多元回歸分析假設(shè)檢驗(yàn)第一頁，共四十二頁，2022年，8月28日1AssumptionsoftheClassicalLinearModel(CLM)Sofar,weknowthatgiventheGauss-Markovassumptions,OLSisBLUE,Inordertodoclassicalhypothesistesting,weneedtoaddanotherassumption(beyondtheGauss-Markovassumptions)Assumethatuisindependentofx1,x2,…,xkanduisnormallydistributedwithzeromeanandvariance

s2:u

~

Normal(0,s2)第二頁，共四十二頁，2022年，8月28日2CLMAssumptions(cont.)UnderCLM,OLSisnotonlyBLUE,butistheminimumvarianceunbiasedestimator,whichmeansthatOLShasthesmallestvarianceamongunbiasedestimators;wenolongerhavetorestrictourcomparisontoestimatorsthatarelinearinyi.WecansummarizethepopulationassumptionsofCLMasfollowsy|x~Normal(b0+b1x1+…+bkxk,s2)Whilefornowwejustassumenormality,clearthatsometimesnotthecaseLargesampleswillletusdropnormality第三頁，共四十二頁，2022年，8月28日3..x1x2ThehomoskedasticnormaldistributionwithasingleexplanatoryvariableE(y|x)=b0+b1xyf(y|x)Normaldistributions第四頁，共四十二頁，2022年，8月28日4NormalSamplingDistributions第五頁，共四十二頁，2022年，8月28日5ThetTest第六頁，共四十二頁，2022年，8月28日6ThetTest(cont)KnowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistestsStartwithanullhypothesisForexample,H0:bj=0Ifacceptnull,thenacceptthatxjhasnoeffectony,controllingforotherx’s第七頁，共四十二頁，2022年，8月28日7ThetTest(cont)第八頁，共四十二頁，2022年，8月28日8tTest:One-SidedAlternatives

Besidesournull,H0,weneedanalternativehypothesis,H1,andasignificancelevelH1maybeone-sided,ortwo-sidedH1:bj>0andH1:bj<0areone-sidedH1:bj

0isatwo-sidedalternativeIfwewanttohaveonlya5%probabilityofrejectingH0ifitisreallytrue,thenwesayoursignificancelevelis5%第九頁，共四十二頁，2022年，8月28日9One-SidedAlternatives(cont)Havingpickedasignificancelevel,a,welookupthe(1–

a)thpercentileinatdistributionwithn–k–1dfandcallthisc,thecriticalvalueWecanrejectthenullhypothesisifthetstatisticisgreaterthanthecriticalvalueIfthetstatisticislessthanthecriticalvaluethenwefailtorejectthenull第十頁，共四十二頁，2022年，8月28日10yi=b0+b1xi1+…

+bkxik+uiH0:bj=0H1:bj>0One-SidedAlternatives(cont)0ca(1-a)Failtorejectreject第十一頁，共四十二頁，2022年，8月28日11AnExample:HourlyWageEquationWagedetermination:(wooldridge,p123)log(wage)=0.284+0.092educ+0.0041exper+0.022tenure

(0.104)(0.007)(0.0017)(0.003)

n=526R2=0.316Whetherthereturntoexper,controllingforeducandtenure,iszerointhepopulation,againstthealternativethatitispositive.H0:bexper=0vs.H1:bexper>0Thetstatisticist=0.0041/0.0017≈2.41Thedegreeoffreedom:df=n-k-1=526-3-1=522Thecriticalvalueof5%is1.645Andthetstatisticislargerthanthecriticalvalue,ie.,2.41>1.645Thatis,wewillrejectthenullhypothesisandbexperisreallypositive.01.645(1-a)Failtoreject5%reject第十二頁，共四十二頁，2022年，8月28日12Anotherexample:StudentPerformanceandSchoolSizeWhethertheschoolsizehaseffectonstudentperformance?math10,mathtestscores,revealthestudentperformancetotcomp,averageannualteachercompensationstaff,thenumberofstaffperonethousandstudentsenroll,studentenrollment,revealtheschoolsize.TheModelEquationmath10=b0+b1totcomp+b2staff+b3enrollH0:benroll=0,H1:benroll<0TheEstimatedEquationmath10=2.274+0.00046totcomp+0.048staff-0.00020enroll

