2regresspostestimation—Postestimationtoolsforregress - PDF document

2regresspostestimation—Postestimationtoolsforregress Thefollowingstandardpostestimationcommandsarealsoavailable:CommandDescription contrastcontrastsandANOVA-stylejointtestsofestimatesestaticAkaike'sandSchwarz'sBayesianinformationcriteria(AICandBIC)estatsummarizesummarystatisticsfortheestimationsampleestatvcevariance–covariancematrixoftheestimators(VCE)estat(svy)postestimationstatisticsforsurveydataestimatescatalogingestimationresultsforecast1dynamicforecastsandsimulationshausmanHausman'sspecicationtestlincompointestimates,standarderrors,testing,andinferenceforlinearcombinationsofcoefcientslinktestlinktestformodelspecicationlrtest2likelihood-ratiotestmarginsmarginalmeans,predictivemargins,marginaleffects,andaveragemarginaleffectsmarginsplotgraphtheresultsfrommargins(proleplots,interactionplots,etc.)nlcompointestimates,standarderrors,testing,andinferencefornonlinearcombinationsofcoefcientspredictpredictions,residuals,inuencestatistics,andotherdiagnosticmeasurespredictnlpointestimates,standarderrors,testing,andinferenceforgeneralizedpredictionspwcomparepairwisecomparisonsofestimatessuestseeminglyunrelatedestimationtestWaldtestsofsimpleandcompositelinearhypothesestestnlWaldtestsofnonlinearhypotheses 1forecastisnotappropriatewithmiorsvyestimationresults.2lrtestisnotappropriatewithsvyestimationresults. 4regresspostestimation—Postestimationtoolsforregress rstandardcalculatesthestandardizedresiduals.rstudentcalculatestheStudentized(jackknifed)residuals.cooksdcalculatestheCook'sDinuencestatistic(Cook1977).leverageorhatcalculatesthediagonalelementsoftheprojection(“hat”)matrix.pr(a,b)calculatesPr(axjb+ujb),theprobabilitythatyjjxjwouldbeobservedintheinterval(a;b).aandbmaybespeciedasnumbersorvariablenames;lbandubarevariablenames;pr(20,30)calculatesPr(20xjb+uj30);pr(lb,ub)calculatesPr(lbxjb+ujub);andpr(20,ub)calculatesPr(20xjb+ujub).amissing(a:)means�1;pr(.,30)calculatesPr(�1xjb+uj30);pr(lb,30)calculatesPr(�1xjb+uj30)inobservationsforwhichlb:andcalculatesPr(lbxjb+uj30)elsewhere.bmissing(b:)means+1;pr(20,.)calculatesPr(+1&#x]TJ/;ྒ ; .56;A T; 10;&#x.516;&#x 0 T; [0;xjb+uj&#x]TJ/;ྒ ; .56;A T; 10;&#x.516;&#x 0 T; [0;20);pr(20,ub)calculatesPr(+1&#x]TJ/;ྒ ; .56;A T; 10;&#x.516;&#x 0 T; [0;xjb+uj&#x]TJ/;ྒ ; .56;A T; 10;&#x.516;&#x 0 T; [0;20)inobservationsforwhichub:andcalculatesPr(20xjb+ujub)elsewhere.e(a,b)calculatesE(xjb+ujjaxjb+ujb),theexpectedvalueofyjjxjconditionalonyjjxjbeingintheinterval(a;b),meaningthatyjjxjistruncated.aandbarespeciedastheyareforpr().ystar(a,b)calculatesE(yj),whereyj=aifxjb+uja,yj=bifxjb+ujb,andyj=xjb+ujotherwise,meaningthatyjiscensored.aandbarespeciedastheyareforpr().dfbeta(varname)calculatestheDFBETAforvarname,thedifferencebetweentheregressioncoefcientwhenthejthobservationisincludedandexcluded,saiddifferencebeingscaledbytheestimatedstandarderrorofthecoefcient.varnamemusthavebeenincludedamongtheregressorsinthepreviouslyttedmodel.Thecalculationisautomaticallyrestrictedtotheestimationsubsample.stdpcalculatesthestandarderroroftheprediction,whichcanbethoughtofasthestandarderrorofthepredictedexpectedvalueormeanfortheobservation'scovariatepattern.Thestandarderrorofthepredictionisalsoreferredtoasthestandarderrorofthettedvalue.stdfcalculatesthestandarderroroftheforecast,whichisthestandarderrorofthepointpredictionfor1observation.Itiscommonlyreferredtoasthestandarderrorofthefutureorforecastvalue.Byconstruction,thestandarderrorsproducedbystdfarealwayslargerthanthoseproducedbystdp;seeMethodsandformulas.stdrcalculatesthestandarderroroftheresiduals.covratiocalculatesCOVRATIO(Belsley,Kuh,andWelsch1980),ameasureoftheinuenceofthejthobservationbasedonconsideringtheeffectonthevariance–covariancematrixoftheestimates.Thecalculationisautomaticallyrestrictedtotheestimationsubsample.dfitscalculatesDFITS(WelschandKuh1977)andattemptstosummarizetheinformationintheleverageversusresidual-squaredplotintoonestatistic.Thecalculationisautomaticallyrestrictedtotheestimationsubsample.welschcalculatesWelschdistance(Welsch1982)andisavariationondfits.Thecalculationisautomaticallyrestrictedtotheestimationsubsample. 