata(a>>dataareinpercentfbrmie.,0.0>=>percent)TheregIssionisestimatedwith43lobseationsbecausetheindependentvaIiabesalaggedonemonthTheregressionoutputisinExhibitlExhibits2through5containcIiticalva>uesfbrselectedteststatistics.

Exhibit1MuMpleRegessi0nOtputtheDepedemViblelstheOneMothMketReturnnExcess0ftheOneM0thTBillRetum

C0echetStat0sticpValue

Inte>cept

DefhuItspadrl

Tennspread-I

PspaItydummy-1

-460

3.04

084

317

NUmberofobservations

TeststatisticfomBreuschPaganBPtest

R2

AdjustedR2Durbin-Watson(DW)

Sumofsquarederrors(SSE)

RegIcssionsumofsquares(SSR)

-436

452

3.4l

4.g7

43l

735

0053

0.016

165

l9,048

l,07>

Aferrespondingtoheintem,squestionsChiesaconcludeswiththefbIlowingstatement:PredictionsfromExhibitlaesubjecttoparameterestimateunceItainty,butnotregressionmodeIunceItainty.,,

Exhibit2CrclValueSfbtheDrbinWtS0nStatStiC005

K=J0

N

l825

l827

l.82g

185

l855

l.857

420

430

440

Exhibit3.Table0ftheStde,SDistribti0nOeTiledP0bbilitieMbrdf

Pr

0.l0l.282

0.05l.645

0.025l.g60

00l2326

0.0052.5760

Exhibit4.Valuesof

Pr0babilityinRightTi

df0.9750950050025

0.000l

00506

02l58

04840

0.0039

01026

035l8

07ll0

l2343.84l

59g>

78l5

g488

5.024

7.378

g.318

ll.l4

Exhibit5Table0ftheRDistbuth0nCticlVuesfbRightHandTailAeEqlt00.05Numerat0r:df1andDen0minat0T:dn

116

l

|

|

dH

dhI23427

l

2

3

4

427

l6l

l8.5l

l0.l3

7.7I

386

200

lg.00

g.55

6.g4

3.02

2l6

lg16

g28

65g

263

225

lg25

9.l2

6.3g

2.39

254

l9.4g

8.53

5.64

l>7

23Regardingtheintem,sQuestion,istheregressionmodeasawholesigncantatthe0.05level?

ANo,becausethecaIculatedFkstatisticislessthanthecritica>vaIuefbrR

BYes,becausethecalculatedFLstatisticisgreaterthanthecriticalvaluefbrR

C.Yes,becausethecalculatedstatisticisgeaterthanthecnticaIvaluefbr

21WhichofthefbowingisChiesa,sbesresponsetoQuestion2egaingseriaIcorrelationintheerrortenn?Ata005>evelofsigIlificance,thetestfbrseria>correIationindicatesthatthereis:

Anoserialcorrel2tionintheeindicatesthatthemeanmonthlyva>uefbrtheexcessstockmarketretumis:

Al43percentIagerduingDemocaticpresidenciesthanRepubicanpesidencies

B3.l7percentlalgerduIingDemocraticpIcsidenciesthanRepublicanpresidencieS

C.3.l7percentlaIgerduringRepublicanpresidenciesthanDemocraticpresidencies

Z6InresponsetoQuestion,theg5percentconfidenceintervafbrtheregressioncoefficientfbrthedefautspIadiscbs2srto:

A0.l3to595.

Bl72to436.

Cl93to4.l5.

27Withrespecttothedefhultspread,theestimatedmode>indicatesthatwhenbusinessconditionsare:

A.strong,expectedexcessreturnswiIlbehigher

117

BweakexpectedexcessetumswiIIbeIower.

C.weakexpectedexcessretumswi>Ibehigher

Z8IsChiesa,sconcudingstatementcorrectrcgadingparametermodeIuncertaintyandregssionmodeluncertainty?

AYes.

B.NopredictionsarenotsuhjecttoparameterestimateuncelTainty

CNopredictionsaresuhjecttoregressionmodeluncertaintyandparametermodeI

uncertalnty

sntuI

1

A.Rb0biMbzXe

BWecantestwhetherthecoefficientontheS&P500Indexretumsisstatisticallysignificant.Ournu>lhypothesisisthatthecoefficientisequa>to0(H0:bl=0);ouraltemativehypothesisisthatthecoefficientisnotequalto0:bl0.WeconstIuctthetestofthenuhypothesisasfbllows:

bI-h--

J

PJ

0

=4.03380.l332

I

where

;Iegessionestimateofb!

bthehypothesizedvaueofthecoefficienthe,0

Itheestimatedstandadeoofb

Becausethisregressionhasl56observatlonsandthreeregreSsioncoefficientsthe/-testhasl56-3=l53degreesoffi.eedom.Atthe0.05significance>evel,thecriticalvaluefbrtheteststatisticisbetweenl.g8and1g7.TheabsoIutevalueof

theteststatisticis40338,therefbre,wecanre>ectthenuIIhypothesisthatbl=0.