(6.113)(0.00010)(0.040)(0.00022)

n=408,R2=0.0541df=408-3-1=404,t=-0.00020/0.00022≈-0.91,c=-1.645-0.91>-1.645,sowecan’trejectthenullhypothesis.-1.645reject-091第十三頁，共四十二頁，2022年，8月28日13One-sidedvsTwo-sidedBecausethetdistributionissymmetric,testingH1:bj<0isstraightforward.ThecriticalvalueisjustthenegativeofbeforeWecanrejectthenullifthetstatistic<–c,andifthetstatistic>than–cthenwefailtorejectthenullForatwo-sidedtest,wesetthecriticalvaluebasedona/2andrejectH1:bj

0iftheabsolutevalueofthetstatistic>c第十四頁，共四十二頁，2022年，8月28日14yi=b0+b1Xi1+…

+bkXik+uiH0:bj=0H1:bj>0c0a/2(1-a)-ca/2Two-SidedAlternativesrejectrejectfailtoreject第十五頁，共四十二頁，2022年，8月28日15SummaryforH0:bj=0Unlessotherwisestated,thealternativeisassumedtobetwo-sidedIfwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthea%level”Ifwefailtorejectthenull,wetypicallysay“xjisstatisticallyinsignificantatthea%level”第十六頁，共四十二頁，2022年，8月28日16AnExample:DeterminantsofCollegeGPA(wooldridge,p128)Variables:colGPA,collegeGPAskipped,theaveragenumberoflecturesmissedperweekACT,achievementtestscorehsGPA,highschoolGPATheestimatedmodel?olGPA=1.39+0.412hsGPA+0.015ACT–0.083skipped

(0.33)(0.094)(0.011)(0.026)

n=141,R2=0.234H0:bskipped=0,H1:bskipped≠

0fd:n-k-1=137,thecriticalvaluet137=1.96Thetstatisticis|-0.083/0.026|=3.19>

t137=1.96,sowewillrejectthenullhypothesisandthebskippedissignanificantlybeyondzero.-1.96reject-3.191.96reject第十七頁，共四十二頁，2022年，8月28日17TestingotherhypothesesAmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0:bj=aj

Inthiscase,theappropriatetstatisticis第十八頁，共四十二頁，2022年，8月28日18AnExample:CampusCrimeandEnrollment(wooldridge,p129)Variablescrime,theannualnumberofcrimesoncollegecampusesenroll,studentenrollment,revealthesizeofcollege.Theregressionmodellog(crime)=b0+b1log(enroll)+uWhetherb1=1,thatisH0:b1=1,H1:b1>1log(crime)=-6.63+1.27log(enroll)(1.03)(0.11)n=97R2=0.585df:n-k-1=95,thecriticalvalueat5%ist95=1.645Thet-statisticis(1.27-1)/0.11≈2.45>t95=1.645Sowerejectthenullhypothesisandtheevidenceprovethatb1>1.第十九頁，共四十二頁，2022年，8月28日19AnExample:HousePricesandAirPollution(wooldridge,p131)Variablesprice,medianhousingprice;nox,theamountofnitrogenoxideintheair,inpartspermillion;dist,aweighteddistanceofthecommunityfromfiveemploymentcenters,inmiles;rooms,theaveragenumberofroomsinhousesinthecommunityStratio,theaveragestudent-teacherratioofschoolsinthecommunity.Theestimatedmodellog(pri?e)=11.08-0.954log(nox)-0.134log(dis)+0.255rooms-0.052stratio(0.32)(0.117)(0.043)(0.019)(0.006)

n=506R2=581ThenullhypothesisH0:blog(nox)=-1,H1:blog(nox)≠-1Thetstatisticis(-0.954-(-1))/0.117=0.393,andthecriticalvaluet=93<1.96,sowecan’trejectthenullhypothesis.第二十頁，共四十二頁，2022年，8月28日20ConfidenceIntervals

