INTERMEDIATE簡單多元回歸推國家發(fā)2015年ChapterOutline本章提SamplingDistributionsoftheOLSEstimatorsTestingHypothesisAboutaSinglePopulationParameter:Thettest單個總體參數(shù)的假設檢驗：t置信區(qū)TestingHypothesesAboutaSingleLinearCombinationoftheParameters參數(shù)線性組合的假設檢驗（一維情形TestingMultipleLinearRestrictions:TheF多個線性約束的假設檢驗：F檢ReportingRegression 報告回歸結(jié) Lecture 本課提樣本分布CLMassumptionsandSamplingDistributionsoftheOLSEstimators經(jīng)典假設與OLS估計量的樣Backgroundreviewofhypothesis假設檢驗的背景知One-sidedandtwo-sidedt單邊與雙邊t檢Calculatingthep計算p SamplingDistribution:Samplingdistributionsplayacentralroleinthedevelopmentofstatisticalandeconometricprocedures.抽樣分布在統(tǒng)計學和計量經(jīng)濟學發(fā)展中具 地Itistheprobabilitydistributionofanestimatoroverall 指一個估計量在其所有可能取值上的概Therearetwoapproachescharacteringsamplingdistributions:an“exact”approachandan“approximate”刻畫抽樣分布有兩種方式：“準確”方 The“exact”approach:derivesaformulaforthedistributionthatholdsforanyvalueof“準確”方式需要對任何n的取值都得到樣本分布的精這樣的分布被稱為準確分布或者有限樣本分Forexample,ifyisnormallydistributed,andy1,y2,…,ynarei.i.d,thentheiraveragehasanexactdistributionofnormal例如，如果y服從正態(tài)分布，且y1,y2,…,yn獨立同分其均值恰好服從正態(tài) SamplingDistribution:theasymptoticThe“approximate”approachusesapproximationstothesamplingdistributionsthatrelyonthesamplesizebeinglarge.“近似”方式對樣本分布進行大樣本下的近Theasymptoticdistribution:Thelargesampleapproximationtothesamplingdistribution.對樣本分布的大樣本近似常稱為Theasymptoticdistributionscanbecountedontoprovidegoodapproximationstotheexactsamplingdistribution,aslongasthesamplesizeis只要樣本量足夠大，漸近分布就是對準確分布的很好 似 SamplingDistribution:theasymptoticTwokeytools:thelawoflargenumbers(LLN),andthecentrallimittheorem(CLT).兩個重要工具：大數(shù)定律，中心極限定Whythesetwotheoremsareimportant?Mostestimatorsencounteredinstatisticsandeconometricscanbewrittenasfunctionsofsampleaverages,hencecanapplythesetwotheoremtogettheasymptoticdistribution.ComebacktotheminChapter SamplingDistributionofOLSEstimatorsWehavediscussedtheexpectedvalueandvariancesoftheOLSestimators,buttoperformstatisticalinference,wewishtoknowthesamplingdistribution.ThesamplingdistributionsoftheOLSestimatorsdependontheunderlyingdistributionoftheerrors. 抽樣在實證使用中的1936起：社會可以通過選取部分有代表性的樣本完成。從發(fā)源，政治、商業(yè)；之后1824-19361895，挪威國 ，AndersNiscolai1936:GeorgeGallopLiteraryDigest240萬vs.GonewiththeWind AssumptionMLR.6假定MLR.6（正態(tài)Sofar,weknowthatgiventheGauss-assumptions,OLSisGsaAnadditionalassumptionneededforclassicalhypothesis為了進行經(jīng)典假設檢驗，需要增加一個AssumptionMLR.6(Normality):Assumethatuiswithzeromeanandvariance2:u~Normal(0,2)假定MLR.6（正態(tài)）：假設u與x1,x2,…,xk獨立，且u服 TheNormalDistribution(TheGaussianDistribution) CLMWhatdoweassumewhennormalityoftheerrortermassumptionisinvoked?Onecanconsideruasthesumofmanydifferentunobservedfactorsaffectingy,hencecaninvoketheCLTtoconcludethatuhasanapproximatenormalItassumesthatallunobservedfactorsaffectyinaseparate,additivefashion.Strong.Toberelaxedwhenlargesampleis CLM經(jīng)典線性模型Classicallinearmodel(CLM)assumptions:AssumptionsMLR.1–MLR.6.假定MLR.1-MLR.6被稱為經(jīng)典線性模SummarizethepopulationassumptionsofCLMasy|x~Normal(0+1x1+…+kxk,Minimumvarianceunbiasedestimator:UnderCLM,OLSisnotonlyBLUE,butalsoMVUE–theOLSestimatorgivesthesmallestvarianceamongallunbiasedestimators.OLSBEO估計量具有 Theorem4.1NormalSamplingUndertheCLMassumptions,conditionalonthesamplevaluesβj~Normalβj,Varj,whereVarj

