課件金融計量學(xué)ch6 slides

?Allthemodelswehavelooked?Allthemodelswehavelookedatthusfarhavebeensingleequationsmodelsthey=Xb+??AllofthevariablescontainedintheXmatrixareassumedtobeyisanENDOGENOUSAnexamplefromeconomicstoillustrate-thedemandandsupplyofa=++(3)equilibrium=quantityofthegood=quantityofthegood=priceofasubstitute=somevariableembodyingthestateof‘IntroductoryEconometricsforFinance’?ChrisBrooksSimultaneousEquationsModels:TheStructuralForm?AssumingthatthemarketSimultaneousEquationsModels:TheStructuralForm?Assumingthatthemarketalwaysclears,anddroppingthetimesubscriptsforsimplicityQ=a+bP+gS+Q=l+mP+kT+ThisisasimultaneousSTRUCTURALFORMofthe?Thepointisthatpriceandquantityaredeterminedsimultaneously(priceaffectsquantityandquantityaffectsprice).?PandQareendogenousvariables,whileSandTare?WecanobtainREDUCEDFORMequationscorrespondingto(4)and(5)bysolvingequations(4)and(5)forPandforQ(separately).‘IntroductoryEconometricsforFinance’?ChrisBrooksObtainingtheReduced?Solvingfora+bP+gS+=ObtainingtheReduced?Solvingfora+bP+gS+=l+mP+kT+?Solvingfor=Q-l-kT-bbbRearrangingmmm?P=l-a+kT-gS-v-b-mb-b-b-‘IntroductoryEconometricsforFinance’?ChrisBrooksObtainingtheReducedForm?Multiplying(7)throughbyObtainingtheReducedForm?Multiplying(7)throughbymQ-ma-mgS-mu=bQ-bl-bkT-mQ-bQ=ma-bl-bkT+mgS+mu-(m-b)Q=(ma-bl)-bkT+mgS+(mu-Q=ma-bl-bkT+mgS+mu-m-m-m-m-?(8)and(9)arethereducedformequationsforPand‘IntroductoryEconometricsforFinance’?ChrisBrooks?Butwhatwouldhappen?Butwhatwouldhappenifwehadestimatedequations(4)and(5),i.e.thestructuralformequations,separatelyusingOLS??BothequationsdependonP.OneoftheCLRMassumptionswasthatE(X=0,whereXisamatrixcontainingallthevariablesontheRHSofthe?Itisclearfrom(8)thatPisrelatedtotheerrorsin(4)and(5)-i.e.itWhatwouldbetheconsequencesfortheOLSestimator,b,thesimultaneity??ifwe‘IntroductoryEconometricsforFinance’?ChrisBrooksSimultaneousEquationsBias?ySimultaneousEquationsBias?y=Xb+(X'X)-1X'(Xb+?So=(X'X)-1X'Xb+(X'X)-1Xb+(X'X)-1X==b+(X'X)-1E(X'?Taking?IftheX’sarenon-stochastic,E(Xu)=0,whichwouldbecaseinsingleequationsystem,sothatE(b)=,b ?a)?general.(biased‘IntroductoryEconometricsforFinance’?ChrisBrooksSimultaneousEquationsBias?Conclusion:ApplicationofOLStostructuralequationswhicharepartofasimultaneoussystemwillleadtobiasedcoefficientestimates.Thisisknownas“simultaneitybias”.?IsSimultaneousEquationsBias?Conclusion:ApplicationofOLStostructuralequationswhicharepartofasimultaneoussystemwillleadtobiasedcoefficientestimates.Thisisknownas“simultaneitybias”.?IstheOLSestimatorstillconsistent,eventhoughitis?No-Infacttheestimatorisinconsistentas?Henceitwouldnotbepossibletoestimateequations(4)and(5)validlyusingOLS.‘IntroductoryEconometricsforFinance’?ChrisBrooksHowtoestimatesimultaneousSoWhatCanWeHowtoestimatesimultaneousSoWhatCanWe?Takingequations(8)and(9),wecanrewritethemQ=p20+p21T+p22S+?WeCANestimateequations(10)&(11)usingOLSsincealltheRHSvariablesareexogenous.?But...weprobablydon’tcarewhatthevaluesofthepcoefficientsare;whatwewantedweretheoriginalparametersinthestructuralequations-a,b,g,l,m,k.