Similarlywecantestwhetherthecoefficientonthechangeintheva>ueoftheUSdollarisstatisticallysignificantinthisregression.OurnuI>hypothesisisthatthecoefficientisequa>to0(H0:b2=0);oura>ternativehypothesisisthatthecoefficientisnotequato0yh:b20.WeconstmctthetestasfbIIows:

418

b7 -b7

0.57680=-I.lZ63

- 0.5I2l

Asbefbre,thetesthasl53degreesoffeedom,andthecriticalvauefbrtheteststatisticisbetweenl98andl97atthe0.05significanceleve>.TheabsoIutevalueoftheteststatisticisl263;therefbre,wecannotectthenuIIhypothesisthatb2=0.|Basedontheabove/-testsweconcIudethatS&P500IndexretumsdoaffbctADM,sretumsbutthatchangesinthevalueoftheUSdollardonotaffbctADM,sretums.

CThestatementisnotcoect.Tomakeitc0Tect,weneedtoaddthequalificationhoIdingAXconsta,totheendofthequotedstatement.

2

A.Ri=b0+bl(B/M)i+b2Sizei=ej

BWecantestwhetherthecoefficientsonthebook-to-marketratioandsizeare

individuaIystatisticallysignificantusingtests.Forthebooktomarketratio,ournu>lhypothesisisthatthecoefficientisequalto0(H0:bl=0),ouraltematlvehypothesisisthatthecoefficientisnotequalto0Hb:bl0.Wecantestthenuhypothesisusingatestconstmctedasfbllows:

bIh 0.0500.920I

--

0.0588I

where

legressionestimateofbl

bIthehypothesizedvalueofthecoeffMenthe0|

|

SBI=theestimatedstandarderrorofb>|Thisregressionhas66obseIationsandthreecoefficientssothe/-testhas66=3=63degreesoffiFeedom.Atthe0.05significancelevelthecriticalvaluefbrtheteststatisticisabout2.0Theabso>utevalueoftheteststatisticis0.g20l;therefbre,wecannotre>ectthenullhypothesisthatbl=0.Wecanconcludethatthebook-to-marketratioisnotuseMinexplainingthecross-sectionalvariationinretumsfbrthissampIe.

|

Weperfbrmthesameanalysistodetenninewhethersize(asmeasuredbythelogofthemarketvalueofequity)canhelpexplainthecross-sectionalvaIiationinassetremms.OurnuIIhypothesisisthatthecoefficientisequalto0(H0:b2=0);ouraltemativehypothesisisthatthecoefficientisnoteqUalto0Hb:b20.Wecantestthenulhypothesisusingatestconstructedasfbllows:

119

-

7

b 0.0I640

=-04680 00350

where

A

bregressionestimateofbz=

b2=thehypothesizedvaIueofthecoefficient(here,0)A

J2theestimatedstandadeoofb2

Again,becausethisgressionhas66observationsandthreecoefficients,thetesthas66363degreesoffTeedom.Atthe0.05significanceIevel,theciticalvaluefbrtheteststatisticisabout2.0.Theabso>utevalueoftheteststatisticis

04686;therefbre,wecannotrejectthenulIhypothesisthatbz=0WecanconcIudethatassetsizeisnotusefUIinexplainingthecross-sectionalvanat1onofassetretumsinthissample.

I

3

A.TheestimatedregressionisAnalystfbIowingj0.284503l9gSize-01895DEiei.Therefbrethepredictionfbthefirstcompanyis

(Analystfb>lowing)i=-0.2845+03l9g(Inl00)-0.l895(0.75)

02845l.47320.2ll.0466

RecaIingthatAnaIystfbowingiisthenaturalIogofwheIiisthe

numberofanalystsfbl!owingcompany;itfbllowsthatl1e.0662.848,approximately.Therefbre,l2.848ll848orabouttwoanalysts.Simiady,thepredictionfbrthesecondcompanyisasfb>lows:

(AnaIystfb>Iowing)i==0.2845+03lgg(lnl000)-0.l8g5(075)

=-0.2845+2.20g8=0.l42l

=l.7832

Thus,l2el.78325.949,approximatelyTherefbre25949l491goraboutfiveanalysts.

Themode>predictsthat5-2=3moreanaIystswiIfbl>owthesecondcompanythanthefirstcompany.

