隔壁計(jì)量的-2015s簡(jiǎn)單多元回歸推斷之一

INTERMEDIATE簡(jiǎn)單多元回歸推國(guó)家發(fā)2015年ChapterOutline本章提SamplingDistributionsoftheOLSEstimatorsTestingHypothesisAboutaSinglePopulationParameter:Thettest單個(gè)總體參數(shù)的假設(shè)檢驗(yàn)：t置信區(qū)TestingHypothesesAboutaSingleLinearCombinationoftheParameters參數(shù)線性組合的假設(shè)檢驗(yàn)（一維情形TestingMultipleLinearRestrictions:TheF多個(gè)線性約束的假設(shè)檢驗(yàn)：F檢ReportingRegression 報(bào)告回歸結(jié) Lecture 本課提樣本分布CLMassumptionsandSamplingDistributionsoftheOLSEstimators經(jīng)典假設(shè)與OLS估計(jì)量的樣Backgroundreviewofhypothesis假設(shè)檢驗(yàn)的背景知One-sidedandtwo-sidedt單邊與雙邊t檢Calculatingthep計(jì)算p SamplingDistribution:Samplingdistributionsplayacentralroleinthedevelopmentofstatisticalandeconometricprocedures.抽樣分布在統(tǒng)計(jì)學(xué)和計(jì)量經(jīng)濟(jì)學(xué)發(fā)展中具 地Itistheprobabilitydistributionofanestimatoroverall 指一個(gè)估計(jì)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(zhǔn)確”方 The“exact”approach:derivesaformulaforthedistributionthatholdsforanyvalueof“準(zhǔn)確”方式需要對(duì)任何n的取值都得到樣本分布的精這樣的分布被稱為準(zhǔn)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(dú)立同分其均值恰好服從正態(tài) SamplingDistribution:theasymptoticThe“approximate”approachusesapproximationstothesamplingdistributionsthatrelyonthesamplesizebeinglarge.“近似”方式對(duì)樣本分布進(jìn)行大樣本下的近Theasymptoticdistribution:Thelargesampleapproximationtothesamplingdistribution.對(duì)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(duì)準(zhǔn)確分布的很好 似 SamplingDistribution:theasymptoticTwokeytools:thelawoflargenumbers(LLN),andthecentrallimittheorem(CLT).兩個(gè)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(shí)證使用中的1936起：社會(huì)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(guó) ，AndersNiscolai1936:GeorgeGallopLiteraryDigest240萬(wàn)vs.GonewiththeWind AssumptionMLR.6假定MLR.6（正態(tài)Sofar,weknowthatgiventheGauss-assumptions,OLSisGsaAnadditionalassumptionneededforclassicalhypothesis為了進(jìn)行經(jīng)典假設(shè)檢驗(yàn)，需要增加一個(gè)AssumptionMLR.6(Normality):Assumethatuiswithzeromeanandvariance2:u~Normal(0,2)假定MLR.6（正態(tài)）：假設(shè)u與x1,x2,…,xk獨(dú)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(jì)量具有 Theorem4.1NormalSamplingUndertheCLMassumptions,conditionalonthesamplevaluesβj~Normalβj,Varj,whereVarj

SST(1-R2) jjsd~Normaljjsd~Normal

j?在CLM假設(shè)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?yàn)樗钦`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(biāo)準(zhǔn)化，定理4.1得證。我們已經(jīng)證明一下關(guān)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(kuò)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(shí)來進(jìn)行假設(shè) 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(duì)一個(gè)特定的j進(jìn)行假設(shè)檢 Background背景知識(shí)回Thehypothesistobetestedisthenull被檢驗(yàn)的假設(shè)稱為零Hypothesistestingentailsusigdatatocomparethenullhypothesiswithasecondhypothesis,i.e.,the假設(shè)檢驗(yàn)利用數(shù)據(jù)將零假設(shè)和另一個(gè)假設(shè)替代假設(shè) Background背景知識(shí)回Thealternativehypothesisspecifieswhatistrueifthenullhypothesisisnot.替代假設(shè)給出在零假設(shè)不成立時(shí)，什么才是正確Goal:usetheevidenceinarandomlyselectedsampleofdatatodecidewhethertorejectthenullhypothesis. Background背景知識(shí)Twokindsofmistakesarepossibleinhypothesis在假設(shè)檢驗(yàn)中存在兩種可能的錯(cuò)TypeIerror:rejectthenullhypothesiswhenitinfacttrue.第一類錯(cuò)誤：當(dāng)零假設(shè)為真 零假設(shè)（棄真TypeIIerror:failtorejectthenullwhenitisactuallyfalse.第二類錯(cuò)誤：當(dāng)零假設(shè)為假時(shí)未 Background背景知識(shí)回HypothesistestingrulesareconstructedtomaketheprobabilityofcommittingtypeIerrorfairlysmall.非常小。TypeIerror.一個(gè)檢驗(yàn)的顯著性水平是發(fā)生第一類錯(cuò)Commonlyspecifiedsignificancelevels:0.1,0.05,0.01.Ifitequals0.05,itmeanstheresearcheriswillingtofalselyrejectthenullat5%ofthetime.通常設(shè)定的顯性為：01,5,1。果為5意味著研究者愿在％的檢中錯(cuò)地 零設(shè)。 Background背景知識(shí)回Thecriticalvalueofateststatisticisthevalueofthestatisticforwhichthetestjustrejectthenullhypothesisatthegivensignificancelevel.檢驗(yàn)統(tǒng)計(jì)量的臨界值是使得零的統(tǒng)計(jì)量的Thesetofvaluesoftheteststatisticforwhichthetestrejectsthenullistherejectionregion,andthevaluesoftheteststatisticforwhichitdoesnotrejectthenullistheacceptanceregion.假設(shè)檢驗(yàn)中，使得零假設(shè) 的檢驗(yàn)統(tǒng)計(jì)量的取值范圍稱域，使得零假設(shè)不能 的檢驗(yàn)統(tǒng)計(jì)量的取值范圍成 BackgroundTheprobabilitythatatestactuallyincorrectlyrejectsthenullhypothesiswhenthenullistrueisthesizeofthetest. Theprobabilitythatatestcorrectlyrejectsthenullwhenthealternativeistrueisthepowerofthetest. Background背景知識(shí)Ateststatistic(T)issomefunctionoftherandomsample.Whenwecomputethestatisticforaparticularsample,weobtainan oftheteststatistic(t). Theorem4.2tDistributionfortheStandardized定理4.2:標(biāo)準(zhǔn)化估計(jì)量的t分

