統(tǒng)計(jì)專業(yè)英語(yǔ) Empirical-Research-Methods-Using-STATA-(ST9008)-Chapter-2

12.Lineardependentvariables2.1Thebasicideaunderlyinglinearregression2.2SinglevariableOLS2.3Correctlyinterpretingthecoefficients2.4Examiningtheresiduals2.5Multipleregression2.6Heteroskedasticity2.7Correlatederrors2.8Multicollinearity2.9Outlyingobservations2.10Medianregression2.11“Looping”22.1Thebasicideaunderlying

linearregressionAsimplelinearregressionaimstocharacterizetherelationbetweenadependentvariableandoneindependentvariableusingastraightlineYouhavealreadyseenhowtofitalinebetweentwovariablesusingthescattercommandLinearregressiondoesthesamethingbutitcanbeextendedtoincludemultipleindependentvariables32.1ThebasicideaForexample,youpredictthatlargercompaniesusuallypayhigherfeesYoucanformalizetheeffectofcompanysizeonpredictedfeesusingasimpleequation:Theparametera0representswhatfeesareexpectedtobeinthecasethatSize=0.Theparametera1capturestheimpactofanincreaseinSizeonexpectedfees.42.1ThebasicideaTheparametersa0anda1areassumedtobethesameforallobservationsandtheyarecalled“regressioncoefficients”Youmayarguethatcompanysizeisnottheonlyvariablethataffectsauditfees.Forexample,thecomplexityoftheauditengagement,orthesizeoftheauditfirmmayalsomatter.Ifyoudonotknowallthefactorsthatinfluencefees,thepredictedfeethatyoucalculatefromtheaboveequationwilldifferfromtheactualfee.52.1ThebasicideaThedeviationbetweenthepredictedfeeandtheactualfeeiscalledthe“residual”.Ingeneral,youmightrepresenttherelationbetweenactualfeesandpredictedfeesinthefollowingway:whererepresentstheresidualterm(i.e.,thedifferencebetweenactualandpredictedfees)62.1ThebasicideaPuttingthetwotogetherwecanexpressactualfeesusingthefollowingequation:Thegoalofregressionanalysisistoestimatetheparametersa0anda1

72.1ThebasicideaOneofthesimplesttechniquestoestimatethecoefficientsisknownasordinaryleastsquares(OLS).TheobjectiveofOLSistomakethedifferencebetweenthepredictedandactualvaluesassmallaspossibleInotherwords,thegoalistominimizethemagnitudeoftheresiduals82.1ThebasicideaGotoMySiteDownload“ols.dta”toyourharddriveandopeninSTATA(use"J:\phd\ols.dta",clear)examinethegraphicalrelationbetweenthetwovariables,twoway(scatteryx)(lfityx)92.1ThebasicideaThislineisfittedbyminimizingthesumofthesquareddifferencesbetweentheobservedandpredictedvaluesofy(knownastheresidualsumofsquare,RSS)

Themainassumptionsrequiredtoobtainthesecoefficientsarethat:TherelationbetweenyandxislinearThexvariableisuncorrelatedwiththeresiduals(i.e.,xisexogenous)Theresidualshaveameanvalueofzero102.2SinglevariableOLS(regress)Insteadofusingthelfitcommandwiththegraph,wecaninsteadusetheregresscommandregressyxThefirstvariable(y)isthedependentvariablewhilethesecond(x)istheindependentvariable112.2SinglevariableOLS(regress)Thisgivesthefollowingoutput:122.2SinglevariableOLS(regress)Thecoefficientestimatesare3.000forthea0parameterand0.500forthea1parameterWecanusethesetopredictthevaluesofYforanygivenvalueofX.Forexample,whenX=5wepredictthatYwillbe:

display3.000091+0.5000909*5

132.2SinglevariableOLS(_b[])Alternatively,wedonotneedtotypethecoefficientestimatesbecauseSTATAwillrememberthemforus.TheyarestoredbySTATAusingthename_b[varname]wherevarnameisreplacedwiththenameoftheindependentvariableortheconstant(_cons)display_b[_cons]+_b[x]*5142.2SinglevariableOLSNotethatthepredictedvalueofywhenxequals5differsfromtheactualvaluelistyifx==5Theactualvalueis5.68comparedtothepredictedvalueof5.50.Thedifferenceforthisobservationisthe“residual”errorthatarisesbecausexisnotaperfectpredictorofy.152.2SinglevariableOLSIfwewanttocomputethepredictedvalueofyforeachvalueofxinourdataset,wecanusethesavedcoefficientsgeny_hat=_b[_cons]+_b[x]*xTheestimatedresidualsarethedifferencebetweentheobservedyvaluesandthepredictedyvaluesgeny_res=y-y_hatlistxy_hatyy_res162.2SinglevariableOLS(predict)Aquickerwaytodothiswouldbetousethepredictcommandafterregresspredictyhatpredictyres,residCheckingthatthisgivesthesameanswer:listyhaty_hatyresy_resYoushouldalsonotethatthevaluesofx,yhatandyrescorrespondwiththosefoundonthescattergraphsortxlistxyy_haty_res172.2SinglevariableOLS182.2SinglevariableOLSNotethatbyconstruction,thereiszerocorrelationbetweenthexvariableandtheresidualstwoway(scattery_resx)(lfity_resx)192.2SinglevariableOLSStandarderrorsTypicallyourdatacomprisesasamplethatistakenfromalargerpopulationThecoefficientsareonlyestimatesofthetruea0anda1valuesthatdescribetheentirepopulationIfweobtainedasecondrandomsamplefromthesamepopulation,wewouldobtaindifferentcoefficientestimatesfora0anda1202.2SinglevariableOLSWethereforeneedawaytodescribethevariabilitythatwouldobtainifweweretoapplyOLStomanydifferentsamplesEquivalently,wewantameasureofhow“precisely”ourcoefficientsareestimatedThesolutionistocalculate“standarderrors”,whicharesimplythesamplestandarddeviationsassociatedwiththeestimatedcoefficientsStandarderrors(SEs)allowustoperformstatisticaltests,e.g.,isourestimateofa1significantlygreaterthanzero?212.2SinglevariableOLSThetechniquesforestimatingstandarderrorsarebasedonadditionalOLSassumptionsHomoscedasticity(i.e.,theresidualshaveaconstantvariance)Non-correlation(i.e.,theresidualsarenotcorrelatedwitheachother)Normality(i.e.,theresidualsarenormallydistributed)222.2SinglevariableOLSThet-statisticisobtainedbydividingthecoefficientestimatebythestandarderror232.2SinglevariableOLSThep-valuesarefromthet-distributionandtheytellyouhowlikelyitisthatyouwouldhaveobservedtheestimatedcoefficientundertheassumptionthatthe“true”coefficientinthepopulationiszero.Thep-valueof0.002tellsyouthatitisveryunlikely(prob=0.2%)thatthetruecoefficientonxiszero.Theconfidenceintervalsmeanyoucanbe95%confidentthatthetruecoefficientofxliesbetween0.233and0.767.

242.2SinglevariableOLSToexplainthisweneedsomenotationcapturesthevariationinyarounditsmeancapturesthevariationthatisnotexplainedbyxcapturesthevariationthatisexplainedbyx252.2SinglevariableOLSThetotalsumofsquares(TSS)=41.27Theexplainedsumofsquares(ESS)=27.51Theresidualsumofsquares(RSS)=13.76NotethatTSS=ESS+RSS.262.2SinglevariableOLSThecolumnlabeleddfcontainsthenumberof“degreesoffreedom”FortheESS,df=k-1wherek=numberofregressioncoefficients(df=2–1)FortheRSS,df=n–kwheren=numberofobservations(=11-2)FortheTSS,df=n-1(=11–1)Thelastcolumn(MS)reportstheESS,RSSandTSSdividedbytheirrespectivedegreesoffreedom272.2SinglevariableOLSThefirstnumbersimplytellsushowmanyobservationsareusedtoestimatethemodelTheotherstatisticsheretellyouhow“well”themodelexplainsthevariationinY282.2SinglevariableOLSTheR-squared=ESS/TSS(=27.51/41.27=0.666)