‘IntroductoryEconometricsforFinance’?ChrisBrooksIdentificationofSimultaneousCanWeRetrievetheOriginalCoefficientsfromthep’s?Shortanswer:sometimes.?identification(識別IdentificationofSimultaneousCanWeRetrievetheOriginalCoefficientsfromthep’s?Shortanswer:sometimes.?identification(識別?Exampleofunidentifiedequations:ConsidersupplyequationsthefollowingdemandQ=a+Q=l+SupplyequationWecannottellwhichis??Theproblemisthatwedonothaveenoughinformationfromtheequationstoestimate4parameters.Noticethatwewouldnothavehadthisproblemwithequations(4)and(5)sincetheyhavedifferentexogenousvariables.‘IntroductoryEconometricsforFinance’?ChrisBrooksWhatDetermineswhetheranEquationisIdentifiedornot??WeWhatDetermineswhetheranEquationisIdentifiedornot??Wecouldhavethreepossible1.Anequationis··like(12)orwecannotgetthestructuralcoefficientsfromthereducedform2.Anequationisexactly··e.g.(4)orcangetuniquestructuralformcoefficient3.Anequationisover-··ExamplegivenMorethanonesetofstructuralcoefficientscouldreducedform.beobtainedfrom‘IntroductoryEconometricsforFinance’?ChrisBrooksWhatDetermineswhetheranWhatDetermineswhetheranEquationisIdentifiedornot?(cont’d)??HowdowetellifanequationisidentifiedorTherearetwoconditionswecouldlook-Theordercondition-isanecessarybutnotsufficientconditionforequationtobe-Therankcondition-isanecessaryandsufficientconditionforidentification.Wespecifythestructuralequationsinamatrixformandconsidertherankofacoefficientmatrix.(weneedsometechnicalalgebrabeyondthisbook,soweskip‘IntroductoryEconometricsforFinance’?ChrisBrooks ?LetGdenotethenumberofstructuralequations.AnequationisjustidentifiedifthenumberofvariablesexcludedfromanequationisG-1.(Note:“excluded”meansNumberofallendogenousandExogenousvariablesthatarenotPresentinthisparticular?IfmorethanG-1areabsent,itisover-identified.IflessthanG-1areabsent,itisnotidentified.?Inthefollowingsystemofequations,theY’sareendogenous,whiletheX’sareexogenous.Determinewhethereachequationisover-,under-,orjust-Y1=a0+a1Y2+a3Y3+a4X1+a5X2+u1Y2=b0+b1Y3+b2X1+u2Y3=g0+g1Y2+(14)-SimultaneousEquationsBiasSimultaneousEquationsBiasG=If#excludedvariables=2,theeqnisjustidentifiedIf#excludedvariables>2,theeqnisover-identifiedIf#excludedvariables<2,theisnotEquation14:NotEquation15:JustidentifiedEquation16:Over-‘IntroductoryEconometricsforFinance’?ChrisBrooksWhyweneedsimultaneousequationsin?IstheWhyweneedsimultaneousequationsin?Isthebankindustryexcessivelyconcentratedandbankindustryneedsmore?ModelofbankingprofitabilityandbankingPriceofPriceoftheoutputAttimetBankBankBankiAttimePriceof‘IntroductoryEconometricsforFinance’?ChrisBrooksTestsforExogeneity:Hausman?Howdowetellwhethervariablesreallyneedtobetreatedasendogenousornot?AndhowTestsforExogeneity:Hausman?Howdowetellwhethervariablesreallyneedtobetreatedasendogenousornot?AndhowtotellwhetherasimultaneousequationsmodelisConsideragainequations(14)-(16).Equation(14)containsY2andY3-butdowereallyneedequationsforthem?WecanformallytestthisusingaHausmantest,whichiscalculatedas??1.Obtainthereducedformequationscorrespondingto(14)-(16).reducedformsturnouttobe:Y2=p20+p21X1+++(17)-Estimatethereducedformequations(17)-(19)usingOLS,andfitted ‘IntroductoryEconometricsforFinance’?ChrisBrooksTestsforExogeneity2.Runtheregressioncorrespondingtoequation3.TestsforExogeneity2.Runtheregressioncorrespondingtoequation3.Runtheregression(14)again,butnowalsoincludingthefitted +a +l +l +11Y=a+a+aY+a10 14.UseanF-testtotestthejointrestrictionthatl2=0,andl3=0.Ifthenullhypothesisisrejected,Y2andY3shouldbetreatedasendogenous.‘IntroductoryEconometricsforFinance’?ChrisBrooks?Considerthefollowingsystemof?ConsiderthefollowingsystemofY1=Y2=b20++g11X1+g12X2+(21-Y3=b30+b31Y1+b32Y2+g31X1+g32X2+?Assumethattheerrortermsarenotcorrelatedwitheachother.CanweestimatetheequationsindividuallyusingOLS??Equation21:Containsnoendogenousvariables,soX1andX2arenotcorrelatedwithu1.SowecanuseOLSon(21).Equation22:ContainsendogenousY1togetherwithexogenousX1andX2.WecanuseOLSon(22)ifalltheRHSvariablesin(22)areuncorrelatedwiththatequation’serrorterm.Infact,Y1isnotcorrelatedwithu2becausethereisnoY2terminequation(21).SowecanuseOLSon(22).?‘IntroductoryEconometricsforFinance’?ChrisBrooks? ? uncorrelatedwithu3.Bysimilarargumentstotheabove,(21)and(22)donotcontainY3,sowecanuseOLSon?ThisisknownasaRECURSIVEorTRIANGULARsystem.Wenothaveasimultaneityproblemhere.?Butinpracticenotmanysystemsofequationswillbe‘IntroductoryEconometricsforFinance’?ChrisBrooksIndirectLeastSquares?3waystoestimatesimultaneousIndirectLeastSquares?3waystoestimatesimultaneousequations:IndirectLeasttwo-squares(over-equations)andinstrumental?CannotuseOLSonstructuralequations,butwecanvalidlyapplyittothereducedformequations.?Ifthesystemisjustidentified,ILSinvolvesestimatingthereducedformequationsusingOLS,andthenusingthemtosubstitutebacktoobtainthestructuralparameters.?However,ILSisnotusedmuch1.Solvingbacktogetthestructuralparameterscanbe‘Introductor2Eoosetrsol’?08temsEstimationofSystemsUsingTwo-StageLeastSquares?Infact,wecanuseEstimationofSystemsUsingTwo-StageLeastSquares?Infact,wecanusethistechniqueforjust-identifiedandover-identified?Twostageleastsquares(2SLSorTSLS)isdoneintwoStage?ObtainandestimatethereducedformequationsusingOLS.Savefittedvaluesforthedependent(?Stage?Estimatethestructuralequations,butreplaceanyRHSendogenousvariableswiththeirstage1fittedvalues.