计量经济学(英文)重点知识点考试必备

计量经济学(英文)重点知识点考试必备计量经济学(英文)重点知识点考试必备计量经济学(英文)重点知识点考试必备计量经济学(英文)重点知识点考试必备编制仅供参考审核批准生效日期地址：电话：传真:邮编:第一章Econometrics（计量经济学）：thesocialscienceinwhichthetoolsofeconomictheory,mathematics,andstatisticalinferenceareappliedtotheanalysisofeconomicphenomena.theresultofacertainoutlookontheroleofeconomics,consistsoftheapplicationofmathematicalstatisticstoeconomicdatatolendempiricalsupporttothemodelsconstructedbymathematicaleconomicsandtoobtainnumericalresults.Econometricanalysisproceedsalongthefollowinglines计量经济学分析步骤Creatingastatementoftheoryorhypothesis.建立一个理论假说Collectingdata.收集数据Specifyingthemathematicalmodeloftheory.设定数学模型Specifyingthestatistical,oreconometric,modeloftheory.设立统计或经济计量模型Estimatingtheparametersofthechoseneconometricmodel.估计经济计量模型参数Checkingformodeladequacy:Modelspecificationtesting.核查模型的适用性：模型设定检验Testingthehypothesisderivedfromthemodel.检验自模型的假设Usingthemodelforpredictionorforecasting.利用模型进行预测Step2：收集数据Threetypesofdata三类可用于分析的数据Timeseries(时间序列数据):Collectedoveraperiodoftime,arecollectedatregularintervals.按时间跨度收集得到Cross-sectional截面数据:Collectedoveraperiodoftime,arecollectedatregularintervals.按时间跨度收集得到Pooleddata合并数据（上两种的结合）Step3：设定数学模型plotscatterdiagramorscattergramwritethemathematicalmodelStep4：设立统计或经济计量模型CLFPRisdependentvariable应变量CUNRisindependentorexplanatoryvariable独立或解释变量（自变量）WegiveacatchallvariableUtostandforalltheseneglectedfactorsInlinearregressionanalysisourprimaryobjectiveistoexplainthebehaviorofthedependentvariableinrelationtothebehaviorofoneormoreothervariables,allowingforthedatathattherelationshipbetweenthemisinexact.线性回归分析的主要目标就是解释一个变量（应变量）与其他一个或多个变量（自变量）只见的行为关系，当然这种关系并非完全正确Step5：估计经济计量模型参数Inshort,theestimatedregressionlinegivestherelationshipbetweenaverageCLFPRandCUNR简言之，估计的回归直线给出了平均应变量和自变量之间的关系Thatis,onaverage,howthedependentvariablerespondstoaunitchangeintheindependentvariable.单位因变量的变化引起的自变量平均变化量的多少。Step6：核查模型的适用性：模型设定检验Thepurposeofdevelopinganeconometricmodelisnottocapturetotalreality,butjustitssalientfeatures.Step7：检验自模型的假设Whydoweperformhypothesistesting?Wewanttofindourwhethertheestimatedmodelmakeseconomicsenseandwhethertheresultsobtainsconformwiththeunderlyingeconomictheory.第二章Themeaningofregression（回归）Regressionanalysisisconcernedwiththestudyoftherelationshipbetweenonevariablecalledthedependentorexplainedvariable,andoneormoreothervariablescalledindependentorexplanatoryvariables.ObjectivesofregressionEstimatethemean,oraverage,andthedependentvaluesgiventheindependentvaluesTesthypothesesaboutthenatureofthedependence-----hypothesessuggestedbytheunderlyingeconomictheoryPredictorforecastthemeanvalueofthedependentvariablegiventhevaluesoftheindependentsOneormoreoftheprecedingobjectivescombinedPopulationRegressionLine（PRL）Inshort,thePRLtellsushowthemean,oraverage,valueofYisrelatedtoeachvalueofXinthewholepopulationThedependenceofYonX,technicallycalledtheregressionofYonX.Howdoweexplainit?Astudent’sS.A.T.score,say,theithindividual,correspondingtoaspecificfamilyincomecanbeexpressedasthesumoftwocomponentsThecomponentcanbecalledthesystematic,ordeterministic,component.MaybecalledthenonsystematicorrandomcomponentWhatisthenatureofU(stochasticerror)term