BWewouldinterpretthepvalueof000236asthesmallestlevelofsiificanceatwhichwecanre}ectanullhypothesisthatthepopulationvalueofthecoefficientis0inatwosidedtest.Clearly,inthisgressionthedebttoequityratioisahighysignificantvariable.

1Theestimatedmode>is

420

`

PecentagedecIineinTSEspreadofcompany0.45005Sizei0.06Ratioofspreads)j+0.29(DecIineinNASDAQspreads)i

Thefbre,thepredictionis

Percentagedec>ineinTSEspread=-0.45+005(>n900,000)-006(13)+02g(l)

=-0.45+0.6g-0.08+0.2g

=0.45

Themodelpredictsthatfbracompanywithaveragesamplecharacteristics,thespreadontheTSEdeclinesby0.15percentfbralpercentdeclineinNASDAQspreads.

5

ATotestthenu>lhypothesisthata>ltheslopecoefficientsintheregressionmodelaIcequalto0(HU:bl=b2=0)againstthealtemativehypothesisthatat>eastones>opecoefficientisnotequalto0wemustuseanRtest

BToconducttheRtest,weneedfburinputs,allofwhicharefbundintheANOVAsectionofthetableinthestatementoftheproblem:

itotalnumberofobservations,

iitotalnumberofregressioncoefficientstobeestimated,l

|

2

i-z/=!

iihsumofsquarederrorsorresiduals, abbreviatedSSE,

and

J

)

-

(

ivregessionsumofsquaIBs, abbreviatedRSS

|

CThetestfbnnulais

RsS/k 00094/2l F= =04394

SSkI0.67362I

ThestatistichasdegreesoffreedomFkklF2,63FromtheRtesttabe,fbrthe0.05significancelevel,thecriticalvaluefbr2,63isabout3.l5sowecannotreiectthehypothesisthattheslopecoefficientsareboth0.ThetwoindependentvaIiablesarejoimlystatisticallyunrelatedtoreturns

DAustedR2isameasureofgoodnessoffitthattakesintoaccountthenumberofindependentvariab>esintheregIBssion,incontrasttoR2.WecanassertthatadiustedR2issmaI>erthanR2=0@0l38withouttheneedtoperfbImanycalcu>ations.[However,adjustedR2canbeshowntoequa>-00l75usinganexpressioninthetextontherelationshipbetweenadiustedandR2.

12I

|

7

8

12Z

AYoubelievethatopeningmarketsactuaIlyreducesremrnvo>ati>ity;ifthatbeliefiscorrect,thenthes>opecoefficientwouldbenegativebl>hypothesisisthatthebeliefisnottme:H0:bl0Thealternativehypothesisisthatthebeliefistme::bl0.

BThecIiticalvaluefbrthestatisticwithg5293degreesofficedomatthe0.05signif}canceIevelinaone-sidedtestisaboutl.66.Fortheone-sidedteststatedinPartA,werC>ectthenul>hypothesisifthe/-statisticonthes>opecoefficientislessthan-l.66.Asthe/-statisticof-2.7604ectthenullBecausethedummyvaiabletakesonava!ueoflwhenfbreigninvestmentisallowed,wecanconcludethatthevolatilitywas>owerwithfbIcigninvestment.

CAccordingtotheestimatedregression,averageretumvolati>itywas0.0l33(theestimatedvalueoftheintercept)befbreJUlyl9g3and0.0058(=00l33-0.0075)afferJulylg93.

ATheappropriateregressionmodelishFb0blPartye!

BThe/-statisticreportedinthetab>efbrthedummyvaIiabletestswhetherthecoefficientonPaItylows:

bI-h-=

!

-00570-0

00466

where

;!egressionestimateofb!

=-I.22

blthehypothesizedvalueofthecoefficientheI0

Jtheestimatedstandamemrofbl

Totwodecimaplaces,thisvalueisthesameasthestatisticreportedinthetablefbrthedummyvaIiable,asexpected.TheproblemspecifiedtwodecimalplacesbecausethereportedICgressionoutputref>ectsrounding;fbrthisreason,weoftencannotexactlyreproducereported/-statistics.

CBecausetheregressionhas77observationsandtwocoefficients,the/-testhas77-2=75degreesoffreedom.Atthe0.05significanceleve>,thecriticalvaluefbrthetwo-tailedteststatisticisaboutl9gTheabsolutevalueoftheteststatisticis

l.2242;therefbe,wedonotrectthenulhypothesisthatb!0.WecanconcludethatthepoliticalpaltyintheWhiteHousedoesnotonaverage,affbcttheannua>retumsoftheoveraIImarketasmeasuredbytheS&P500.