SST(1-R2) jjsd~Normaljjsd~Normal

j?在CLM假設下，條件于解釋變量的樣本值有?j~

,

jj

-

sd

~Normalj?服從正態(tài)分布，因為它是誤差的線性組j BriefProofofTheorem證明提E(?Var(??isnormallydistributed,Theorem4.1isprovedafterstandardization.Weshownthatthefollowingrelationholds

i

(β+β

+...+β

+u

1

where?ijistheresidualfromregressingxjontherestofE(?Var(?? 經(jīng)過標準化，定理4.1得證。我們已經(jīng)證明一下關系成

ij

(β+β

+...+β

+u

1

ij BriefProofofTheorem證明提,2?+? ,2r?ijxm=0,m=1,...,j-1,j+1,...,kweget?

=βj+

ij?2

+? r?ijxm=0,m=1,...,j-1,j+,,，?

=βj+

?

BriefProofofTheorem證明提

?ij?2

?

ij

ij?j

Theorem4.1NormalSampling定理4.1正態(tài)樣本分Theorem4.1canbeextended.Anylinearcombinationsof?0,?1,...,?kisalsonormallydistributed,andany1subsetof?0,?,...,?1hasajointnormal可以擴展定理4.1?0,?1,...,?k的任意線性組合服從正態(tài)分布?0?1,...,?k任意子集服從聯(lián)合正態(tài)分布。Wewillusethesefactsinhypothesis利用這些事實來進行假設 TestingHypothesesaboutaSingleConsiderapopulationy01x1...kxkwhichsatisfiestheCLMWenowstudyhowtotesthypothesesabouta考慮總體中滿足CLM假定的模y01x1...kxk我們現(xiàn)在研究如何對一個特定的j進行假設檢 Background背景知識回Thehypothesistobetestedisthenull被檢驗的假設稱為零Hypothesistestingentailsusigdatatocomparethenullhypothesiswithasecondhypothesis,i.e.,the假設檢驗利用數(shù)據(jù)將零假設和另一個假設替代假設 Background背景知識回Thealternativehypothesisspecifieswhatistrueifthenullhypothesisisnot.替代假設給出在零假設不成立時，什么才是正確Goal:usetheevidenceinarandomlyselectedsampleofdatatodecidewhethertorejectthenullhypothesis. Background背景知識Twokindsofmistakesarepossibleinhypothesis在假設檢驗中存在兩種可能的錯TypeIerror:rejectthenullhypothesiswhenitinfacttrue.第一類錯誤：當零假設為真 零假設（棄真TypeIIerror:failtorejectthenullwhenitisactuallyfalse.第二類錯誤：當零假設為假時未 Background背景知識回HypothesistestingrulesareconstructedtomaketheprobabilityofcommittingtypeIerrorfairlysmall.非常小。TypeIerror.一個檢驗的顯著性水平是發(fā)生第一類錯Commonlyspecifiedsignificancelevels:0.1,0.05,0.01.Ifitequals0.05,itmeanstheresearcheriswillingtofalselyrejectthenullat5%ofthetime.通常設定的顯性為：01,5,1。果為5意味著研究者愿在％的檢中錯地 零設。 Background背景知識回Thecriticalvalueofateststatisticisthevalueofthestatisticforwhichthetestjustrejectthenullhypothesisatthegivensignificancelevel.檢驗統(tǒng)計量的臨界值是使得零的統(tǒng)計量的Thesetofvaluesoftheteststatisticforwhichthetestrejectsthenullistherejectionregion,andthevaluesoftheteststatisticforwhichitdoesnotrejectthenullistheacceptanceregion.假設檢驗中，使得零假設 的檢驗統(tǒng)計量的取值范圍稱域，使得零假設不能 的檢驗統(tǒng)計量的取值范圍成 BackgroundTheprobabilitythatatestactuallyincorrectlyrejectsthenullhypothesiswhenthenullistrueisthesizeofthetest. Theprobabilitythatatestcorrectlyrejectsthenullwhenthealternativeistrueisthepowerofthetest. Background背景知識Ateststatistic(T)issomefunctionoftherandomsample.Whenwecomputethestatisticforaparticularsample,weobtainan oftheteststatistic(t). Theorem4.2tDistributionfortheStandardized定理4.2:標準化估計量的t分