Theerrortermmayrepresenttheinfluenceofthosevariablesthatarenotexplicitlyincludedinthemodel.误差项代表了未纳入模型变量的影响Someintrinsicrandomnessinthemathscoreisboundtooccurthatcannotbeexplainedevenweincludeallrelevantvariables.即使模型包括了决定性数学分数的所有变量，内在随机性也不可避免，这是做任何努力都无法解释的。Umayalsorepresenterrorsofmeasurement.U还代表了度量误差TheprincipleofOckham’srazor-thedescriptionbekeptassimpleaspossibleuntilprovedinadequate-wouldsuggestthatwekeepourregressionmodelassimpleaspossible.“奥卡姆剃刀原则”，描述应该尽可能简单，只要不遗漏重要信息。这表明回归模型应尽可能简单。HowdoweestimatethePRF（populationregressionfunction）

Unfortunately,inpractice,Werarelyhavetheentirepopulationinourdisposal,oftenwehaveonlyasamplefromthispopulation.GrantedthattheSRFisonlyanapproximationofPRF.CanwefindamethodoraprocedurethatwillmakethisapproximationascloseaspossibleSRF仅仅是PRF的近似，那么能不能找到一种方法使这种近似尽可能接近真实呢Specialmeaningof“linear”Linearityinthevariables变量线性TheconditionalmeanvalueofthedependentvariableisalinearfunctionoftheindependentvariablesLinearityintheParameters参数线性Theconditionalmeanofthedependentvariableisalinearfunctionoftheparameters,theB’s;itmayormaynotbelinearinthevariables.第三章UnlesswearewillingtoassumehowthestochasticUtermsaregenerated,wewillnotbeabletotellhowgoodanSRFisasanestimateofthetruePRF.只有假定了随机误差的生成过程，才能判定SRF对PRF拟合的是好是坏。ClassicalLinearRegressionModelAssumption1:Theregressionmodelislinearintheparameters.Itmayormaynotbelinearinthevariables.回归模型是参数线性的，但不一定是变量线性的。Assumption2:TheexplanatoryvariablesXisuncorrelatedwiththedisturbancetermU.X’sarenonstochastic,Uisstochastic.解释变量X与扰动误差项u不相关.X是非随机的，U是随机的。Assumption3:GiventhevalueofXi,theexpected,ormeanvalueofthedisturbancetermUiszero.给定Xi，扰动项的期望或均值为零。DisturbanceUrepresentallthosefactorsthatarenotspecificallyintroducedinthemodel干扰项U代表了所有未纳入模型的影响因素。Assumption4:ThevarianceofeachUiisconstant,orhomoscedastic.U的方差为常数，或同方差。Homoscedasticity（同方差）:ThisassumptionsimplymeansthattheconditionaldistributionofeachYpopulationcorrespondingtothegivenvalueofXhasthesamevariance.该假定表明，与给定的X相对应的每个Y的条件分布具有同方差。TheindividualYvaluesarespreadaroundtheirmeanvalueswiththesamevariance.即每个Y值以相同的方差分布在其均值周围。Assumption5:Thereisnocorrelationbetweentwoerrorterms,thisistheassumptionofno-autocorrelation.无自相关假定，即两个误差项之间不相关。Assumption6:Theregressionmodeliscorrectlyspecified.回归模型是正确假定的。Thereisnospecificationbiasorspecificationerrorinthemodel.实证分析的模型不存在设定偏差或设定误差。Thisassumptioncanbeexplainedinformallyasfollows.Aneconometricinvestigationbeginswiththespecificationoftheeconometricmodelunderlyingthephenomenonofinterest.3.VariancesandStandarderrorsofOLSestimators普通最小二乘估计量的方差与标准误:OneimmediateresultoftheassumptionsintroducedisthattheyenableustoestimatethevariancesandstandarderrorsoftheOLSestimatorsgiveninEq.