6regresspostestimation—Postestimationtoolsforregress Nowallthisisamostunsatisfactorystateofaffairs.Pointswithlargeresidualsmay,butneednot,havealargeeffectonourresults,andpointswithsmallresidualsmaystillhavealargeeffect.Pointswithhighleveragemay,butneednot,havealargeeffectonourresults,andpointswithlowleveragemaystillhavealargeeffect.Canyounotidentifytheinuentialpointsandsimplyhavethecomputerlistthemforyou?Youcan,butyouwillhavetodenewhatyoumeanby“inuential”.“Inuential”isdenedwithrespecttosomestatistic.Forinstance,youmightaskwhichpointsinyourdatahavealargeeffectonyourestimateda,whichpointshavealargeeffectonyourestimatedb,whichpointshavealargeeffectonyourestimatedstandarderrorofb,andsoon,butdonotbesurprisedwhentheanswerstothesequestionsaredifferent.Inanycase,obtainingsuchmeasuresisnotdifcult—allyouhavetodoisttheregressionexcludingeachobservationoneatatimeandrecordthestatisticofinterestwhich,inthedayofthemoderncomputer,isnottooonerous.Moreover,youcansaveconsiderablecomputertimebydoingalgebraaheadoftimeandworkingoutformulasthatwillcalculatethesameanswersasifyouraneachoftheregressions.(Ignorethequestionofpairsofobservationsthat,together,exertundueinuence,andtriples,andsoon,whichremainslargelyunsolvedandforwhichthebruteforcet-every-possible-regressionprocedureisnotaviablealternative.)FittedvaluesandresidualsTypingpredictnewvarwithnooptionscreatesnewvarcontainingthettedvalues.Typingpredictnewvar,residcreatesnewvarcontainingtheresiduals. Example1Continuingwithexample1from[R]regress,wewishtotthefollowingmodel:mpg=0+1weight+2foreign+.usehttp://www.stata-press.com/data/r13/auto(1978AutomobileData).regressmpgweightforeignSource SSdfMSNumberofobs=74 F(2,71)=69.75Model 1619.28772809.643849Prob�F=0.0000Residual 824.1717617111.608053R-squared=0.6627 AdjR-squared=0.6532Total 2443.459467333.4720474RootMSE=3.4071 mpg Coef.Std.Err.t�P|t|[95%Conf.Interval] weight -.0065879.0006371-10.340.000-.0078583-.0053175foreign -1.6500291.075994-1.530.130-3.7955.4954422_cons 41.67972.16554719.250.00037.3617245.99768 Thatdone,wecannowobtainthepredictedvaluesfromtheregression.Wewillstoretheminanewvariablecalledpmpgbytypingpredictpmpg.Becausepredictproducesnooutput,wewillfollowthatbysummarizingourpredictedandobservedvalues. 10regresspostestimation—Postestimationtoolsforregress StandardizedandStudentizedresidualsThetermsstandardizedandStudentizedresidualshavemeantdifferentthingstodifferentauthors.InStata,predictdenesthestandardizedresidualasbei=ei=(sp 1�hi)andtheStudentizedresidualasri=ei=(s(i)p 1�hi),wheres(i)istherootmeansquarederrorofaregressionwiththeithobservationremoved.Stata'sdenitionoftheStudentizedresidualisthesameastheonegiveninBollenandJackman(1990,264)andiswhatChatterjeeandHadi(1988,74)callthe“externallyStudentized”residual.Stata's“standardized”residualisthesameaswhatChatterjeeandHadi(1988,74)callthe“internallyStudentized”residual.StandardizedandStudentizedresidualsareattemptstoadjustresidualsfortheirstandarderrors.Althoughtheitheoreticalresidualsarehomoskedasticbyassumption(thatis,theyallhavethesamevariance),thecalculatedeiarenot.Infact,Var(ei)=2(1�hi)wherehiaretheleveragemeasuresobtainedfromthediagonalelementsofhatmatrix.Thusobservationswiththegreatestleveragehavecorrespondingresidualswiththesmallestvariance.Standardizedresidualsusetherootmeansquarederroroftheregressionfor.Studentizedresidualsusetherootmeansquarederrorofaregressionomittingtheobservationinquestionfor.Ingeneral,Studentizedresidualsarepreferabletostandardizedresidualsforpurposesofoutlieridentication.Studentizedresidualscanbeinterpretedasthetstatisticfortestingthesignicanceofadummyvariableequalto1intheobservationinquestionand0elsewhere(Belsley,Kuh,andWelsch1980).Suchadummyvariablewouldeffectivelyabsorbtheobservationandsoremoveitsinuenceindeterminingtheothercoefcientsinthemodel.Cautionmustbeexercisedhere,however,becauseofthesimultaneoustestingproblem.Youcannotsimplylisttheresidualsthatwouldbeindividuallysignicantatthe5%level—theirjointsignicancewouldbefarless(theirjointsignicancelevelwouldbefargreater). Example5:standardizedandStudentizedresidualsIntheTerminologysectionofRemarksandexamplesforpredict,wedistinguishedresidualsfromleverageandspeculatedontheimpactofanobservationwithasmallresidualbutlargeleverage.Ifweadjusttheresidualsfortheirstandarderrors,however,theadjustedresidualwouldbe(relatively)largerandperhapslargeenoughsothatwecouldsimplyexaminetheadjustedresiduals.Takingourpriceonweightandforeign##c.mpgmodelfromexample1of[R]regresspostestimationdiagnosticplots,wecanobtainthein-samplestandardizedandStudentizedresidualsbytyping.usehttp://www.stata-press.com/data/r13/auto,clear(1978AutomobileData).regresspriceweightforeign##c.mpg(outputomitted).predictestaife(sample),rstandard.predictestuife(sample),rstudent 12regresspostestimation—Postestimationtoolsforregress Example6:DFITSinuencemeasureContinuingwithourmodelofpriceonweightandforeign##c.mpg,wecanobtaintheDFITSinuencemeasure:.predicteife(sample),resid.predictdfits,dfitsWedidnotspecifyife(sample)incomputingtheDFITSstatistic.DFITSisavailableonlyovertheestimationsample,sospecifyingife(sample)wouldhavebeenredundant.Itwouldhavedonenoharm,butitwouldnothavechangedtheresults.Ourmodelhask=5independentvariables(kincludestheconstant)andn=74observations;followingthe2p k=ncutoffadvice,wetype.listmakepriceedfitsifabs(dfits)�2*sqrt(5/74),divider make price e dfits 12. Cad.Eldorado 14,500 7271.96 .9564455 13. Cad.Seville 15,906 5036.348 1.356619 24. FordFiesta 4,389 3164.872 .5724172 27. Linc.MarkV 13,594 3109.193 .5200413 28. Linc.Versailles 13,466 6560.912 .8760136 42. Plym.Arrow 4,647 -3312.968 -.9384231 WecalculateCook'sdistanceandlisttheobservationsgreaterthanthesuggested4=ncutoff:.predictcooksdife(sample),cooksd.listmakepriceecooksdifcooksd�4/74,divider make price e cooksd 12. Cad.Eldorado 14,500 7271.96 .1492676 13. Cad.Seville 15,906 5036.348 .3328515 24. FordFiesta 4,389 3164.872 .0638815 28. Linc.Versailles 13,466 6560.912 .1308004 42. Plym.Arrow 4,647 -3312.968 .1700736 Hereweusedife(sample)becauseCook'sdistanceisnotrestrictedtotheestimationsamplebydefault.Itisworthcomparingthislistwiththeprecedingone.Finally,weuseWelschdistanceandthesuggested3p kcutoff:.predictwd,welsch.listmakepriceewdifabs(wd)�3*sqrt(5),divider make price e wd 12. Cad.Eldorado 14,500 7271.96 8.394372 13. Cad.Seville 15,906 5036.348 12.81125 28. Linc.Versailles 13,466 6560.912 7.703005 42. Plym.Arrow 4,647 -3312.968 -8.981481 Herewedidnotneedtospecifyife(sample)becausewelschautomaticallyrestrictsthepredictiontotheestimationsample. 