‘IntroductoryEconometricsforFinance’?ChrisBrooksEstimationofSystemsUsing2SLS(cont’d)Example:Estimateequations(14)-Y1=a0+a1Y2+a3Y3+a4X1EstimationofSystemsUsing2SLS(cont’d)Example:Estimateequations(14)-Y1=a0+a1Y2+a3Y3+a4X1+a5Y2=b0+b1Y3+b2X1++StageEstimatethereducedformY3=g0+g1Y2+(17)-(19)individuallybyOLS?obtainthefitted. StageReplacetheRHSendogenousvariableswiththeirstage1estimated?Y1=a0+a1Y+a3Y+a4X+a5X+23121Y2=b0+b1Y+bX+ Y3=g0+g1Y+ (24)-? willnotbecorrelatedwithuwillnotbecorrelatedwith,23212andY3willnotbecorrelatedwithu3?Note:2SLSisconsistentbutbiasedEstimatingthesystems:Instrumental?directlyequationsEstimatingthesystems:Instrumental?directlyequationsisthattheendogenousvariablesarecorrelatedwiththe?OnesolutiontothiswouldbenottouseY2orY3,butrathertousesomeothervariablesinstead.?Wewanttheseothervariablestobe(highly)correlatedwithY2andY3,butnotcorrelatedwiththeerrors-theyarecalledINSTRUMENTS.?SaywefoundsuitableinstrumentsforY2andY3,z2andz3respectively.Wedonotusetheinstrumentsdirectly,butrunregressionsoftheformY2=l1+l2z2+Y3=l3+l4z3+(27)&‘IntroductoryEconometricsforFinance’?ChrisBrooksObtainthefittedvaluesfrom(27)Obtainthefittedvaluesfrom(27)&(28),YandY,andreplace?232Y3withtheseinthestructural?Wedonotusetheinstrumentsdirectlyinthestructural?Itistypicaltousemorethanoneinstrumentperendogenous?Iftheinstrumentsarethevariablesinthereducedformequations,thenIVisequivalentto2SLS.‘IntroductoryEconometricsforFinance’?ChrisBrooksAnExampleoftheUseof2SLS:ModellingtheBid-AskSpreadandVolumeforOptions??AnExampleoftheUseof2SLS:ModellingtheBid-AskSpreadandVolumeforOptions??GeorgeandLongstaff(1993)Istradingactivityrelatedtothesizeofthebid/askHowdospreadsvaryacross?HowMighttheBid/AskSpreadbeConsider3Marketmakersequalisespreadsacrossoptions(administrativeThespreadmightbeaconstantproportionoftheoptionvalue(inventoryholdingcosts).Marketmakersmightequalisemarginalcostsacrossoptionsirrespectiveoftradingvolume(riskcosts).‘IntroductoryEconometricsforFinance’?ChrisBrooksTheInfluenceofTick-SizeRulesTheInfluenceofTick-SizeRuleson?TheCBOE(ChicagoBoardOptionsExchange)limitsthetick$1/8foroptionsworth$3or$1/16foroptionsworthlessthan?Thespreadislikelytodependontrading...butalsotradingvolumeislikelytodependonthe?Sotherewillbeasimultaneous‘IntroductoryEconometricsforFinance’?ChrisBrooksThe?Alltradingdaysduring1989The?Alltradingdaysduring1989areusedforTheaveragebid&askpricesarecalculatedforeachtime2:00pm–2:15pmCentralStandardtime.?optionduring?ThefollowingarethendroppedfromthesampleforthatAnyoptionsthatdonothavebid/askquotesreportedduringthe?Anyoptionswithfewerthan10tradesduringthe?Theoptionpriceisdefinedastheaverageofthebid&the?Wegetatotalof2456observations.Thisisapooled‘IntroductoryEconometricsforFinance’?ChrisBrooksTheCalbid-askTimesbetweentradesfor?FortheCBAi=a0+TheCalbid-askTimesbetweentradesfor?FortheCBAi=a0+a1CDUMi+a2Ci+a3CLi+a4Ti+++and+andiMeasureshowspreadaffectstradingvolumeAndsymmetricallyfortheputs:?bid-askPBA=b+bPDUM+bP+bPL+bT+b+i01i2 4 ii‘IntroductoryEconometricsforFinance’?ChrisBrooksResultsCallBid-AskSpreadandTradingVolume=a0+a1CDUMi+a2Ci+a3CLi+++ResultsCallBid-AskSpreadandTradingVolume=a0+a1CDUMi+a2Ci+a3CLi+++TMCL+gCBA+g+++i01i23 iAdj.