(2.16)and(2.17).4.Weshouldknow:VariancesoftheestimatorsStandarderrorsoftheestimators5.WhatisthevalueofσThehomoscedasticσisestimatedfromformula6.StandardErroroftheRegression(SER)回归标准误IssimplythestandarddeviationoftheYvaluesabouttheestimatedregressionline.Y值偏离估计回归的标准差。7.SummaryofmathS.A.T.scorefunctionInterpretationThestandarddeviation,orstandarderror,is0.000245,isameasureofvariabilityofb2fromsampletosample.Ifwecansaythatourcomputedb2lieswithinacertainnumberofstandarddeviationunitsfromthetrueB2,wecanstatewithsomeconfidencehowgoodthecomputedSRFisasanestimatorofthetruePRF.2）SamplingDistribution抽样分布Oncewedeterminethesamplingdistributionofourtwoestimators,thetaskofhypothesistestingbecomesstraightforward.一旦确定了两个估计量的抽样分布，那么假设检验就是举手之劳的事情。8.WhydoweuseOLS

ThepropertiesofOLSestimatorsThemethodofOLSisusedpopularlynotonlybecauseitiseasytousebutalsobecauseithassomestrongtheoreticalproperties.OLS法得到广泛使用，不仅是因为它简单易行，还因为它具有很强的理论性质。9.Gauss-Markovtheorem高斯-马尔科夫定理Giventheassumptionsoftheclassicallinearregressionmodel(CLRM),theOLSestimatorshaveminimumvarianceintheclassoflinearestimators.TheOLSestimatorsareBLUE(bestlinearunbiasedestimators)满足古典线性模型的基本假定，则在所有线性据计量中，OLS估计两具有最小方差性，即OLS是最优线性无偏估计量（BLUE）BLUEproperty最优线性无偏估计量的性质B1andB2arelinearestimators.B1和B2是线性估计量Theyareunbiased,thatisE(b1)=B1,E(b2)=B2.B1和B2是无偏估计两TheOLSestimatoroftheerrorvarianceisunbiased.误差方差的OLS估计量是无偏的b1andb2areefficientestimators.B1和B2是有效估计量Var(b1)islessthanthevarianceofanyotherlinearunbiasedestimatorofB1Var(b2)islessthanthevarianceofanyotherlinearunbiasedestimatorofB2MonteCarlosimulation蒙特卡洛模拟DotheexperimentatlabDoitbyExcell.=NORMINV(RAND(),0,2)Doitbymatlab.=NORMINV(uniform(),MU,SIGMA)DoitbyStata.=invnorm(uniform())CentralLimitTheorem’s中心极限定理Ifthereisalargenumberofindependentandidenticallydistributed(iid)randomvariables,then,withafewexceptions,thedistributionoftheirsumtendstobeanormaldistributionasthenumberofsuchvariablesincreasesindefinitely.随着变量个数的无限增加，独立同分布随机变量近似服从正态分布RecallU,theerrortermrepresentstheinfluenceofallthoseforcesthataffectYbutarenotspecificallyincludedintheregressionmodelbecausetherearesomanyofthemandtheindividualeffectofanyonesuchforceonYmaybetoominor.误差项代表了未纳入回归模型的其他所有因素的影响。因为在这些影响中，每种因素对Y的影响都很微弱Ifalltheseforcesarerandom,ifweletUrepresentthesumofalltheseforces,thenbyinvokingtheCLT,wecanassumethattheerrortermUfollowsthenormaldistribution.如果所有这些影响因素都是随机的，用U代表所有这些影响因素之和，那么根据中心极限定理，可以假定误差项服从正态分布。Anotherpropertyofnormaldistribution另一个正态分布的性质Anylinearfunctionofanormallydistributedvariableisitselfnormallydistributed.正态变量的性质函数仍服从正态分布。Hypothesistesting假设检验HavingknownthedistributionofOLSestimatorsb1andb2,wecanproceedthetopicofhypothesistesting.Nullhypothesis零假设“zero”nullhypothesisisdeliberatelychosentofindoutwhetherYisrelatedtoXalall,whichisalsocalledstrawmanhypothesis.