Anotherwaytouseclassicalstatisticaltestingistoconstructaconfidenceintervalusingthesamecriticalvalueaswasusedforatwo-sidedtestA(1-a)%confidenceintervalisdefinedas第二十一頁，共四十二頁，2022年，8月28日21Computingp-valuesforttests

Analternativetotheclassicalapproachistoask,“whatisthesmallestsignificancelevelatwhichthenullwouldberejected?”So,computethetstatistic,andthenlookupwhatpercentileitisintheappropriatetdistribution–thisisthep-value

p-valueistheprobabilitywewouldobservethetstatisticwedid,ifthenullweretrue第二十二頁，共四十二頁，2022年，8月28日22Stataandp-values,ttests,etc.Mostcomputerpackageswillcomputethep-valueforyou,assumingatwo-sidedtestIfyoureallywantaone-sidedalternative,justdividethetwo-sidedp-valueby2Stataprovidesthetstatistic,p-value,and95%confidenceintervalforH0:bj=0foryou,incolumnslabeled“t”,“P>|t|”and“[95%Conf.Interval]”,respectively第二十三頁，共四十二頁，2022年，8月28日23TestingaLinearCombination

Supposeinsteadoftestingwhetherb1isequaltoaconstant,youwanttotestifitisequaltoanotherparameter,thatisH0:b1=b2,orb1-b2=0Usesamebasicprocedureforformingatstatistic第二十四頁，共四十二頁，2022年，8月28日24TestingLinearCombination(cont)第二十五頁，共四十二頁，2022年，8月28日25TestingaLinearCombo(cont)So,touseformula,needs12,whichstandardoutputdoesnothaveManypackageswillhaveanoptiontogetit,orwilljustperformthetestforyouInStata,afterregyx1x2…xkyouwouldtypetestx1=x2togetap-valueforthetestMoregenerally,youcanalwaysrestatetheproblemtogetthetestyouwant第二十六頁，共四十二頁，2022年，8月28日26Example:SupposeyouareinterestedintheeffectofcampaignexpendituresonoutcomesModelisvoteA=b0+b1log(expendA)+b2log(expendB)+b3prtystrA+uH0:b1=-b2,orH0:q1=b1+b2=0b1=q1

–

b2,sosubstituteinandrearrange

voteA=b0+q1log(expendA)+b2log(expendB-expendA)+b3prtystrA+u第二十七頁，共四十二頁，2022年，8月28日27Example(cont):Thisisthesamemodelasoriginally,butnowyougetastandarderrorforb1

–

b2=q1directlyfromthebasicregressionAnylinearcombinationofparameterscouldbetestedinasimilarmannerOtherexamplesofhypothesesaboutasinglelinearcombinationofparameters:b1=1+b2;b1=5b2;b1=-1/2b2;etc第二十八頁，共四十二頁，2022年，8月28日28MultipleLinearRestrictionsEverythingwe’vedonesofarhasinvolvedtestingasinglelinearrestriction,(e.g.b1=0orb1=b2)However,wemaywanttojointlytestmultiplehypothesesaboutourparametersAtypicalexampleistesting“exclusionrestrictions”

–wewanttoknowifagroupofparametersareallequaltozero第二十九頁，共四十二頁，2022年，8月28日29TestingExclusionRestrictionsNowthenullhypothesismightbesomethinglikeH0:bk-q+1=0,...,bk=0ThealternativeisjustH1:H0isnottrueCan’tjustcheckeachtstatisticseparately,becausewewanttoknowiftheqparametersarejointlysignificantatagivenlevel–itispossiblefornonetobeindividuallysignificantatthatlevel第三十頁，共四十二頁，2022年，8月28日30ExclusionRestrictions(cont)Todothetestweneedtoestimatethe“restrictedmodel”withoutxk-q+1,,…,xkincluded,aswellasthe“unrestrictedmodel”withallx’sincludedIntuitively,wewanttoknowifthechangeinSSRisbigenoughtowarrantinclusionofxk-q+1,,…,xk