jsej

?jjjj

? Thet Knowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistests知道標(biāo)準(zhǔn)化估計(jì)量的樣本分布后，便可以進(jìn)行Startwithanullhypothesis，e.g.,H0:由零假設(shè)出H0:Ifwedonotrejectthenull,thenwedonotrejectthathasnopartialeffectony,aftercontrollingforother ThetTestjToperformourtestwefirstneedtoform"the"tstatisticfor?j?? ?

sejWewillthenuseourtstatisticalongwitharejectionruledeterminewhethertoacceptthenullhypothesis,為了進(jìn)行檢驗(yàn),我們首先要構(gòu)造j

的t? ?

sej ThetTestThet measureshowmanyjdeviatioj ?isawayfromjt統(tǒng)計(jì)量t?度量了估?相對(duì)0偏離了標(biāo)準(zhǔn)差。 jItssignisthesame j它的符號(hào)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(xué)生表現(xiàn)與學(xué)校Question:DoeslargerclasssizeresultsinpoorerstudentUse408highschoolsinMichiganforyear1993,performthefollowingregression:應(yīng) ^math10=2.274 p+0.048staff– math10:percentageofstudentspassingtheMEAPstandardized10mathp:averageannualteacher’sstaff:#ofstaffper1000enroll:student Example:StudentPerformanceandSchool例子：學(xué)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(xué)生表現(xiàn)與學(xué)校規(guī)Whetherbetter-paidteachersleadstobetterstudentperformance？WecantestH0 p=0versusH1 Thecalculatedtstatisticequals4.6.Since4.6>2.326,thereforerejectingtheH0at1%level.算得t統(tǒng)計(jì)量為4.6。因?yàn)?.6>2.326，在1%顯著性水平 TheTwo-sidedH1:j0isatwo-sidedalternative.Underthisalternative,wehavenotspecifiedthesignofthepartialeffectofxjony.規(guī)定xj對(duì)y影響的符號(hào)。Foratwo-sidedtest,wesetthecriticalvaluebasedc.cisthe97.5thpercentileinthetdistributionwithn-k-1degreesoffreedomif 零假設(shè)。當(dāng)時(shí)，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(huì)對(duì)學(xué)生表現(xiàn)產(chǎn)生影響^math10=2.274 p+0.048staff– Canformhypothesesof:H0:staff=0,H1:≠構(gòu)造如下檢驗(yàn)：H0staff0H1staffThecalculatedtratiois1.2The5%criticalvalueofstandardnormalis1.96.Since1.2<1.96,wefailtorejectthenull。算得t值為。準(zhǔn)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(rèn)為替代假設(shè)是雙邊 Ifwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthe%level”如 了零假設(shè)，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(duì)形如H0:j=aj的假設(shè)進(jìn)行檢驗(yàn)，需要Inthiscase,theappropriatetstatistic此時(shí)，恰當(dāng)?shù)膖統(tǒng)計(jì)量jjt

se

0forthestandard當(dāng)進(jìn)行標(biāo)準(zhǔn)檢驗(yàn)時(shí)aj Example:CampusCrimeand例子：校 與錄Question:Will1%increaseinenrollmentincreasecampuscrimebymorethan1%?問題：錄取量提高1%是否會(huì)導(dǎo)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(bào)告（97個(gè)觀察值）的數(shù)據(jù)，估計(jì)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(duì)于95自由度的t分布，1%顯著水平下單邊檢驗(yàn)的臨界值為2.37。因?yàn)?.37<2.45，零假 Computingp-valuesfort計(jì)算t檢驗(yàn)的pThestepsinclassicalhypothesis經(jīng)典假設(shè)檢驗(yàn)的Statethenullandthealternative表述零假設(shè)和替代DecideasignificancelevelandfindthecriticalCalculatethetstatisticbasedonthesampletComparethetstatisticwiththecriticalvaluetodecidewhethertorejectthenull比較t值與臨界值，決定是 零假設(shè) Computingp-valuesfort計(jì)算t檢驗(yàn)的pSupposeat40degreesoffreedom,acalculatedtratiois2