ATheregressionmode>isasfb>lows:

(AnaIyStfbIIowing)i=b0+b>Sizei+b2(D/E)i+b3S&Pl+e!

where(Ana>ystfbIowing)isthenaturaIlogof(l+numberofanaIystsfbllowingcompany/);Size!isthenaturalIogofthemarketcapita>izationofcompany/inmiI>ionsofdollars;(D/E)#isthedebt-to-equityratiofbrcompanyj,andS&Pjisadummyvariab>ewithava>ueoflifthecompany/belongstotheS&P500Indexand0othewise.

BTheappropriatenuIIandaltemativehypothesesareH0:b30andL:b0,respective>y.

CThe/-statistictotestthenu>>hypothesiscanbecomputedasfbIIowsA

b- 02!80.58,8

--

0.00I9$&$

ThisvaIueisofcourse,thesameasthevaluereportedinthetab>e.Theregressionhas500observationsand4regressioncoefficients,sothe!-testhas500-1=1g6degreesofhPeedomAtthe0.05significancelevel,thecriticalva>uefbrthetestBtatisticisbetweenlg6andl.g7.Becausethevalue.oftheteststatisticis4.58g8

wecanrQiectthenuI>hypothesisthatb3=0.Thusacompany,smembershipintheSP500appeastosignificantyinfuencethenumberofanaIystswhocoverthatcompany.

DTheestimatedmodeIis

AnaIysMbIIowingi0.007502648Sizei-0829DEi+0.12>8S&B+e/

< >

Therefbrethepredictionfbrnumberofanalystsfb>>owingtheindicatedcompanythatisnotpartoftheS&P500Indexis

(Analystfb>lowing)!=-00075+02648[>n10,000)-0.l829(Z/3)+0.42l8(0)

=-0.0075+2.4389-0.l2lg+0

=230g5

RecallingthatAnaIystfblowingjisthenatural!ogofl,whereisthenumberofanalystsfbIowingcompanyMtensues(codingthecompanyunderconsiderationaslthatl!e2.30,5l006g,approximately.Therefbre,thepredictionisthatll0.069l9.06g,oraboutnineanalysts.

Similarly,thepredictionfbrthecompanythatisinc>udedintheS&P500Indexis

(Analystfbllowing)i=-00075+02648(Inl0,000)-0182g(2/3)+042l8(l)

0.00752.3890.l2lg0.42l8

=2.73l3

423

24

9

CodingthecompanythatdoesbelongtotheSP500as2,l2e273l3l5353Therefbre,thepredictionisthat2l5.353l4353,oraboutl4anaIysts.

EThereisnoinconsistencyinthecoefficientonthesizevariabIedifferingbetweenthetworegressions.Theregressioncoefficientonanindependentvariab>einamultipleregressionmodeImeasurestheexpectedneteffCctontheexpectedvalueofthedependentvariab>efbraone-unitincreaseinthatindependentvariab!e,afferaccountingfbranyeffbctsoftheotherindependentvariab>esontheexpectedvaIueofthedependentvariab>e.TheearlierregressionhadonefewerindependentvariabIe;affertheeffbctofS&P500membershipontheexpectedvaIueofthedependentvariabIeistakenintoaccount,itistobeexpectedthattheeffbctofthesizevariabeonthedependentvariablewillchange.WhattheegessionsappeartoindicateisthattheneteffbctofthesizevariableontheexpectedanalystfbllowingdilninisheswhenS&P500membershipistakenintoaccount.

AInawellspecifiedregression,thediffbrencesbetweentheactualandpredictedrelationshipshouldberandom;theenPorsshouldnotdependonthevalueoftheindependentvariable.InthisregressiontheerorsseemlaIgerfbrsmallervauesofthebooktomarketratio.Thisfindingindicatesthatwemayhaveconditionalheteroskedasticityintheerrorsandconsequently,thestandarderrorsmaybeincorIct.WecannotpIoceedwithhypothesistestinguntilwetestfbrand,ifnecessaIy,coectfbrheteroskedasticity.

BAtestfbrheteIoskedasticityistoregressthesquaredresidualsfiFomtheestimatedregressionequationontheindependentvariab>esintheregression.AsseeninSection4.l.2,BreuschandPaganshowedthat,underthenu>lhypothesisofnoconditionalheteIDskedasticity,xR2homtheregressionofthesquaredresidualsontheindependentvariabIesfiomtheorigina>regression)wi>Ibeafrandomvariabe,withthenumberofdecesoffreedomequaItothenumbeofindependentvariab>esintheregression.