Soxexplains66%ofthevariationiny.Unfortunately,manyresearchersinaccounting(andotherfields)evaluatethequalityofamodelbylookingonlyattheR-squared.Thisisnotonlyinvaliditisalsoverydangerous(Iwillexplainwhylater)

292.2SinglevariableOLSOneproblemwiththeR-squaredisthatitwillalwaysincreaseevenwhenanindependentvariableisaddedthathasverylittleexplanatorypower.Addinganothervariableisnotalwaysagoodideaasyouloseonedegreeoffreedomforeachadditionalcoefficientthatneedstobeestimated.Addinginsignificantvariablescanbeespeciallyinefficientifyouareworkingwithasmallsamplesize.TheadjustedR-squaredcorrectsforthisbyaccountingforthenumberofmodelparameters,k,thatneedtobeestimated:AdjR-squared=1-(1-R2)(n-1)/(n-k)=1-(1-.666)(10)/9=0.629InfacttheadjustedR-squaredcaneventakeonnegativevalues.Forexample,supposethatyandxareuncorrelatedinwhichcasetheunadjustedR-squarediszero:AdjR-squared=1-(n-1)/(n-2)=(n-2-n+1)/(n-2)=-1/(n-2)302.2SinglevariableOLSYoumightthinkthatanotherwaytomeasurethefitofthemodelistoadduptheresiduals.However,bydefinitiontheresidualswillsumtozero.Analternativeistosquaretheresiduals,addthemup(givingtheRSS)andthentakethesquareroot.RootMSE=squarerootofRSS/n-k=[13.76/(11-2)]0.5=1.236OnewaytointerprettherootMSEisthatitshowshowfarawayonaveragethemodelisfromexplainingyTheF-statistic=(ESS/k-1)/(RSS/n-k)=(27.51/1)/(13.76/9)=17.99theFstatisticisusedtotestwhethertheR-squaredissignificantlygreaterthanzero(i.e.,aretheindependentvariablesjointlysignificant?)Prob>FgivestheprobabilitythattheR-squaredwecalculatedwillbeobservedifthetrueR-squaredinthepopulationisactuallyequaltozeroThisFtestisusedtotesttheoverallstatisticalsignificanceoftheregressionmodel31Classexercise2aOpenyourFees.dtafileandrunthefollowingtworegressions:auditfeesontotalassetsthelogofauditfeesonthelogoftotalassetsWhatdoestheoutputofyourregressionmean?Whichmodelappearstohavethebetter“fit”322.3CorrectlyinterpretingthecoefficientsSofarwehaveconsideredthecasewheretheindependentvariableiscontinuous.Interpretationofresultsisevenmorestraightforwardwhentheindependentvariableisadummy.regauditfeesbig6ttestauditfees,by(big6)332.3CorrectlyinterpretingthecoefficientsSupposewewishtotestwhethertheBig6feepremiumissignificantlydifferentbetweenlistedandnon-listedcompanies342.3Correctlyinterpretingthecoefficientsgenlisted=0replacelisted=1ifcompanytype==2|companytype==3|companytype==5regauditfeesbig6iflisted==0ttestauditfeesiflisted==0,by(big6)regauditfeesbig6iflisted==1ttestauditfeesiflisted==1,by(big6)genlisted_big6=listed*big6regauditfeesbig6listedlisted_big6352.3CorrectlyinterpretingthecoefficientsSomestudiesreportthe“economic”significanceoftheestimatedcoefficientsaswellasthestatisticalsignificanceEconomicsignificancereferstothemagnitudeoftheimpactofxonyThereisnosinglewaytoevaluate“economicsignificance”butmanystudiesdescribethechangeinthepredictedvalueofyasxincreasesfromthe25thpercentiletothe75th(orasxchangesbyonestandarddeviationarounditsmean)362.3CorrectlyinterpretingthecoefficientsForexample,wecancalculatetheexpectedchangeinauditfeesascompanysizeincreasesfromthe25thto75thpercentilesregauditfeestotalassetssumtotalassetsifauditfees<.,detailgenfees_low=_b[_cons]+_b[totalassets]*r(p25)genfees_high=_b[_cons]+_b[totalassets]*r(p75)sumfees_lowfees_high37Classexercise2bEstimatetheauditfeemodelinlogsratherthaninabsolutevaluesCalculatetheexpectedchangeinauditfeesascompanysizeincreasesfromthe25thto75thpercentilesCompareyourresultsforeconomicsignificancetothoseweobtainedwhenthefeemodelwasestimatedusingtheabsolutevaluesoffeesandassets.Hint:youwillneedtotaketheexponentialofthepredictedlogoffeesinordertomakethiscomparison.382.3CorrectlyinterpretingthecoefficientsWhenevaluatingtheeconomicsignificanceofadummyvariablecoefficient,weusuallydosousingthevalueszeroandoneratherthanpercentilesForexamplereglnafbig6genfees_nb6=exp(_b[_cons])genfees_b6=exp(_b[_cons]+_b[big6])sumfees_nb6fees_b6392.3CorrectlyinterpretingthecoefficientsSupposewebelievethattheimpactofaBig6auditonfeesdependsuponthesizeofthecompanyUsually,wewouldquantifythisimpactusingarangeofvaluesforlnta(e.g.,aslntaincreasesfromthe25thtothe75thpercentile)402.3CorrectlyinterpretingthecoefficientsForexample:genbig6_lnta=big6*lnta reglnafbig6lntabig6_lntasumlntaiflnaf<.&big6<.,detailgenbig6_low=_b[big6]+_b[big6_lnta]*r(p25)genbig6_high=_b[big6]+_b[big6_lnta]*r(p75)genbig6_mean=_b[big6]+_b[big6_lnta]*r(mean)sumbig6_lowbig6_highbig6_mean41Itisamazinghowmanystudiesgiveamisleadinginterpretationofthecoefficientswhenusinginteractionterms.Forexample,Blackwelletal.4243Classquestions:Theoretically,howshouldauditingaffecttheinterestratethatthecompanyhastopay?Empirically,howdowemeasuretheimpactofauditingontheinterestrateusingeq.(1)?4445Classquestion:Atwhatvaluesoftotalassets($000)istheeffectoftheAuditDummyontheinterestrate:negative,zero,positive?4647Classquestions:Whatisthemeanvalueoftotalassetswithintheirsample?Howdoesauditingaffecttheinterestratefortheaveragecompanyintheirsample?4849Verifythattheaboveclaimis“true”.SupposeBlackwelletal.hadreportedtheimpactforafirmwith$11minassetsandanotherfirmwith$15minassets.Howwouldthishavechangedtheconclusionsdrawn?Doyouthinkthepaperwouldhavebeenpublishediftheauthorshadmadethiscomparison?50512.4ExaminingtheresidualsGotoMySiteDownload“anscombe.dta”toyourharddriveuse"J:\phd\anscombe.dta",clearRunthefollowingregressionsregy1x1regy2x2regy3x3regy4x4Notethattheoutputfromtheseregressionsisvirtuallyidenticalintercept=3.0(t-stat=2.67)xcoefficient=0.5(t-stat=4.24)R-squared=66%52Classexercise2cIfyoudidnotknowaboutregressionassumptionsorregressiondiagnosticsyouwouldprobablystopyouranalysisatthispoint,concludingthatyouhaveagoodfitforallfourmodels.Infact,onlyoneofthesefourmodelsiswellspecified.Drawscattergraphsforeachofthesefourassociations(e.g.,twoway(scattery1x1)(lfity1x1)).Ofthefourmodels,whichdoyouthinkisthewellspecifiedone?Drawscattergraphsfortheresidualsagainstthexvariableforeachofthefourregressions–isthereapattern?WhichoftheOLSassumptionsareviolatedinthesefourregressions?532.4ExaminingtheresidualsUnfortunately,accountingresearchersoftenjudgewhetheramodelis“well-specified”solelyintermsofitsexplanatorypower(i.e.,theR-squared).Manyresearchersfailtoreportothertypesofdiagnostictestsistheresignificantheteroscedasticity?isthereanypatterntotheresiduals?arethereanyproblemsofoutliers?542.4ExaminingtheresidualsForexample,manyauditfeestudiesclaimthattheirmodelsarewell-specifiedbecausetheyhavehighR2