jsej

?jjjj

? Thet Knowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistests知道標準化估計量的樣本分布后，便可以進行Startwithanullhypothesis，e.g.,H0:由零假設出H0:Ifwedonotrejectthenull,thenwedonotrejectthathasnopartialeffectony,aftercontrollingforother ThetTestjToperformourtestwefirstneedtoform"the"tstatisticfor?j?? ?

sejWewillthenuseourtstatisticalongwitharejectionruledeterminewhethertoacceptthenullhypothesis,為了進行檢驗,我們首先要構造j

的t? ?

sej ThetTestThet measureshowmanyjdeviatioj ?isawayfromjt統(tǒng)計量t?度量了估?相對0偏離了標準差。 jItssignisthesame j它的符號與?jNoticewearetestinghypothesesaboutthepopulationparameters,nottestinghypothesesabouttheestimatesfromaparticularsample. tTest:One-Sided Besidesournull,H0,weneedanalternativehypothesis,H1,andasignificancelevelH1maybeone-sided,ortwo-H1:bj>0andH1:bj<0areone-H1bj0是雙邊替代 Ifwewanttohaveonlya5%probabilityofrejectingH0ifitisreallytrue,thenwesayoursignificancelevelis5% One-SidedAlternatives單邊替代假 Havingpickedasignificancelevel,,welookupthe(1–)thpercentileinatdistributionwithn–k–1degreeoffreedomandcallthisc,thecriticalvalue.取定顯著性水平后，找到自由度為n–k–1的 One-SidedAlternativesBecausetdistributionisIfH0:bj=0versusH1:bj>werejectH0iftbj>c,failtorejectH0iftbjIfH0:bj=0versusH1:bjwerejectH0iftbj<-c,failstorejectH0iftbj≥- One-SidedAlternatives單邊替代假yi=0+1xi1+…+kxik+H0:j= H1:j>Failto Thetdistributionversusnormalt分布與正態(tài)Noticethatasthedegreeoffreedominthetdistributiongetslarge,thetdistributionapproachesthestandardnormal Example:StudentPerformanceandSchool例子：學生表現(xiàn)與學校Question:DoeslargerclasssizeresultsinpoorerstudentUse408highschoolsinMichiganforyear1993,performthefollowingregression:應 ^math10=2.274 p+0.048staff– math10:percentageofstudentspassingtheMEAPstandardized10mathp:averageannualteacher’sstaff:#ofstaffper1000enroll:student Example:StudentPerformanceandSchool例子：學生表H0:βenroll=0versusH1Computethett=-0.0002/0.00022=-Sincen-k-1=404,weusethestandardnormalcriticalvalue.Atthe5%level,thecriticalvalueis–1.65.由于--144Because-0.91>–1.65,wefailtorejectthe由于-0.91>-1.65，我們不 零假 Example:StudentPerformanceandSchool例子：學生表現(xiàn)與學校規(guī)Whetherbetter-paidteachersleadstobetterstudentperformance？WecantestH0 p=0versusH1 Thecalculatedtstatisticequals4.6.Since4.6>2.326,thereforerejectingtheH0at1%level.算得t統(tǒng)計量為4.6。因為4.6>2.326，在1%顯著性水平 TheTwo-sidedH1:j0isatwo-sidedalternative.Underthisalternative,wehavenotspecifiedthesignofthepartialeffectofxjony.規(guī)定xj對y影響的符號。Foratwo-sidedtest,wesetthecriticalvaluebasedc.cisthe97.5thpercentileinthetdistributionwithn-k-1degreesoffreedomif 零假設。當時，c是n-k-1 自由度的t分布的97.5分位數(shù) Two-Sided雙邊替代yi=0+1Xi1+…+kXik+H0:j= H1:j≠failto-