14regresspostestimation—Postestimationtoolsforregress DFBETAinuencestatisticsSyntaxfordfbetadfbetaindepvarindepvar:::,stub(name)MenufordfbetaStatistics�Linearmodelsandrelated�Regressiondiagnostics�DFBETAsDescriptionfordfbetadfbetawillcalculateone,morethanone,oralltheDFBETAsafterregress.AlthoughpredictwillalsocalculateDFBETAs,predictcandothisforonlyonevariableatatime.dfbetaisaconveniencetoolforthosewhowanttocalculateDFBETAsformultiplevariables.Thenamesforthenewvariablescreatedarechosenautomaticallyandbeginwiththeletters dfbeta .Optionfordfbetastub(name)speciestheleadingcharactersdfbetausestonamethenewvariablestobegenerated.Thedefaultisstub( dfbeta ).RemarksandexamplesfordfbetaDFBETAsareperhapsthemostdirectinuencemeasureofinteresttomodelbuilders.DFBETAsfocusononecoefcientandmeasurethedifferencebetweentheregressioncoefcientwhentheithobservationisincludedandexcluded,thedifferencebeingscaledbytheestimatedstandarderrorofthecoefcient.Belsley,Kuh,andWelsch(1980,28)suggestobservationswithjDFBETAij�2=p nasdeservingspecialattention,butitisalsocommonpracticetouse1(BollenandJackman1990,267),meaningthattheobservationshiftedtheestimateatleastonestandarderror. Example8:DFBETAsinuencemeasure;thedfbeta()optionUsingourmodelofpriceonweightandforeign##c.mpg,let'srstaskwhichobservationshavethegreatestimpactonthedeterminationofthecoefcienton1.foreign.Wewillusethesuggested2=p ncutoff:.usehttp://www.stata-press.com/data/r13/auto,clear(1978AutomobileData).regresspriceweightforeign##c.mpg(outputomitted) 16regresspostestimation—Postestimationtoolsforregress .dfbeta_dfbeta_7:dfbeta(weight)_dfbeta_8:dfbeta(1.foreign)_dfbeta_9:dfbeta(mpg)_dfbeta_10:dfbeta(1.foreign#c.mpg).dfbetampgweight_dfbeta_11:dfbeta(weight)_dfbeta_12:dfbeta(mpg)dfbetawouldhavemadeupdifferentnamesforthenewvariables.dfbetaneverreplacesexistingvariables—itinsteadmakesupadifferentname,soweneedtopayattentiontodfbeta'soutput. TestsforviolationofassumptionsSyntaxforestathettestestathett estvarlist,r hsno rmaljii djfs tatm test(spec)MenuforestatStatistics�Postestimation�ReportsandstatisticsDescriptionforestathettestestathettestperformsthreeversionsoftheBreusch–Pagan(1979)andCook–Weisberg(1983)testforheteroskedasticity.Allthreeversionsofthistestpresentevidenceagainstthenullhypothesisthatt=0inVar(e)=2exp(zt).Inthenormalversion,performedbydefault,thenullhypothesisalsoincludestheassumptionthattheregressiondisturbancesareindependent-normaldrawswithvariance2.Thenormalityassumptionisdroppedfromthenullhypothesisintheiidandfstatversions,whichrespectivelyproducethescoreandFtestsdiscussedinMethodsandformulas.Ifvarlistisnotspecied,thettedvaluesareusedforz.Ifvarlistortherhsoptionisspecied,thevariablesspeciedareusedforz.Optionsforestathettestrhsspeciesthattestsforheteroskedasticitybeperformedfortheright-hand-side(explanatory)variablesofthettedregressionmodel.Therhsoptionmaybecombinedwithavarlist.normal,thedefault,causesestathettesttocomputetheoriginalBreusch–Pagan/Cook–Weisbergtest,whichassumesthattheregressiondisturbancesarenormallydistributed.iidcausesestathettesttocomputetheNR2versionofthescoretestthatdropsthenormalityassumption.fstatcausesestathettesttocomputetheF-statisticversionthatdropsthenormalityassumption. regresspostestimation—Postestimationtoolsforregress17 mtest(spec)speciesthatmultipletestingbeperformed.Theargumentspecieshowp-valuesareadjusted.Thefollowingspecications,spec,aresupported:b onferroniBonferroni'smultipletestingadjustmenth olmHolm'smultipletestingadjustments idakSid´ak'smultipletestingadjustmentnoadj ustnoadjustmentismadeformultipletestingmtestmaybespeciedwithoutanargument.Thisisequivalenttospecifyingmtest(noadjust);thatis,testsfortheindividualvariablesshouldbeperformedwithunadjustedp-values.Bydefault,estathettestdoesnotperformmultipletesting.mtestmaynotbespeciedwithiidorfstat.Syntaxforestatimtestestatimt est,p reservewh