Carsonetal.(2003):552.4ExaminingtheresidualsGu(2007)pointsoutthat:econometriciansconsiderR2valuestoberelativelyunimportant(accountingresearchersputfartoomuchemphasisonthemagnitudeoftheR2)regressionR2sshouldnotbecomparedacrossdifferentsamplesincontrastthereisalargeaccountingliteraturethatusesR2stodeterminewhetherthevaluerelevanceofaccountinginformationhaschangedovertime 56Usingeithereq.(1)or(2),wewillobtainexactlythesamecoefficientestimatesbecausetheeconomicmodelisthesameIfeq.(1)iswell-specified,soalsoiseq.(2)Ifeq.(1)ismis-specified,soalsoiseq.(2)However,theR2ofeq.(1)willbeverydifferentfromtheR2ofeq.(2)Itiseasytoshowthatthesame“economic”modelcanyieldverydifferentR2dependingonhowthevariablesaretransformed:57Example:use"J:\phd\Fees.dta",cleargenlnaf=ln(auditfees)genlnta=ln(totalassets)sortcompanyidyearendbycompanyid:genlnaf_lag=lnaf[_n-1]egenmiss=rmiss(lnaflntalnaf_lag)genchlnaf=lnaf-lnaf_lagreglnaflntalnaf_lagifmiss==0regchlnaflntalnaf_lagifmiss==0Thelntacoefficientsareexactlythesameinthetwomodels.Thelnaf_lagcoefficientineq.(2)equalsthelnaf_lagcoefficientineq.(1)minusone.TheR2ismuchhigherineq.(1)thaneq.(2).ThehighR2ineq.(1)doesnotimplythatthemodeliswell-specified.ThelowR2ineq.(2)doesnotimplythatthemodelismis-specified.Eitherbothequationsarewell-specifiedortheyarebothmis-specified.TheR2tellsusnothingaboutwhetherourhypothesisaboutthedeterminantsofYiscorrect.582.4ExaminingtheresidualsInsteadofrelyingonlyontheR2,anexaminationoftheresidualscanhelpusidentifywhetherthemodeliswellspecified.Forexamplecomparetheauditfeemodelwhichisnotlogged:regauditfeestotalassetspredictres1,residtwoway(scatterres1totalassets,msize(tiny))(lfitres1totalassets)Withtheloggedauditfeemodelreglnaflntapredictres2,residtwoway(scatterres2lnta,msize(tiny))(lfitres2lnta)Noticethattheresidualsaremore“spherical”displayinglessofanobviouspatternintheloggedmodel.592.4ExaminingtheresidualsInordertoobtainunbiasedstandarderrorswehavetoassumethattheresidualsarenormallydistributedWecantestthisusingahistogramoftheresidualshistres1thisdoesnotgiveuswhatweneedbecausetherearesevereoutlierssumres1,detailhistres1ifres1>-22&res1<208,normalxlabel(-25(25)210)