COnemethodtocoIectfbrheteroskedasticityistouserobuststandarderrors.Thismethdusestheparameterest1matesfi.omthelinearregressionmodelbutconPectsthestandarderrorsoftheestimatedpammeterstoaccountfbrtheheteroskedasticity.ManystatisticalsoftwarepackagescaneasiIycomputerobuststandardermrs.

10TheteststatisticisR2,whereisthenumberofObservationsandR2istheR2oftheregressionofsquaredresidualsSo,theteststatisticis52x0.l4l=7.332.Underthenu>lhypothesisofnoconditionalheteroskcdasticitythisteststatisticisafrandomvMablcTherearethreedegreesoffreedom,thenumberofindependentvariablesintheregssionAppendixC,attheendofthisvolume,showsthatfbraonetailedtest,thetest

statisticcticalvaluefbavariabIcfiDmadistnbutionwith3degreesofficedomatthe0.05significanceIeveIis7.8>5.TheteststatisticfiomtheBreusch=Pagantestis7332.So,wecannotrectthehypothesisofnoconditionalheteIoskedasticityatthe005levelTherefbre,wedonotneedtocoectfbrconditionaIheteroskedasticity.

I1.

A.Theteststatisticis2,whereisthenumberofobservationsandR2istheR2ofthegressionofsquaIdresidualsSotheteststatisticis750x000645.Under

I2

I3

thenuhypothesisofnoconditionalheteoskedasticity,thisteststatiSticisafrandomvaiabIe.BecausetheregressionhasonIyoneindependentvariablethenumberofdegeesofficedomisequaltolAppendixC,attheendofthisvoume,showsthatfbraone-taiIedtest,theteststatisticcriticalva>uefbravariabIefibma

distnbutionwithonedegreeofeedomatthe005significancelevelis3.81lTheteststatisticis4.5.So,wecanrectthehypothesisofnoconditionalheteroskedasticityatthe0.05levelTherefbre,weneedtocorrectfbrconditiona>heteroskedasticity.

BTwodiffbrentmethodscanbeusedtocorIctfbrtheeffbctsofcondhiona

heteroskedasticityinlinearregressionmodels.ThefirstmethodinvolvescomputingrobuststandaderrorsThismethodcorrectsthestandarderrorsofthe>inearregressionmode>,sestimatedparameterstoaccountfbrtheconditionalheteroskedasticity.Thesecondmethodisgeneralizedeastsquals.Thismethodmodifiestheoriginalequationinanattempttoeliminatetheheteroskedasticity.Thenew,modifiedregressionequationisthenestimatedundertheassumptionthatheteroskedasticityisnoIongeraproblem

Manystatisticalsoftwarepackagescaneasiycomputerobuststa1darderrorsthefirstmethod)andwerecommendusingthem.

ABecausethevalueoftheDurbinWatsonstatisticisessthan2,wecansaythattheregressionresiduaIsarepositivelyconPelated.Becausethisstatisticisfairlycloseto2howeveLwecannotsaywithoutastatisticaltestifthesenalcorIdationisstatisticaIysignificant.

BFromJanuaIylg87throughDecember2002,therearel6years,orl6xl2lg2monthlyretumsThusthesamp>eanalyzedisquite>aIge.TherefbretheDurbinWatsonstatisticisapproximateyequalto2l,whereisthesamplecondationbetweentheregssionresidualsfomonepeiodandthosefomthepreviousperiod

DW189532

So,rlDW2ll8953200524ConsistentwithouranswertoPaItA,

thecoIeationcoefficientispositive.

CAppendixEindicatesthatthecriticalvaluesandfbrl00obseIvationswhenthereisoneindependentvariab>earel.65and16grespectively.Basedontheinfbnnationvenintheproblem,thecIiticalvaluesandMbrabout200

obseIationswhenthereisoneindependentvariab>eareaboutl.74andl.78,respectively.BecausetheDWstatisticof18g53fbrourgssionisabovewefailtoectthenuIIhypothesisofnopositiveseaIcorreation.Therefbweconcudethattheeisnoevidenceofpositiveserialcorreationfbrtheerrortenn

AThisprob!emisknownasmu>ticol>ineaIity.Whensome>inearcombinationsoftheindependentvariabIesinaregressionmodelaIhighycoelated,thestandardenorsoftheindependentcoefficientestlmatesbecomequite>arge,eventhoughtheregIcssionequationmayfitratherwe>l.

125

M

15

26

BThechoiceofindependentvanabIespesetsmuIticoIIinealityconcensbecausemarketvaIueofequityappearsinbothvariabIes.

CTheclassicsymptomofmu>tico>>inearityisahighR2(andsignificantRstatistic)eventhoughthe/-statisticsontheestimateds!opecoefficientsareinsignificant.HereasignificantRstatisticdoesnotaccompanytheinsignificant/-statistics,sotheclassicsymptomisnotpresent.