iteMenuforestatStatistics�Postestimation�ReportsandstatisticsDescriptionforestatimtestestatimtestperformsaninformationmatrixtestfortheregressionmodelandanorthogonalde-compositionintotestsforheteroskedasticity,skewness,andkurtosisduetoCameronandTrivedi(1990);White'stestforhomoskedasticityagainstunrestrictedformsofheteroskedasticity(1980)isavailableasanoption.White'stestisusuallysimilartothersttermoftheCameron–Trivedidecomposition.Optionsforestatimtestpreservespeciesthatthedatainmemorybepreserved,allvariablesandcasesthatarenotneededinthecalculationsbedropped,andattheconclusiontheoriginaldataberestored.Thisoptioniscostlyforlargedatasets.However,becauseestatimtesthastoperformanauxiliaryregressiononk(k+1)=2temporaryvariables,wherekisthenumberofregressors,itmaynotbeabletoperformthetestotherwise.whitespeciesthatWhite'soriginalheteroskedasticitytestalsobeperformed.Syntaxforestatovtestestatovt est,r hsMenuforestatStatistics�Postestimation�Reportsandstatistics regresspostestimation—Postestimationtoolsforregress19 Remarksandexamplesforestathettest,estatimtest,estatovtest,andestatszroeterWeintroducesomeregressiondiagnosticcommandsthataredesignedtotestforcertainviolationsthatrvfplot(see[R]regresspostestimationdiagnosticplots)lessformallyattemptstodetect.estatovtestprovidesRamsey'stestforomittedvariables—apatternintheresiduals.estathettestprovidesatestforheteroskedasticity—theincreasingordecreasingvariationintheresidualswithttedvalues,withrespecttotheexplanatoryvariables,orwithrespecttoyetothervariables.Thescoretestimplementedinestathettest(BreuschandPagan1979;CookandWeisberg1983)performsascoretestofthenullhypothesisthatb=0againstthealternativehypothesisofmultiplicativeheteroskedasticity.estatszroeterprovidesaranktestforheteroskedasticity,whichisanalternativetothescoretestcomputedbyestathettest.Finally,estatimtestcomputesaninformationmatrixtest,includinganorthogonaldecompositionintotestsforheteroskedasticity,skewness,andkurtosis(CameronandTrivedi1990).TheheteroskedasticitytestcomputedbyestatimtestissimilartothegeneraltestforheteroskedasticitythatwasproposedbyWhite(1980).CameronandTrivedi(2010,chap.3)discussmostofthesetestsandprovidesmoreexamples. Example10:estatovtest,estathettest,estatszroeter,andestatimtestWeuseourmodelofpriceonweightandforeign##c.mpg..usehttp://www.stata-press.com/data/r13/auto,clear(1978AutomobileData).regresspriceweightforeign##c.mpg(outputomitted).estatovtestRamseyRESETtestusingpowersofthefittedvaluesofpriceHo:modelhasnoomittedvariablesF(3,66)=7.77Prob�F=0.0002.estathettestBreusch-Pagan/Cook-WeisbergtestforheteroskedasticityHo:ConstantvarianceVariables:fittedvaluesofpricechi2(1)=6.50Prob�chi2=0.0108Testingforheteroskedasticityintheright-hand-sidevariablesisrequestedbyspecifyingtherhsoption.Byspecifyingthemtest(bonferroni)option,werequestthattestsbeconductedforeachofthevariables,withaBonferroniadjustmentforthep-valuestoaccommodateourtestingmultiplehypotheses. regresspostestimation—Postestimationtoolsforregress21 werenotamanual,havingfoundevidenceofomittedvariables,wewouldneverhaveruntheestathettest,estatszroeter,andestatimtestcommands,atleastnotuntilwesolvedtheomitted-variableproblem. Technicalnoteestatovtestandestathettestbothperformtwoavorsoftheirrespectivetests.Bydefault,estatovtestlooksforevidenceofomittedvariablesbyttingtheoriginalmodelaugmentedbyby2,by3,andby4,whicharethettedvaluesfromtheoriginalmodel.Undertheassumptionofnomisspecication,thecoefcientsonthepowersofthettedvalueswillbezero.Withtherhsoption,estatovtestinsteadaugmentstheoriginalmodelwithpowers(secondthroughfourth)oftheexplanatoryvariables(exceptfordummyvariables).estathettest,bydefault,looksforheteroskedasticitybymodelingthevarianceasafunctionofthettedvalues.If,however,wespecifyavariableorvariables,thevariancewillbemodeledasafunctionofthespeciedvariables.Inourexample,ifwehad,apriori,somereasontosuspectheteroskedasticityandthattheheteroskedasticityisafunctionofacar'sweight,thenusingatestthatfocusesonweightwouldbemorepowerfulthanthemoregeneraltestssuchasWhite'stestorthersttermintheCameron–Trivedidecompositiontest.estathettest,bydefault,computestheoriginalBreusch–Pagan/Cook–Weisbergtest,whichincludestheassumptionofnormallydistributederrors.Koenker(1981)derivedanNR2versionofthistestthatdropsthenormalityassumption.Wooldridge(2013)givesanF-statisticversionthatdoesnotrequirethenormalityassumption. Storedresultsforestathettest,estatimtest,andestatovtestestathetteststoresthefollowingresultsforthe(multivariate)scoretestinr():Scalarsr(chi2)2teststatisticr(df)#dffortheasymptotic2distributionunderH0r(p)p-valueestathettest,fstatstoresresultsforthe(multivariate)scoretestinr():Scalarsr(F)teststatisticr(df m)#dfofthetestfortheFdistributionunderH0r(df r)#dfoftheresidualsfortheFdistributionunderH0r(p)p-valueestathettest(ifmtestisspecied)andestatszroeterstorethefollowinginr():Matricesr(mtest)amatrixoftestresults,withrowscorrespondingtotheunivariatetestsmtest[.,1]2teststatisticmtest[.,2]#dfmtest[.,3]unadjustedp-valuemtest[.,4]adjustedp-value(ifanmtest()adjustmentmethodisspecied)Macrosr(mtmethod)adjustmentmethodforp-values regresspostestimation—Postestimationtoolsforregress23 Dataanalystsrelyonthesefactstocheckinformallyforthepresenceofmulticollinearity.estatvif,anothercommandforuseafterregress,calculatesthevarianceinationfactorsandtolerancesforeachoftheindependentvariables.Theoutputshowsthevarianceinationfactorstogetherwiththeirreciprocals.Someanalystscomparethereciprocalswithapredeterminedtolerance.Inthecomparison,ifthereciprocaloftheVIFissmallerthanthetolerance,theassociatedpredictorvariableisremovedfromtheregressionmodel.However,mostanalystsrelyoninformalrulesofthumbappliedtotheVIF;seeChatterjeeandHadi(2012).Accordingtotheserules,thereisevidenceofmulticollinearityif1.ThelargestVIFisgreaterthan10(somechooseamoreconservativethresholdvalueof30).2.ThemeanofalltheVIFsisconsiderablylargerthan1. Example11:estatvifWeexaminearegressionmodeltusingtheubiquitousautomobiledataset:.usehttp://www.stata-press.com/data/r13/auto(1978AutomobileData).regresspricempgrep78trunkheadroomlengthturndisplgear_ratioSource SSdfMSNumberofobs=69 F(8,60)=6.33Model 264102049833012756.2Prob�F=0.0000Residual 312694909605211581.82R-squared=0.4579 AdjR-squared=0.3856Total 576796959688482308.22RootMSE=2282.9 price Coef.Std.Err.t�P|t|[95%Conf.Interval] mpg -144.8482.12751-1.760.083-309.119519.43948rep78 727.5783337.61072.160.03552.256381402.9trunk 44.02061108.1410.410.685-172.2935260.3347headroom -807.0996435.5802-1.850.069-1678.3964.19062length -8.68891434.89848-0.250.804-78.4962661.11843turn -177.9064137.3455-1.300.200-452.638396.82551displacement 30.731467.5769524.060.00015.575345.88762gear_ratio 1500.1191110.9591.350.182-722.13033722.368_cons 6691.9767457.9060.900.373-8226.05821610.01 .estatvifVariable VIF1/VIF length 8.220.121614displacement 6.500.153860turn 4.850.205997gear_ratio 3.450.290068mpg 3.030.330171trunk 2.880.347444headroom 1.800.554917rep78 1.460.686147 MeanVIF 4.02Theresultsaremixed.AlthoughwehavenoVIFsgreaterthan10,themeanVIFisgreaterthan1,thoughnotconsiderablyso.Wecouldcontinuetheinvestigationofcollinearity,butgiventhatotherauthorsadvisethatcollinearityisaproblemonlywhenVIFsexistthataregreaterthan30(contradictingourruleabove),wewillnotdosohere. regresspostestimation—Postestimationtoolsforregress25 .estatvifVariable VIF1/VIF midarm 1.010.992831thigh 1.010.992831 MeanVIF 