histres2sumres2,detailhistres2ifres2>-2&res2<1.8,normalxlabel(-2(0.5)2)Theresidualsaremuchclosertotheassumednormaldistributionwhenthevariablesaremeasuredinlogs6061Classexercise2dFollowingPongandWhittington(1994)estimatetherawvalueofauditfeesasafunctionofrawassetsandassetssquaredExaminetheresidualsDoyouthinkthismodelisbetterspecifiedthantheoneinlogs?622.5MultipleregressionResearchersuse“multipleregression”whentheybelievethatYisaffectedbymultipleindependentvariables:Y=a0+a1X1+a2X2+eWhyisitimportanttocontrolformultiplefactorsthatinfluenceY?632.5MultipleregressionSupposethe“true”modelis:Y=a0+a1X1+a2X2+ewhereX1andX2isuncorrelatedwiththeerror,eSupposetheOLSmodelthatweestimateis:Y=a0+a1X1+uwhereu=a2X2+eOLSimposestheassumptionthatX1isuncorrelatedwiththeresidualterm,u.SinceX1isuncorrelatedwithe,theassumptionthatX1isuncorrelatedwithuisequivalenttoassumingthatX1isuncorrelatedX2.642.5MultipleregressionIfX1iscorrelatedwithX2theOLSestimateofa1isbiased.ThemagnitudeofthebiasdependsuponthestrengthofthecorrelationbetweenX1andX2.Ofcourse,weoftendonotknowwhetherthemodelweestimateisthe“true”modelInotherwords,weareunsurewhetherthereisanomittedvariable(X2)thataffectsYandthatiscorrelatedwithourvariableofinterest(X1)652.5MultipleregressionWecanjudgewhetherornotthereislikelytobeacorrelatedomittedvariableproblemusing:theorypriorempiricalstudies662.5MultipleregressionPreviously,whenwewereusingsimpleregressionwithoneindependentvariable,wecheckedwhethertherewasapatternbetweentheresidualsandtheindependentvariablelnaf=a0+a1lnta+res1twoway(scatterres1lnta)(lfitres1lnta)Whenweareusingmultipleregression,wewanttotestwhetherthereisapatternbetweentheresidualsandtherighthandsideoftheequationasawholeTherighthandsideoftheequation“asawhole”isthesamethingasthepredictedvalueofthedependentvariable672.5MultipleregressionSoweshouldexaminewhetherthereisapatternbetweentheresidualsandthepredictedvaluesofthedependentvariableForexample,let’sestimateamodelwhereauditfeesdependoncompanysize,auditfirmsize,andwhetherthecompanyislistedonastockmarketgenlisted=0replacelisted=1ifcompanytype==2|companytype==3|companytype==5reglnaflntabig6listedpredictlnaf_hatpredictlnaf_res,residtwoway(scatterlnaf_reslnaf_hat)(lfitlnaf_reslnaf_hat)682.5Multipleregression(rvfplot)Infact,thoseniceguysatSTATAhavegivenusacommandwhichenablesustoshort-cuthavingtousethepredictcommandforcalculatingtheresidualsandthefittedvaluesreglnaflntabig6listedrvfplotrvfstandsforresidualsversusfitted692.6Heteroscedasticity(hettest)TheOLStechniquesforestimatingstandarderrorsarebasedonanassumptionthatthevarianceoftheerrorsisthesameforallvaluesoftheindependentvariables(homoscedasticity)Inmanycases,thehomoscedasticityassumptionisclearlyviolated.Forexample:regauditfeesnonauditfeestotalassetsbig6listedrvfplotthehomoscedasticityassumptioncanbetestedusingthehettestcommandafterwedotheregressionregauditfeesnonauditfeestotalassetsbig6listedhettestHeteroscedasticitydoesnotbiasthecoefficientestimatesbutitdoesbiasthestandarderrorsofthecoefficients702.6Heteroscedasticity(robust)Heteroscedasticityisoftencausedbyusingadependentvariablethatisnotsymmetricforexampletheauditfeesvariableishighlyskewedduetothefactthatithasalowerboundofzeromuchoftheheterosedasticitycanoftenberemovedbytransformingthedependentvariable(e.g.,usethelogofauditfeesinsteadoftherawvalues)Whenyoufindthatthereisheteroscedasticity,youneedtoadjustthestandarderrorsusingtheHuber/White/sandwichestimatorInSTATAitiseasytodothisadjustmentusingtherobustoptionregauditfeesnonauditfeestotalassetsbig6listed,robustComparetheadjustedandunadjustedresultsregauditfeesnonauditfeestotalassetsbig6listedWhatisdifferent?Whatisthesame?71Classexercise2eEsimatetheauditfeemodelinlogsratherthanabsolutevaluesUsingrvfplot,assesswhethertheresidualsappeartobenon-constantUsinghettest,provideaformaltestforheteroscedasticityComparethecoefficientsandt-statisticswhenyouestimatethestandarderrorswithandwithoutadjustingforheteroscedasticity.722.7CorrelatederrorsTheOLStechniquesforestimatingstandarderrorsarebasedonanassumptionthattheerrorsarenotcorrelatedThisassumptionistypicallyviolatedwhenweuserepeatedannualobservationsonthesamecompaniesTheresidualsofagivenfirmarecorrelatedacrossyears(“timeseriesdependence”)73Time-seriesdependenceTime-seriesdependenceisnearlyalwaysaproblemwhenresearchersuse“paneldata”Paneldata=datathatarepooledforthesamecompaniesacrosstimeInpaneldata,therearelikelytobeunobservedcompany-specificcharacteristicsthatarerelativelyconstantovertimeCompanyYearA1996A1997B1996B199774Let’sstartwithasimpleregressionmodelwheretheerrorsareassumedtobeuncorrelatedWenowrelaxtheassumptionofindependenterrorsbyassumingthattheerrortermhasanunobservedcompany-specificcomponentthatdoesnotvaryovertimeandanidiosyncraticcomponentthatisuniquetoeachcompany-yearobservation:Similarly,wecanassumethattheXvariablehasacompany-specificcomponentthatdoesnotvaryovertimeandanidiosyncraticcomponent:75Time-seriesdependenceInthiscase,theOLSstandarderrorstendtobebiaseddownwardsandthemagnitudeofthisbiasisincreasinginthenumberofyearswithinthepanel.Tounderstandtheintuition,considertheextremecasewheretheresidualsandindependentvariablesareperfectlycorrelatedacrosstime.Inthiscase,eachadditionalyearprovidesnoadditionalinformationandwillhavenoeffectonthetruestandarderrorHowever,undertheincorrectassumptionoftime-seriesindependence,itisassumedthateachadditionalyearprovidesadditionalobservationsandtheestimatedstandarderrorswillshrinkaccordinglyandincorrectlyThisproblemcanbeavoidedbyadjustingthestandarderrorsfortheclusteringofyearlyobservationsacrossagivencompany76Time-seriesdependenceTounderstandallthis,itishelpfultoreviewthefollowingexampleFirst,Iestimatethemodelusingjustoneobservationforeachcompany(intheyear1998)genfye=date(yearend,"MDY")genyear=year(fye)dropifyear!=1998sortcompanyiddropifcompanyid==companyid[_n-1]reglnaflntabig6listed,robust77Time-seriesdependenceNowIcreateadatasetinwhicheachobservationisduplicatedEachduplicatedobservationprovidesnoadditionalinformationandwillhavenoeffectonthetruestandarderrorbutitwillreducetheestimatedstandarderror(i.e.,theestimatedstandarderrorwillbebiaseddownwards)save"J:\phd\Fees98.dta",replaceappendusing"J:\phd\Fees98.dta"reglnaflntabig6listed,robustWhat’shappenedtothecoefficientsandt-statistics?78Time-seriesdependence