ATotestthenulIhypothesisthatalloftheregressioncoeffIcientsexceptfbrtheinterceptinthemuItipleregressionmode>areequaIto0(f/b:bl=bz=b3=0)agamstthealtemativehypothesisthatatleastonesIopecoefficientisnotequaIto0,wemustuseanRtest

RSS/k 0!720/]

SSI0.563I9.703

TheRstatistichasdegreesoffreedomFMlH3,l52FromtheFBtesttablethecIiticalvaluefbr3,l202.68andFUl52wilbeessthanF(3,120),sowecanre>ectatthe005significanceleve>thenul>hypothesisthatthes!opecoefficientsareal0.Changesinthethreeindependentvariablesaejointystatisticallyre>atedtoretums.

BNoneofthestatisticsaresignificant,butthefLstatisticissignificant.Thissuggeststhepossibilityofmulticollinearityintheindependentvariables.

C.TheapparentmulticoIinearityisverylikelyrelatedtotheinclusionofbotheretumsontheSP500IndextheItumsonava!ueweightedindexofalthecompanieslistedontheNYSEAMEXandNASDAQasindependentvariables.ThevalueweightigofthelatterindexgivingrelativeyhighweightstolagercompaniessuchasthoseincludedintheS&P500maymakeoneretumsenesanapproximate>inearflmctionoftheother.Bydroppingoneortheotherofthesetwovariab>eswemightexpecttoe>iminatethemuItico>Iineanty.

AYourcolleagueisdicatingthatyouhaveomittedanimportantvaIiablefYomtheregression.Thispmblemisca>ledtheomittedvariablebiasIftheomittedvaIiableisconclatedwithincudedvariable,theestimatedvaluesoftheIcgressioncoefficientswouldbebiasedandinconsistent.Moreovenheestimatesofstandardenorsofthosecociemswoudasobeinconsistent.So,wecannotuseeitherthe

coefficientestimaiBsortheestimatesoftheirstandardenbrstoperfbrmstatisticaltests.

BAcompansonofenewestmateswiththeo!igi;tesceadyindicatesthattheoiginamodsuffbredfibmtheomittedvariabiasduetotheexclusionofcompanysizefiomthatmodelAsthe/-statisticsofincontrasttotheoriginaImodeLinwhichitisnotsignificantatthe5percentlevelTvaIueoftheestimatedcoecientoftheSP500dummy

l6

substantialIydecIinesfioml2222to0.42>8.ThesechangesimpIythatsizeshouldbeincludedinthemodel

AYouneedtouseaquaitativedependentvaIiable.YoucouldgiveavaIueofltothisdummyvariablefbra>istingintheUnitedStatesandava>ueof0fbrnotlistingintheUnitedStates.

BBecauseyouareusingaqua!itativedependentvaIiable,IinearregIcssionisnottherighttechniquetoestimatethemodelOnepossibilityistouseeitheraprobitora>ogitmodel.Bothmodelsareidentical,exceptthatthe>ogitmode>isbasedonlogisticdistributionwhiletheprobitmodelisbasedonnormaIdistIibutionAnotherpossibiiistousediscIiminantanalysis.

I7Ciscorrect.ThepIcdictedinitiaItumIRis:

IR=00177+(0.0l50x6)+(0.435x0.01)-(0000gx40)+(0.05x070)

=0154l

I8.Biscorrect.Theg5confidenceinteais0.4350.0202xI.g603950475

l9Ciscorrect.TotestHansen,sbeiefaboutthediIctionandmagnitudeoftheinitiaI

retum,thetestshouldbeaone-tailedtest.ThealtemativehypothesisisHl:h/is-l.645.Theteststatisticissignificant,andthenu>lhypothesiscanberC>ectedatthe0.05levelofsignificance

Z0CisconPect.ThemultipleRsquaredfbrtheregressionis0.36;thus,themodelexplains36percentofthevariationinthedependentvariable.Thecorrelationbetweentheprediptedandactualvaluesofthedependentvariableisthesqua!crootoftheR-squared

or0.60

21Aiscorrect.ChaPgisconFectbecausethepresenceofconditionalheteroskedasticityresutsinconsistentparameterestimates,butbiasedupordownstandardeors,/-statistics,andFEstatistics

ZZAiscorrect.ChangiscoectbecauseacorrelatedomittedvariablewillresutinbiasedandinconsistentparameterestimatesandinconsistentstandarderIors.

23Biscorrect.

ThetestisusedtodetennineiftheregIcssionmodelasawholeissignificant.