0

c ExampleStudentPerformanceandSchoolSize例：Whetherthenumberofteachershasimpactsonstudentperformance？教師數(shù)目會對學生表現(xiàn)產(chǎn)生影響^math10=2.274 p+0.048staff– Canformhypothesesof:H0:staff=0,H1:≠構造如下檢驗：H0staff0H1staffThecalculatedtratiois1.2The5%criticalvalueofstandardnormalis1.96.Since1.2<1.96,wefailtorejectthenull。算得t值為。準正分布著性平下臨為我們能 零假。 SummaryforH0:j= Unlessotherwisestated,thealternativeisassumedtobe除非特 ，我們總認為替代假設是雙邊 Ifwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthe%level”如 了零假設，我們通常說“xj在%水平下顯著 Ifwefailtorejectthenull,wetypicallysay“xjisstatisticallyinsignificantatthe%level” TestingotherAmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0:j=aj如果我們想對形如H0:j=aj的假設進行檢驗，需要Inthiscase,theappropriatetstatistic此時，恰當?shù)膖統(tǒng)計量jjt

se

0forthestandard當進行標準檢驗時aj Example:CampusCrimeand例子：校 與錄Question:Will1%increaseinenrollmentincreasecampuscrimebymorethan1%?問題：錄取量提高1%是否會導致校 增加超過Supposetotalnumberofcrimesisdetermined假 總數(shù)由下式?jīng)Qcrime=exp()·enroll1·Onecan可以估log(crime)=+ Example:CampusCrimeand例子：校 與錄AndtestH0:1=1v.sH1:1>UsingdatafromtheFBI’suniformCrimereports(97observations),theestimatedequationis利用 報告（97個觀察值）的數(shù)據(jù)，估計得到方log(crime)=-6.63+ Thecorrecttratio=(1.27-1)/0.11=2.45.The1%one-sidedcriticalvalueforatdistributionwith95degreesoffreedomis2.37，Since2.37<2.45,rejectthenull.t值=(1.27-1)/0.11=2.45。對于95自由度的t分布，1%顯著水平下單邊檢驗的臨界值為2.37。因為2.37<2.45，零假 Computingp-valuesfort計算t檢驗的pThestepsinclassicalhypothesis經(jīng)典假設檢驗的Statethenullandthealternative表述零假設和替代DecideasignificancelevelandfindthecriticalCalculatethetstatisticbasedonthesampletComparethetstatisticwiththecriticalvaluetodecidewhethertorejectthenull比較t值與臨界值，決定是 零假設 Computingp-valuesfort計算t檢驗的pSupposeat40degreesoffreedom,acalculatedtratiois2