1.01Notehowthecoefcientschangeandhowtheestimatedstandarderrorsforeachoftheregressioncoefcientsbecomemuchsmaller.ThecalculatedvalueofR2fortheoverallregressionforthesubsetmodeldoesnotappreciablydeclinewhenweremovethecorrelatedpredictor.Removinganindependentvariablefromthemodelisonewaytodealwithmulticollinearity.Othermethodsincluderidgeregression,weightedleastsquares,andrestrictingtheuseofthettedmodeltodatathatfollowthesamepatternofmulticollinearity.Ineconomicstudies,itissometimespossibletoestimatetheregressioncoefcientsfromdifferentsubsetsofthedatabyusingcross-sectionandtimeseries. AllexamplesabovedemonstratedtheuseofcenteredVIFs.AspointedoutbyBelsley(1991),thecenteredVIFsmayfailtodiscovercollinearityinvolvingtheconstantterm.OnesolutionistousetheuncenteredVIFsinstead.AccordingtothedenitionoftheuncenteredVIFs,theconstantisviewedasalegitimateexplanatoryvariableinaregressionmodel,whichallowsonetoobtaintheVIFvaluefortheconstantterm. Example13:estatvif,withstrongevidenceofcollinearitywiththeconstanttermConsidertheextremeexampleinwhichoneoftheregressorsishighlycorrelatedwiththeconstant.WesimulatethedataandexaminebothcenteredanduncenteredVIFdiagnosticsafterttedregressionmodelasfollows..usehttp://www.stata-press.com/data/r13/extreme_collin.regressyonexzSource SSdfMSNumberofobs=100 F(3,96)=2710.27Model 223801.985374600.6617Prob�F=0.0000Residual 2642.421249627.5252213R-squared=0.9883 AdjR-squared=0.9880Total 226444.406992287.31723RootMSE=5.2464 y Coef.Std.Err.t�P|t|[95%Conf.Interval] one -3.27858210.5621-0.310.757-24.2441917.68702x 2.038696.024267384.010.0001.9905262.086866z 4.863137.268103618.140.0004.3309565.395319_cons 9.76007510.509350.930.355-11.1008230.62097 .estatvifVariable VIF1/VIF z 1.030.968488x 1.030.971307one 1.000.995425 MeanVIF 1.02 28regresspostestimation—Postestimationtoolsforregress .estatesizeEffectsizesforlinearmodels Source Eta-Squareddf[95%Conf.Interval] Model .12357363.0399862.2041365 smoke .07693451.0193577.1579213race .09083942.0233037.1700334 Theomegaoptioncausesestatesizetoreport!2andpartial!2..estatesize,omegaEffectsizesforlinearmodels Source Omega-Squareddf[95%Conf.Interval] Model .10936133.0244184.1912306 smoke .07194491.0140569.1533695race .08101062.0127448.1610608 Example15:CalculatingeffectsizeforanANOVAmodelWecanuseestatesizeafterANOVAmodelsaswell..anovabwtsmokeraceNumberofobs=189R-squared=0.1236RootMSE=687.999AdjR-squared=0.1094Source PartialSSdfMSFProb�F Model 12346897.634115632.548.690.0000 smoke 7298536.5717298536.5715.420.0001race 8749453.324374726.659.240.0001 Residual 87568400.9185473342.708 Total 99915298.6188531464.354.estatesizeEffectsizesforlinearmodels Source Eta-Squareddf[95%Conf.Interval] Model .12357363.0399862.2041365 smoke .07693451.0193577.1579213race .09083942.0233037.1700334 30regresspostestimation—Postestimationtoolsforregress DenotethepreviouslyestimatedcoefcientvectorbybanditsestimatedvariancematrixbyV.predictworksbyrecallingvariousaspectsofthemodel,suchasb,andcombiningthatinformationwiththedatacurrentlyinmemory.Letxjbethejthobservationcurrentlyinmemory,andlets2bethemeansquarederroroftheregression.Iftheuserspeciedweightsinregress,thenX0XinthefollowingformulasisreplacedbyX0DX,whereDisdenedinCoefcientestimationandANOVAtableunderMethodsandformulasin[R]regress.LetV=s2(X0X)�1.Letkbethenumberofindependentvariablesincludingtheintercept,ifany,andletyjbetheobservedvalueofthedependentvariable.Thepredictedvalue(xboption)isdenedasbyj=xjb.Let`jrepresentalowerboundforanobservationjandujrepresentanupperbound.Theprobabilitythatyjjxjwouldbeobservedintheinterval(`j;uj)—thepr(`,u)option—isP(`j;uj)=Pr(`jxjb+ejuj)=uj�byj s�`j�byj swhereforthepr(`,u),e(`,u),andystar(`,u)options,`jandujcanbeanywhereintherange(�1;+1).Theoptione(`,u)computestheexpectedvalueofyjjxjconditionalonyjjxjbeingintheinterval(`j;uj),thatis,whenyjjxjistruncated.ItcanbeexpressedasE(`j;uj)=E(xjb+ejj`jxjb+ejuj)=byj�suj�byj