之所以选择这样一个假设是为了确定Y是否与X有关，也称为稻草人假设。Weneedsomeformaltestingproceduretorejectorreceivethenullhypothesisandmaketheskepticalguysshutup.需要正规的检验过程拒绝或接受零假设 IfournullhypothesisisB2=0andthecomputedb2=0.0013,wecanfindouttheprobabilityofobtainingsuchavaluefromtheZ,thestandardnormaldistribution.如果零假设为B2=0，计算得到b2=0.0013，那么根据标准正态分布Z，能够求得获此b2值的概率Iftheprobabilityisverysmall,wecanrejectthenullhypothesis.如果这个概率非常小，则拒绝零假设。Iftheprobabilityislarger,say,greaterthan10percent,wemaynotrejectthenullhypothesis.如果这概率比较大，比如大于10%，就不拒绝零假设。Wedon’tknowtheσ2Wemustknowthetrueσ2,butwecanestimateitbyusingWhatwillhappenifwereplaceσbyitsestimatorσ-hatLetusassumethatα,thelevelofsignificanceortheprobabilityofcommittingatypeIerror,isfixedat5percent.假定α，显著水平成犯第一类错误的概率为5%。redarea=rejectionregionfor2-sidedtest(1-a)(1-a)t0f(t)-tctca/2a/2LoopandballThisisa95%confidenceintervalforB2给出了B2的一个95%的置信区间。inrepeatedapplications95outof100suchintervalswillincludethetrueB2重复上述过程，100个这样的区间中将有95个包括真实的B2。Suchaconfidenceintervalisknownastheregionofacceptance(ofH0)andtheareaoutsidetheconfidenceintervalisknownastherejectionregion(ofH0)用假设检验的语言把这样的置信区间称为（H0的）接受区域，把置信区间以外的区间成为（H0的）拒绝区域回归系数的假设检验目的：简单线性回归中，检验X对Y是否真有显著影响基本概念回顾:临界值与概率、大概率事件与小概率事件相对于显著性水平的临界值为:（单侧）或（双侧）计算的统计量为:统计量t统计量t0（大概率事件）（小概率事件）ConclusionsSincethisintervaldoesnotincludethenull-hypothesizedvalueof0.因为这个区间没有包括零假设值0。WecanrejectthenullhypothesisthatannualfamilyincomeisnotrelatedtomathS.A.T.Scores.所以拒绝假设：家庭年收入对数学SAT没有影响。Putpositively,incomedoeshavearelationshiptomathS.A.T.scores.换言之，收入确实与数学SAT有关系。AcautionarynoteAlthoughthestatementgivenistrue,wecannotsaythattheprobabilityis95percentthattheparticularintervalincludesB2,forthisintervalisnotarandominterval,itisfixed,therefore,theprobabilityiseither1ore0thattheintervalincludesB2.虽然式子3.26为真，但不能说某个特定区间式3.27包括真实B2的概率为95%，因为与式子3.26不同，式3.27是固定的，而不是一根随机区间，所以区间3.27包括B2的概率为1或0.Wecanonlysaythatifweconstruct100intervalslikethisinterval,95outof100suchintervalswillincludethetrueB2.我们只能说，如果建立100个像式3.27这样的区间，则有95个区间包括真实的B2.WecannotguaranteethatthisparticularintervalwillnecessarilyincludesB2.并不能保证某个区间一定有B2.Thetestofsignificanceapproachtohypothesistesting假设检验的显著性检验方法Hypothesistestingisthatofateststatisticandthesamplingdistributionoftheteststatisticunderthenullhypothesis,H0.假设检验方法涉及两个重要的概念检验统计量和零假设下检验统计量的抽样分布。ThedecisiontoacceptorrejectH0ismadeonthebasisofthevalueoftheteststatisticobtainedfromthesampledata.根据从样本数据求得的检验统计量的值决定接受或拒绝零假设。TtestWecanusethetvaluecomputedhereadtheteststatistic,whichfollowsthetdistributionwith(n-2)d.f.可以计算出t值作为检验统计量，它服从自由度为（n-2）的t分布。Insteadofarbitrarilychoosingtheαvalue,wecanfindthepvalue(theexactlevelofsignificance)andrejectthenullhypothesisifthecomputedPvalueissufficientlylow.为了避免选择显著水平的随意性，通常求出p值（精确的显著水平），如果计算的p值充分小，则拒绝零假设。ConclusionsInthecaseoftwo-sidedttest双边检验情况中Ifthecomputed|t|,theabsolutevalueoft,exceedsthecriticaltvalueatthechosenlevelofsignificance,wecanrejectthenullhypothesis.