panel data 处理.docx

panel data 处理.docx

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paneldata处理

Lecture4

4.Analysingdecisionmaking

4.1Learningoutcomes

∙Binarydependentvariables

∙Logitandprobit

∙Eventstudies

∙Application:

Understandmergersandacquisitions

∙WorkingwithSTATA:

Datamanagement

4.2Decisionsandmarketresponses

Thislecturerfocusesontwoaspectsofdecisionmaking:

first,weexplorewhyfirmsmakecertaindecisionsusingQualitativeResponseModels（QRM）;second,wetrytoassesstheimpactonthosedecisionsonthefirm.Bothaspectsareveryimportantinempiricalfinance.Duetomypriorresearch,Iwillmainlyfocusonmergersandacquisitions（M&A）.Hence,wetrytoanswerthequestionswhytofirmsmergeandwhatisthemarketresponsetomergerannouncements.Thelatterquestionhasreceivedagreatdealofattentionsincethe1960s,asthereisempiricalevidencesuggestingthatshareholdersofacquirersdon’tbenefitfromM&A.ThisiscalledtheMergerParadox.Putdifferently,thequestionis:

whydofirmsmergeiftheydestroyshareholdervalue.Thereareafewexplanations（e.g.agencytheoryandempirebuilding）–butthereisstillalottoexplore.

ThefollowingsectionwillderivetheQRMforabinarychoicevariable–i.e.havingtwooptions（“tobuyornottobuy”）.Section4.4developstheEventStudymethodology,whichdetectsmarketresponsestodecisions.Finally,weapplyourknowledgetoarealdatasetonM&AtransactionsintheUS,whichisbasedonmypreviouswork.

4.3Binarydependentvariables

Inmanycases,wetrytounderstandbinarychoice–yesornodecisions.Modellingbinarychoicesisverydifferentfromanalysingacontinuousrandomvariable.Inparticular,theobservedchoicelabelledythasonlytwo（discrete）outcomes,namely0（negativeevent）and1（positiveevent）.Accordingly,assumingthatytisdrawnfromanormaldistributionisnotplausible.Sowehavetolookforalternatives.Inaddition,wehavetobeawareofthefactthatweonlyobservetheoutcomeofdecisionmaking,namelywhetherafirmactsornot.However,wedonotobservetheinternaldecisionprocess.Forinstance,wedon’tknowwhetheradecisionwas“clear-cut”ornot.Hence,wehavetodealwithalatentvariableproblem,forthedecisionprocesswillbeablackboxforoutsiders.

Theliteraturesuggeststwoalterativedensityfunctions;theprobitmodelandthelogitmodel.Iwillfocusonthelatteranddescribethemethodology.Hayashi（2000）discussestheprobitmodelinchapter7ifyouareinterested.Understandinglogitissufficient,asthemethodologyisthesame.

Wecandescribethequalitativeresponsemodel（QRM）usingthefollowingparameteriseddensityfunction.

（1）

WecanobservetheoutcomeytandKindependentvariablesxt.ThecolumnvectorΘreferstotheparametersofthemodel,whichwetrytoestimate.Moreover,weneedanalternativedensityfunction–thelogisticdistributionΛdefinedasfollows.

（2）

Thevariableztisunobserved（internaldecisionmaking）andinfluencedbyKindependentvariablesxtassumingastandardlinearmodel.Notethat（3）isascalar.

（3）

Let’sillustratethelogisticdistributionusingEXCEL.Whathappensifztapproaches+/-∞（seereviewquestions）?

Figure1showsthelogisticdistribution.

Figure1:

Logisticdistribution

ThelogisticdistributionΛ（zt）lookslikeaprobability.Infact,Λ（zt）istheprobabilitythatytisequalto1,andthenegativeeventyt=0hastheprobability1-Λ（zt）.UsingaBernoullidistribution,wecanrewriteparameteriseddensityfunction

（1）asfollows.

（4）

Nowwehavethedensityfunctionforeverybinarychoiceyt.TodeterminetheunknownparametervectorΘ–putdifferentlytoestimateΘ,weapplytheprincipalofMaximumLikelihood（ML）.TheideaistoderivealikelihoodfunctionL（Θ）thatdescribestheplausibilitytoobservethesample（knownytandxt）assumingtheparametervectorΘ.Let’sassumeweobserveTrealisationsofytandxtandbyassumethatytisiidandfollowsthedensityfunction（4）,weobtainthefollowinglikelihoodfunction.

（5）

Asthenametellsyou,MLtriestomaximisetheplausibility（likelihood）andselectedtheparametervectorΘforwhichthelikelihoodfunctionL（Θ）reachesamaximum.Hence,wehavetoderivethegradient（vectorofpartialderivatives）of（5）withrespecttotheparametervectorΘ.Tomakethecalculationeasier,weapplytheln（.）functiontobothsidesofexpression（5）.Thisdefinesthelog-likelihoodfunctionl（Θ）.

（6）

Thefollowingstepsaremarkedwitha*,whichindicatesthatitwon’tbeassessed.Thefirststepistofocusonthelogisticdistribution

（2）andtakethepartialderivativewithrespecttotheparametervectorΘ.

（7）

Considerthefollowingrelationship:

（8）

Hence,itfollowsfrom（7）:

（9）

Nowwegobacktoequation（6）andtakethepartialderivativewithrespecttotheparametervectorΘusingourresult（9）.

（10）

STATAusestheloglikelihoodfunction（6）andnumericmethodstoderivetheMLestimator.Theissueisthatthereisnoguaranteethatthenumericsolutionisreallythemaximumofthefunction.

4.4Eventstudies

Famaetal.（1969）andBallandBrown（1968）introducedtheevent-studymethodintoempiricalfinance.Sincethenithasbeenusedwidelytoassesstheimpactofeventsandfirms’decisionsonshareprices.BasedontheEfficientMarketHypothesis（EMH）,onecanarguethatsharepricesshouldreflectpublicandprivateinformationinstantaneously.Hence,byobservingshareprices,weshouldbeabletoassesstheeconomicimplicationsofdecisionmaking.MoststudieshaveanalysedM&Aannouncements,asM&Ahasasignificantimpactonacquirersandtargetfirms（e.g.restructuringandsynergies）.

Event-studiesmeasurethisimpactofeventsbycomparingtheshareprice（actuallythestockreturn）duringaneventwindowwiththeexpectedstockreturn（thenormalreturn）.Nowadaysempiricalstudiesselectveryshorteventwindows–usually+/-1dayaroundtheannouncementday,asstockmarketsarehighlyefficient（mainlyjustifiedbyelectronictradingandaccesstoinformationthroughtheinternet）.

4.4.1Randomwalkhypothesis

Inlecture1,wesawthatiftheEMHisfulfilled,theMarkovPropertyappliesandhencesharepricesPlfollowingarandomwalk.Timeduringtheestimationperiodislabelledbyl{1,2,…,L},andduringtheeventperiodweuset{1,2,…,T}.Themainideaistoderiveanormalreturnbasedontheestimationwindow,whichisusuallybeforetheevent（itiscommontouse-250,...,-50days）.Thisnormalreturnisthenforecastedfortheeventwindowduringwhichwetrytoobservethepriceimpactofevents.

Followingtherandomwalkhypothesis,wecanarguethatachangeinpricesisonlyduetopublicinformation.Publicinformationisregardedasawhite-noiseprocessel.Thisprocessofnewlyavailablepublicinformationdoesnotexhibitserialcorrelation.Hence,publicinformationisnotpredictable.

（11）

Thewhite-noiseprocesshasthefollowingproperties.

（12）

4.4.2Theconstant-mean-returnmodel（CMR）

Masulis（1980）developedtheCMRmodel,whichhasbeenwidelyusedinempiricalfinance.Analternativemodelistheso-calledmarketmodel,whichisbasedontheCAPM.Inpractice,theCAPMandtheassociatedstochasticmarketmodeldoesnotperformtoowell.Quiteoftentheestimatedbetacoefficientsdonotmakemuchsense;hence,IrecommendusingtheCMRiftheeventperiodisshort.

Takingthefirstdifferenceofequation（11）anddividingbyPl-1representsstockreturnsRl.Let’sassumethatthesampleconsistsofndifferentstocksiandtheestimationwindowisreferstol{1,2,…,L};hence,oneobtainsthefollowingexpression.

（13）

where:

and

Stockreturnsfollowawhite-noiseprocess.Notethateilisthewhite-noiseprocesswithmeanzero.Thus,irepresentsthemeanfunctionofRilwhichissupposedtobeconstantovertime.Thisistheexpectedstockreturnoffirmi.Forconvenience,Iputexpression（13）inmatrixnotation.

（14）

Inequation（14）allvectorsarecolumnvectorswithdimensionn1.NotethatImaintaintherandomwalkhypothesisand,hence,thenullhypothesisthattheeventhasnoimpactonshareprices.Followingthislogic,Equation14definesthenormalreturn.Thisisthereturnonewouldexpect,iftheeventdidnotoccur.Estimatingequation14isstraightforward.

（15）

Lstandsforthelengthoftheestimationperiod,whichisthesameforallstocks.AndRlisan1dimensionalvectorthatcollectsfortimel{1,2,…,L}oftheestimationwindowthereturnofeachstocki.Moreover,Iderivethevarianceofthenormalreturndirectlyfromexpression（15）.

（16）

Atthatpoint,notethatsuccessivereturnsofstockiaresupposedtobeuncorrelatedovertime.Therefore,itispossibletodrawthevarianceoperatorunderthesumoperator.Theresultingvariancevector2isobviouslyan1dimensionalvectorthatallowsfordifferencesinvariancesamongstocksiandstatesthatthevariancesremainunchangedovertimel{1;2;…;L}.IncontrasttotheCMRmodelproposedbyMasulis（1980）,Iusethevariance（16）toaccountfortheinaccuracyofestimatingnormalreturns.

4.4.3Abnormalreturns

Havingdeterminedthenormalreturn,wecandefinetheabnormalreturnt*causedbytheevent（decision）wewanttostudy.TheabnormalreturnissimplythedifferencebetweentheobservedreturnvectorRtduringtheeventwindowt{1,2,…,T}andthepartofthisobservedreturnthatcanbepredictedusingtheCMRmodel.

（17）

Underthenullhypothesisthattheeventhasnoeconomicimpact,onecannowderivethestatisticalpropertiesoftheabnormalreturns.

Finding1

Underthenullhypothesis,theconditionaldistributionoftheestimatedabnormalreturnvectort*afterhavingobservedthedatamatrix（observedreturnsduringthe