第三十一頁，共四十二頁，2022年，8月28日31TheFstatisticTheFstatisticisalwayspositive,sincetheSSRfromtherestrictedmodelcan’tbelessthantheSSRfromtheunrestrictedEssentiallytheFstatisticismeasuringtherelativeincreaseinSSRwhenmovingfromtheunrestrictedtorestrictedmodel

q=numberofrestrictions,ordfr

–

dfur

n–k

–1=dfur第三十二頁，共四十二頁，2022年，8月28日32TheFstatistic(cont)TodecideiftheincreaseinSSRwhenwemovetoarestrictedmodelis“bigenough”torejecttheexclusions,weneedtoknowaboutthesamplingdistributionofourFstatNotsurprisingly,F~Fq,n-k-1,whereqisreferredtoasthenumeratordegreesoffreedomandn–k–1asthedenominatordegreesoffreedom第三十三頁，共四十二頁，2022年，8月28日330c(1-a)f(F)FTheFstatistic(cont)rejectfailtorejectRejectH0atasignificancelevelifF>ca第三十四頁，共四十二頁，2022年，8月28日34Example:thedeterminationsofleaguebaseballplayer’ssalaries(wooldridge,p143)Theregressionmodellog(salary)=b0+b1year+b2gamesyr+b3bavg+b4hrunsyr+b5rbisyr+usalary,the1993totalsalaryyears,yearsintheleaguegamesyr,averagegamesplayedperyearbavg,careerbattingaveragehrunsyr,homerunsperyearrbisyr,runsbattledinperyearThenullhypothesisisH0:b3=0,b4=0,b5=0,whichiscalledmultiplehypothesestestorjointhypothesestest.ThealternativehypothesisisH1:H0isnottrue.Theunrestrictedmodel:log(salary)=11.19+0.0689year+0.0126gamesyr+0.00098bavg+0.0144hrunsyr+0.0108rbisyr

(0.29)(0.0689)(0.0026)(0.00110)(0.0161)(0.0072)

n=353,SSR=183.186,R2=0.6278Therestrictedmodellog(salary)=11.22+0.0713year+0.0202gamesyr

(0.11)(0.0125)(0.0013)

n=353,SSR=198.311,R2=0.5971第三十五頁，共四十二頁，2022年，8月28日35Example:thedeterminationsofleaguebaseballplayer’ssalaries,cont.Therestrictednumberandthedegreeofthefreedomofrestrictedmodelisq=3;Thedegreeoffreedomofunrestrictedmodelis353-5-1=347;ThentheFstatisticis第三十六頁，共四十二頁，2022年，8月28日36TheR2formoftheFstatisticBecausetheSSR’smaybelargeandunwieldy,analternativeformoftheformulaisusefulWeusethefactthatSSR=SST(1–R2)foranyregression,socansubstituteinforSSRuandSSRur第三十七頁，共四十二頁，2022年，8月28日37Example:Parent’sEducationinaBirthWeightEquation(wooldridge,p150)Variablesbwght,birthweightinpounds;cigs,averagenumberofcigarettesthemothersmokedperdayduringpregnancy;parity,thebirthorderofthischild;faminc,annualfamilyincome;motheduc,yearsofschoolingforthemother;fatheduc,yearsofschoolingforthefather.Model:bwght=b0+b1cigs+b2parity+b3faminc+b4motheduc+b5fatheduc+uWhethertheparents’educationhasanyeffectonbirthweight?ThisisstatedasH0:b4=0,b5=0,soq=2.bwght=cigs+1.788parity+0.056faminc-0.370motheduc+0.472fatheduc

(3.728)(0.110)(0.659)(0.037)(0.320)(0.283)

n=1191R2=0.0387bwght=cigs+1.832parity+0.067faminc

(1.656)(0.109)(0.658)(0.032)