如果计算得到的|t|值超过临界t值，则拒绝零假设。PvalueThePvalueofthattstatisticof5.4354isabout0.0006.t统计量（5.4354）的p值（概率值）约为0.0006。Thesmallerthepvalue,themoreconfidentwearewhenrejectthenullhypothesis.p值越小，在拒绝零假设的时候就越有自信。ThusifweweretorejectthenullhypothesisthatthetrueslopecoefficientiszeroatthisPvalue,wewouldbewronginsixoutoftenthousandoccasions.如果在这个p值水平之上拒绝零假设：真实的斜率系数为0，则犯错误的机会有万分之六。HowcanwecomputedtWefirstcomputethetvalueasifthenullhypothesiswerethatB2=0,westillgetthet首先计算在零假设B2=0下的t值Sincethisvalueexceedsanyofthecriticalvaluesshownintheprecedingtable,followingtheruleslaiddown.t值大与上表给出的任何临界值，附录D表D-2列出的规则，WecanrejectthehypothesisthatannualfamilyincomehasnorelationshiptomathS.A.T.Scores.拒绝零假设：家庭年收入对数学SAT没有影响。Howgoodisthefittedregressionline:thecoefficientofdeterminationr2Onthebasisofttestboththeestimatedinterceptandslopecoefficientsarestatisticallysignificant(i.e.significantlydifferentfromzero)suggeststhattheSRFseemsto“fit”thedata“reasonably”well.根据t检验，估计的斜率和结局都是统计显著的，这说明样本回归函数式3.16很好地拟合了样本数据。CoefficientofdeterminationCanwedevelopanoverallmeasureof“goodnessoffit”thatwilltellushowwelltheestimatedregressionlinefitstheactualYvalues?能否建立一个“拟合优度”的判定规则，从而辨别估计的回归线拟合真实Y值的优劣程度呢？Suchameasurehasbeendevelopedandisknownasthecoefficientofdetermination.称之为判定系数。RecallRearrangeitDecomposition1、2、3、Indeviationforms1、2、Squarebothsidesandsum=thetotalvariationoftheactualYvaluesabouttheirsamplingmeanYbar,whichmaybecalledthetotalsumofsquares(TSS)总平方和，真实Y值围绕其均值的总变异=ThetotalvariationoftheestimatedYvaluesabouttheirmeanvalue,Yhatbar,whichmaybecalledappropriatelythesumofsquaresduetoregression(i.e.,duetotheexplanatoryvariables),orsimplycalledtheexplainedsumofsquares(ESS)解释平方和，估计的Y值围绕气均值的变异，也称回归平方和（由解释变量解释的部分）PutsimplyThetotalvariationintheobservedYvaluesabouttheirmeanvaluecanbepartitionedintotwoparts,oneattributabletotheregressionlineandtheothertorandomforces,becausenotallactualYobservationslieonthefittedline.Y值与其均值的总离差可以分解为两部分：一部分归于回归线，另一部分归于随机因素，因为不是所有的真实观察值Y都落在你和直线上。ESSvsRSSIfthechosenSRFfitsthedataquitewell,ESSshouldbemuchlargerthanRSS.如果选择的SRF很好的拟合了样本数据，则SEE远大于RSS。IftheSRFfitsthedatapoorlyRSSwillbemuchlargerthanESS.如果SRF拟合的不好，则RSS远大于ESS。Letusdefine定义R2样本判定系数R2measurestheproportionorpercentageofthetotalvariationinYexplainedbytheregressionmodel样本判定系数度量了回归模型对Y变异的解释比例（或百分比）R2isthecoefficientofdeterminationandisthemostcommonlyusedmeasureofthegoodnessoffitofaregressionline.样本判定系数通常用来度量回归线的拟合优度。PropertiesofR2itisanon-negativequantity.非负性itslimitsare0≤R2≤1sinceapart(ESS)cannotbegreaterthanthewhole(TSS).0≤R2≤1，因为部分（ESS）不可能大于整体（TSS）。AnR2of1meansa“perfectfit”fortheentirevariationinYisexplainedbytheregression.若R2=1，则表示完全拟合，即线性模型完全解释Y的变异。AnR2ofzeromeansnorelationshipbetweenYandXwhatsoever.若R2=0，则表示Y与X之间无任何关系。ReportingtheresultsExplanationThefiguresinthefirstsetofparenthesesaretheestimatedstandarderrors(se)oftheestimatedregressioncoefficients.