s�`j�byj s uj�byj s�`j�byj swhereisthenormaldensityandisthecumulativenormal.Youcanalsocomputeystar(`;u)—theexpectedvalueofyjjxj,whereyjisassumedcensoredat`janduj:yj=8:`jifxjb+ej`jxjb+uif`jxjb+ejujujifxjb+ejujThiscomputationcanbeexpressedinseveralways,butthemostintuitiveformulationinvolvesacombinationofthetwostatisticsjustdened:yj=P(�1;`j)`j+P(`j;uj)E(`j;uj)+P(uj;+1)ujAdiagonalelementoftheprojectionmatrix(hat)or(leverage)isgivenbyhj=xj(X0X)�1x0jThestandarderroroftheprediction(thestdpoption)isdenedasspj=q xjVx0jandcanalsobewrittenasspj=sp hj.Thestandarderroroftheforecast(stdf)isdenedassfj=sp 1+hj. 32regresspostestimation—Postestimationtoolsforregress Special­interestpostestimationcommandsTheomitted-variabletest(Ramsey1969)reportedbyestatovtesttstheregressionyi=xib+zit+uiandthenperformsastandardFtestoft=0.Thedefaulttestuseszi=(by2i;by3i;by4i).Ifrhsisspecied,zi=(x21i;x31i;x41i;x22i;:::;x4mi).Ineithercase,thevariablesarenormalizedtohaveminimum0andmaximum1beforepowersarecalculated.Thetestforheteroskedasticity(BreuschandPagan1979;CookandWeisberg1983)modelsVar(ei)=2exp(zt),wherezisavariablelistspeciedbytheuser,thelistofright-hand-sidevariables,orthettedvaluesxb.Thetestisoft=0.Mechanically,estathettesttstheaugmentedregressionbe2i=b2=a+zit+vi.TheoriginalBreusch–Pagan/Cook–WeisbergversionofthetestassumesthattheeiarenormallydistributedunderthenullhypothesiswhichimpliesthatthescoreteststatisticSisequaltothemodelsumofsquaresfromtheaugmentedregressiondividedby2.Underthenullhypothesis,Shasthe2distributionwithmdegreesoffreedom,wheremisthenumberofcolumnsofz.Koenker(1981)derivedascoretestofthenullhypothesisthatt=0undertheassumptionthattheeiareindependentandidenticallydistributed(i.i.d.).KoenkershowedthatS=NR2hasalarge-sample2distributionwithmdegreesoffreedom,whereNisthenumberofobservationsandR2istheR-squaredintheaugmentedregressionandmisthenumberofcolumnsofz.estathettest,iidproducesthisversionofthetest.Wooldridge(2013)showedthatanFtestoft=0intheaugmentedregressioncanalsobeusedundertheassumptionthattheeiarei.i.d.estathettest,fstatproducesthisversionofthetest.Szroeter'sclassoftestsforhomoskedasticityagainstthealternativethattheresidualvarianceincreasesinsomevariablexisdenedintermsofH=Pni=1h(xi)e2i Pni=1e2iwhereh(x)issomeweightfunctionthatincreasesinx(Szroeter1978).Hisaweightedaverageoftheh(x),withthesquaredresidualsservingasweights.Underhomoskedasticity,Hshouldbeapproximatelyequaltotheunweightedaverageofh(x).LargevaluesofHsuggestthate2itendstobelargewhereh(x)islarge;thatis,thevarianceindeedincreasesinx,whereassmallvaluesofHsuggestthatthevarianceactuallydecreasesinx.estatszroeterusesh(xi)=rank(xiinx1:::xn);seeJudgeetal.[1985,452]fordetails.estatszroeterdisplaysanormalizedversionofH,Q=r 6n n2�1HwhichisapproximatelyN(0;1)distributedunderthenull(homoskedasticity).estathettestandestatszroeterprovideadjustmentsofp-valuesformultipletesting.Thesupportedmethodsaredescribedin[R]test.estatimtestperformstheinformationmatrixtestfortheregressionmodel,aswellasanorthogonaldecompositionintotestsforheteroskedasticity1,nonnormalskewness2,andnonnormalkurtosis3(CameronandTrivedi1990;LongandTrivedi1993).Thedecompositionisobtainedviathreeauxiliaryregressions.Letebetheregressionresiduals,b2bethemaximumlikelihoodestimateof2intheregression,nbethenumberofobservations,Xbethesetofkvariablesspeciedwithestatimtest,andR2unbetheuncenteredR2fromaregression.1isobtainedasnR2unfromaregressionofe2�b2onthecrossproductsofthevariablesinX.2iscomputedasnR2unfromaregressionofe3�3b2eonX.Finally,3isobtainedasnR2unfromaregressionofe4�6b2e2�3b4 regresspostestimation—Postestimationtoolsforregress35 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