-(--(-(--(-Adj.Note:t-ratiosinSource:GeorgeandLongstaff(1993).ReprintedwithpermissiontheSchoolofBusinessAdministration,Universityof‘IntroductoryEconometricsforFinance’?ChrisBrooksResultsPutBid-AskSpreadandTradingVolume=b0+b1PDUMi+b2Pi+b3PLi+ResultsPutBid-AskSpreadandTradingVolume=b0+b1PDUMi+b2Pi+b3PLi++b5++dT2+dM2+PL=+dPBA+di01i23 iAdj.-(--(-(--(-Adj.Note:t-ratiosinparentheses.Source:GeorgeandLongstaff(1993).ReprintedtheSchoolofBusinessAdministration,UniversityofWashington.withpermission‘IntroductoryEconometricsforFinance’?ChrisBrooksAdjustedR2?AdjustedR2?a3andb3measuretheeffectoftradingactivityontheBid-Askspreadvariationsbetweenoptionscanbeexplainedbyreferencetotheleveloftradingactivity,deltas,timetomaturityetc.Thereisa2wayrelationshipbetweenvolumeandthespread.‘IntroductoryEconometricsforFinance’?ChrisBrooks?AnaturalgeneralisationofautoregressivemodelspopularisedbySimsAVARisinasenseasystemsregressionmodeli.e.thereismorethanonedependentvariable.Acombinationofunivariatetimeseriesmodelsandsimultaneousequationsmodels.?y2?AnaturalgeneralisationofautoregressivemodelspopularisedbySimsAVARisinasenseasystemsregressionmodeli.e.thereismorethanonedependentvariable.Acombinationofunivariatetimeseriesmodelsandsimultaneousequationsmodels.?y2?SimplestcaseisabivariateVAR(only2eachdependsonthepreviouskvaluesofboth++==whereuitisaniiddisturbancetermwithE(uit)=0,i=1,2;E(u1tTheanalysiscouldbeextendedtoaVAR(g)model,orsothattherearevariablesandg?A‘IntroductoryA‘IntroductoryEconometricsforFinance’?ChrisBrooksVectorAutoregressiveModels:NotationandConcepts?OneimportantfeatureofVARsisthecompactnesswithwhichwecanwritethenotation.Forexample,considerthecasefromabovewherek=1.VectorAutoregressiveModels:NotationandConcepts?OneimportantfeatureofVARsisthecompactnesswithwhichwecanwritethenotation.Forexample,considerthecasefromabovewherek=1.++?Wecanwritethis=+b21y2t-1+a21y1t-yya1t1 b =++2t20212tborevenmorecompactly+=+‘IntroductoryEconometricsforFinance’?ChrisBrooksVectorAutoregressiveModels:NotationandConcepts?ThismodelcanbeextendedVectorAutoregressiveModels:NotationandConcepts?Thismodelcanbeextendedtothecasewherethereareklagsofeachvariableineachequation:yt-yt-yt-kyt=++‘IntroductoryEconometricsforFinance’?ChrisBrooks?ThemaindifferencebetweenaVARandasimultaneousequationsystemisthatitdoesnot?ThemaindifferencebetweenaVARandasimultaneousequationsystemisthatitdoesnotdistinguishbetweenendogenousvariablesandpredeterminedvariables.Eachvariableistreatedasanendogenousvariableandexplainedbyitsownlaggedvaluesandthelaggedvaluesofallothervariablesinthesystem.‘IntroductoryEconometricsforFinance’?ChrisBrooksVectorAutoregressiveModelsComparedwithStructuralEquationsModels?AdvantagesVectorAutoregressiveModelsComparedwithStructuralEquationsModels?AdvantagesofVARDonotneedtospecifywhichvariablesareendogenousorexogenous-allareAllowsthevalueofavariabletodependonmorethanjustitsownlagsorcombinationsofwhitenoisete