第一行括号内的数值表示估计回归系数的标准误Thoseinthesecondsetofparenthesesaretheestimatedtvaluecomputedunderthenullhypothesisthatthepopulationvalueofeachregressioncoefficientindividuallyiszero.Tvaluesaresimplycomputedtheratiosoftheestimatedcoefficienttotheirstandarderrors.第二行括号内的数值表示在零假设下（每个回归系数的真实值为零），根据式3.22估计的t值（即估计的系数与其标准误之比）thoseinthethirdsetofparenthesesarepvaluesofthecomputedtvalues.第三行括号内的数值表示获得t值的p值。AsamatterofconventionFromnowon,ifwedonotspecifyaspecificnullhypothesis,thenwewillassumethatitisthezeronullhypothesis.从现在起，如果没有设定特殊的零假设，习惯地规定零假设为：总体参数为零。PvalueByquotingthePvalueswecandeterminetheexactlevelofsignificanceoftheestimatedtvalue.通过列出的p值能够确定t值的精确显著水平。ThelowerthePvalue,thegreatertheevidenceagainstthenullhypothesis,thelowerlikelihoodthecoefficientiszero.p值越低，拒绝假设的证据就越充分。AwarningWhendecidingwhethertorejectornotrejectanullhypothesis,determinebeforehandwhatlevelofthepvalueyouarewillingtoacceptandthencomparethecomputedpvaluewiththecriticalPvalue.当拒绝或不拒绝原假设时，需要鱼线确定一个接受的p值水平（即临界p值），然后把计算的p值进行比较。IfthecomputedPvalueissmallerthanthecriticalPvalue,thenullhypothesiscanberejected.如果计算的p值小于临界p值，则拒绝原假设。IfitisgreaterthanthecriticalPvaluethenullhypothesismaynotberejected.如果计算的p值大雨临界p值，则不能拒绝原假设。Errorterm:normalitytestOurstatisticaltestingprocedureisbasedontheassumptionthattheerrortermUiisnormallydistributed.这一统计检验过程是建立在误差项ui服从正态分布的基础上。normalitytest:JBtest雅克-贝拉检验SrepresentsskewnessandKrepresentskurtosisS为偏度，K为峰度TheJBstatisticfollowstheChi-squaredistributionwith2d.f.Asymptotically.在正态性假设下,给出的JB统计量渐近服从自由度为2的卡方分布。IfthecomputedChi-squarevalueexceedsthecriticalChi-squarevaluefor2d.f.atthechosenlevelofsignificance,werejectthenullhypothesisofnormaldistribution.如果在选定的显著水平下，根据式3.47计算的卡方值超过临界的卡方值，则拒绝正态分布的零假设IfitdoesnotexceedthecriticalChi-squarevalue,wemaynotrejectthenullhypothesis.如果没有超过临界的卡方值，则不能拒绝零假设。第四章Whyshouldweintroducemultipleregressionmodel为什么介绍多元回归模型Becausemultipleinfluences(i.e.,variable)mayaffectthedependentvariable.TheThree-variableregressionmodel三变量线性回归模型Thethree-variablePRFtoitsnon-stochasticform：三变量PRF的非随机形式

：TheconditionalmeanvalueofYt,conditionaluponthegivenorfixedvaluesofthevariablesX2andX3给定X2、X3取值下Y的条件均值WeobtaintheaverageormeanvalueofYforthefixedvaluesofXvariables.给定解释变量X取值条件下，得到的Y的均值Thethree-variablePRFtoitsstochasticform三变量PRF的随机形式AnyindividualYvaluecanbeexpressedasthesumoftwocomponentsAnyindividualYvaluecanbeexpressedasthesumoftwocomponents：任何一个Y值可以表示成两部分之和asystematicordeterministic，components，Whichissimplyitsmeanvalue系统成分或确定性成分也就是Y的均值Ut,whichisthenonsystematicorrandomcomponentdeterminedbyfactorsotherthanX2andX3.非系统成分或随即成分Ut，由除X2,X3以外的因素决定。Themeaningofpartialregressioncoefficient偏回归系数的含义TheregressioncoefficientsB2andB3areknownaspartialregressionorpartialslopecoefficients.B2,B3称为偏回归系数或偏斜率系数ThemeaningofPartialregressioncoefficientisasfollows:B2measuresthechangeinthemeanvalueofY,E(Y),perunitchangeinX2,holdingthevalueofX3constant.B2度量了在X3保持不变的情况下，X2单位变动引起Y均值E(Y)的变化量。