课件金融计量学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(识别IdentificationofSimultaneousCanWeRetrievetheOriginalCoefficientsfromthep’s?Shortanswer:sometimes.•identification(识别•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(ˆStage•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/askquotesreportedduringthe¼Anyoptionswithfewerthan10tradesduringthe•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