FMeansquare:regressionMSRMeansquarederrorMSE

MSESSEnkllg,0484274460

MSRSSRl07l3357

F35744608004

27

428

TheciticaIvaIuefbrdegeesoffTeedomof3and427with0.05onetaiIisF2.63fiomExhibit5.Theca>cuIatedFisgreaterthanthecriticaIvalue,andChiesashou>drqectthenuIIhypothesisthataregressioncoecientsareequaItozero

24Biscorrect.TheDurbin-Watsontestusedtotestfbrseationintheerrortenn,anditsvaIuereportedinExhibit1isl65.FornoseIia>corre>ation,DWisapploximatelyequaItoZIfDWeeo.emsePvesem!ycoqeBecausetheDWl65isIessthan1827fbr43lseeExhibit2,ChiesashouIdrqectthenul

hypothesisofnoseIiaIcorrelationandconc>udethatthereisevidenceofpositiveseIiaIcorreIationamongtheeITorterms.

Z5.BiscorrectThecoefficientfbrthePrespartydummyvariab>e(3.l7)representstheincrementinthemeanvalueofthedependentvariablerelatedtotheDemocraticPartyholdingthepresidency.Inthiscase,theexcessstockmarketretumis3.17percentgreaterinDemocraticpresidenciesthaninRepub>icanpresidencies.

Z6BiscorrectTheconfidenceintervaliscomputedaslslx95,.FromExhibit1,l3.04andl452,resutinginastandarderroroflI3.044.520.673.ThecliticalvaIuefbr/fibmExhibit3is196fbrp=0.025.Theconfidenceintervalfbr!is3.040.673xl.963.01l3lg08orfrom172092to435g08.

27.Ciscorect.ThedefhutspreadistypicaylaIgerwhenbusinessconditionsarepoor,i.e.ageatepobabilityofdefhutbytheborrower.ThepositivsignfbrdefhultspIadseeExhibit1indicatesthatexpectedBtumsarepositivelyreatedtodefhultspadsmeaningthatexcessretumsaregreaterwhenbusinessconditionsarepoor.

Z8Ciscorrect.Predictionsinamultipleregessionmodearesuhiecttobothparameterestimateuncertaintyandregressionmode>uncertainty.

CAD

SEK

AUD

KRW

NZD

Canadiandollar

SwedishkIDna

Austra>iandoIlar

Koreanwon

NewZea>anddolIar

PacticePobIems

Copyright2012CFAInstitute

ThefoowihgihfOmatineIatestQuestionS161

EdSmithisanewtraineeinthefbreignexchange(FX)servicesdepartmentofama}orgloba!bankSmith,sfbcusistoassistseniorFXtrader,FelizMehmet,CFAMehmetmentionsthatanIndian

corpoIntec>ientexportingtotheUnitedKingdomwantstoestimatethepotentialhedgingcostfbrasa>eclosinginoneyear.SmithistodeterminethCpremium/discountfbranannual(360day)fbrwaIdcontractusingtheexchangeratedatapresentedinExhibit>

Exhibit1SelectCurrencyDatarGBPandINR

Spot(INR/GBP)-.

Annual(360-day)Libor(GBP)

Annual(360-day)Libor(INR)

79.50g3

5.3

7.52%

Mehmetisalso>ookingattwopossibletradestodetelminetheirprofItpotentia>.ThefirsttlQadeinvo>vesapossibletriangu>ararbitragetradeusingtheSwiss,USandBraziliancurIcncies,tobeexecutedbasedonadea>er,sbid/offerlatequoteof05l6l/0.5l63inCHF/BRLandtheinterbankspotratequotespresentedinExhibit2.

604

EXhibit2.InterbankMarketoI0teS

CrrecyPai

CHFSD

BRL/USD

BidOfr

0.9099/09l0l

l.77g0/l.7792

|

MehmetisaIsoconsideringacarytradeinvolvingtheUSDandtheEuro.HeanticipatesitwiI

generateahigherretumthanbuyingaone-yeardomesticnoteatthecurrentmarketquoteduetolowUSinterestlatesandhispredictionsofexchangeatesinoneyearTohelpMehmetassessthecarrytrade,MehmetprovidesSmithwithseIectedcurrentmarketdataandhisoneyearfbrecastsinExhibit3.

