2015 CFA Level 2 Book 1 Ethics Econ QA
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Transcript of 2015 CFA Level 2 Book 1 Ethics Econ QA
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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
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2.Usethedatasamp>ebelowtoanswerthefbIowingquestions
)(}p
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7
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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
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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
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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
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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
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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
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|
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
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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
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-
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