(robustcluster())Wecanobtaincorrectstandarderrorsintheduplicatedatasetusingtherobustcluster()optionwhichadjuststhestandarderrorsforclusteringofobservations(heretheyareduplicated)foreachcompanyreglnaflntabig6listed,robustcluster(companyid)

What’shappenedtothecoefficientsandt-statistics?79Time-seriesdependenceInrealitytheobservationsofagivencompanyarenotexactlythesamefromoneyeartothenext(i.e.,theyarenotexactduplicates).However,theobservationsofagivencompanyoftendonotchangemuchfromoneyeartothenext.Forexample,acompany’ssizeandthefeesthatitpaysmaynotchangemuchovertime(i.e.,thereisastrongunobservedcompany-specificcomponenttothevariables).Failingtoaccountforthisinpaneldatatendstooverstatethemagnitudeofthet-statistics.80Time-seriesdependenceItiseasytodemonstratethattheresidualsofagivencompanytendtobeveryhighlycorrelatedovertimeFirst,startagainwiththeoriginaldatause"J:\phd\Fees.dta",cleargenfye=date(yearend,"MDY")genyear=year(fye)genlnaf=ln(auditfees)genlnta=ln(totalassets)save"J:\phd\Fees1.dta",replaceEstimatethefeemodelandobtaintheresidualsforeachcompany-yearobservationreglnaflntapredictres,resid81Time-seriesdependenceReshapethedatasothatwehaveeachcompanyasarowandt