Exhibit3Sp0tRatesandIterestRatesrPr0p0sedCarryTrade

T0day,s0ne-yearLib0r

USD

CAD

EUR

0.80%

>7>%

2.20%

Currencypai

(Price/Base)Sp0tratet0dayPmjectedsp0trateiI0Ieyear

CAD/USD

EUR/CAD

l0055

0.72>8

L0006

0.727g

FinaIy,MehmetasksSmithtoassistwithatradeinvoMngaUSmultinationa!customeroperatinginEucompanywithaAAcreditratingandstrivestoexecuteitscurencytmdesatthemostfavorablebidoffbrspread.BecauseitsJapanesesubsidiaockinatradeinvoIvingtheJapaneseyenandtheEuroasearlyaspossib!ethenextmoming,pIcfbrabyby8:05AMNewYorktime

Atlunch,SmithandotheFXtraineesdiscusshowbesttoanayzecuITencymaIketvolatilityfiomongoingnanciaIcrisesThegmupagesthatatheoreticaIexpanationofexchangeratemovements,suchaSthefinmeworkoftheinternationalparityconditions,shou>dbeapplicab>eacrossatradingenvionments.TheynotesuchanalysisshouIdenabIetraderstoanticipatefhturespotexchangerates.Buttheydisageonwhichparityconditionbestpdictsexchangerates,voicingseveaIdifferentassessments.SmithconcIudesthediscussiononparitycoHditionsbystatingtothetrainees:

lbelievethatinthecurIntenvironmentbothcoveredanduncoveredintestrateparityconditionsareineffbct.,,

Theconversationnextshifistoexchangeateassessmenttools,specificalIythetechniquesoftheIMFConsu

MacIDeconomicBaIancefbcusesonthestocksofoutstandingassetsandIiabi>ities

2Statement2

ReducedFormhasaweaknessinunderestimatingfUtureappreciationofundervalued

currencles

3.Statement3

605

ExternalSustainabiIitycentersonaustmentsIeadingtoIongtemequiIibiuminthecapitalaccount.

lBaseduponExhibit,thefbrwardpremiumdiscountfba360dayNRGBPfbrwardcontractisc/Oscs/to:

A.-I.546.

B.>546

C.I.576

2.BasedonhihlL,theosappopiaterecommendationregaMingthetriangulararbitragetradeisto:

A.decIinethetrade,noabitrageprofitsarepossibIe

B.executethetrade,buyBRLintheinterbankmarketandseIIittothedeaIer

C.executethetrade,buyBRLfiomthedea!erandseIitintheinterbankmarket.

3.BasedonhihiL,thepotentiaIaIIinUSDretumonthecalytradeiscOsesto:

Al04%.

B.l.40%

Cl.84%.

4.ThefactoreMiyto!eadtoanarIDwbidoffbrspadfbrtheindustriaIcompany,sneededcurIncytIadeiSthe:

Atimingofitstrade.

B.company,screditrating

C.pairofcuITenciesinvoIved

5.IfSmith,sstatementonparityconditionsiscorrect,flltuespotexchangeratesareostobefbrecastby:

Acurrentspotrates

B.fbrwardexchangerates.

C.inf>ationratediffCrentia>s.

6.Whichofthefb>IowingstatementsgivenbyTrainee#lindescribingtheapproachesusedbyCGERisosaccuInte?

A.Statementl

BStatement2

C.Statement3

0L0-T

ThefowinginfOmatineIatestQuestins7-132

ConnorWagenerastudentattheUniversityofCantebuPyinNewZealandhasbeenaskedto

plpaeapresentationonfbreignexchangeratesfbrhisnternationaIBusinesscourse.Wagenerhas606

I

abasicunderstandingofexchangeratesbutwouldlikeapractitioner,sperspective,andhehasal.rangedaninterviewwithcurrencytraderHannahMcFaddenDuringtheinterview,WagenerasksMcFadden:

ouIdyouexpIainwhatdrivesexchangerates?I,mcuriousastowhyourNewZeaIanddoIIarwas

affbctedbytheEuropeandebtcrisisin20>>andwhatotherfactorsimpactit.,,

ustratehowexchangelHtesareIinkedtoexpectedinf>ation,interestratediffbences,andfbrwardexchangemtesasweascurrentandexpectedfiturespotratesMcPaddenstates:

StatementlFortunateIy,theintemationaparityconditionmostrelevantfbFXcarIytradesdoesnota>waysho>d.,,

McFaddencontinuesherdiscussion:

FXcarytradersgoIongi.ebuyhighyieIdcunnciesandfimdtheirpositionbyshorting,thatisborrowingin,Iow-yie>dcurrenciesUnfbrtunately,cIqashesincurrencyva!uescanoccurwhichcreatefinanciaIcrisesastlndersunwindtheirpositions.Forexamp>e,in2008,theNewZeaIanddo!>arwasnegativeIyimpactedwhenhigh>y>eveIagedcarrytradeswereunwound.Inadditiontoinvestors,consumersandbusinessownerscana!soaff