Likewise,B3measureschangeinthemeanvalueofYperunitchangeinX3holdingthevalueofX2constant.同样的，B2度量了X2保持不变的情况下，X3单位变动引起Y均值E(Y)的变化量。Uniqueness：特殊性质Inthemultipleregressionmodel在多元回归模型中wewanttofindoutwhatpartofthechangeintheaveragevalueofYcanbedirectlyattributabletoX2andwhatparttoX3.我们想要知道的是Y均值的变动有多大比例“直接”来源于X2，多大比例“直接”来源于X3。Aexample：ThemeaningofB2B2=-1.2indicatesthatthemeanvalueofYdecreaseby1.2perunitincreaseinX2whenX3isheldconstant,inthisexampleitisheldconstantatthevalueof10.B2是斜率，表示当X3为常数时，X2每增加1个单位，Y的均值将减少1.2个单位——本例中，X3为常数10ThemeaningofB3HeretheslopecoefficientB3=0.8meansthatthemeanvalueofYincreaseby0.8perunitincreaseinX3whenX2isheldconstant.Hereitisheldconstantatthevalueof5.斜率B3=0.8，表示X2为常量时，X3每增加1个单位，Y的平均值增加0.8个单位，（这里假设X2等于5）4、Inshort，Apartialregressioncoefficientreflectsthe(partial)effectofoneexplanatoryvariableonthemeanvalueofthedependentvariablewhenthevaluesofotherexplanatoryvariablesincludedinthemodelareheldconstant.总之，偏回归系数反映了当模型中其他解释变量为常量时，某个解释变量对应变量均值的影响。5、uniquenessThisuniquefeatureofmultipleregressionenablesusnotonlytoincludemorethanoneexplanatoryvariableinthemodelbutalsoto“isolate”or“disentangel”theeffectofeachXvariableonYfromtheotherXvariablesincludedinthemodel.多元回归的这个独特性质不但能够引入多个解释变量，而且能够“分离”出每个解释变量X对应变量Y的影响。Assumptionsofthemultiplelinearregressionmodel多元线性回归模型的若干假定Inordertoestimatetheregressioncoefficientsofthemultipleregressionmodel,wewillcontinuetooperatewithintheframeworkoftheclassicallinearregressionmodel(CLRM)tousetheordinaryleastsquares(OLS)toestimatethecoefficients.为了对多元回归模型的参数进行估计，我们沿用古典线性回归模型的基本框架，并利用普通最小二乘法（OLS）进行参数估计。A4.1Theregressionmodelislinearintheparametersandiscorrectlyspecified.A4.2X2andX3areuncorrelatedwiththedisturbancetermU.IfX2andX3arenon-stochastic,thisassumptionisautomaticallyfulfilled.A4.3TheerrortermUhasazeromeanvalueA4.4Homoscedasticity,thevarianceofUisconstant.A4.5NoautocorrelationexistsbetweentheerrortermUiandUjA4.6NoexactcollinearityexistsbetweenX2andX3Thereisnoexactlinearrelationshipbetweenthetwoexplanatoryvariables.A4.7TheerrortermUfollowsthenormaldistributionwithmeanzeroandvarianceσ2Whywemakeassumptions？Wemaketheseassumptionstofacilitatethedevelopmentofthesubject.为了确保能够使用OLS法估计模型的参数NoMulticollinearity：无多重共线性ThereisnoexactlinearrelationshipbetweentheexplanatoryvariablesX2andX3.Thisistheassumptionofnocollinearityornomulticollinearity.解释变量X2,X3不存在严格的共线性，这个假定也称为无共线性或者无多重共线性假设Noperfectcollinearitymeansthatavariable,say,X2,cannotbeexpressedasanexactlinearfunctionofanothervariable无完全共线性通俗的解释是，变量X2不能表示为另一变量X3的线性函数TroublesomeThisisoneequationwithtwounknownsweneedtwo(independent)equationstoobtainuniqueestimatesofB2andB3(wehaveonlyoneA,butwehavetwoBtosolve.)NowevenifwecanestimateandobtainanestimateofA,thereisnowaythatwecangetindividualestimatesofB2andB3fromtheestimatedA.WecannotassestheindividualeffectofX2andX3onY.Butthisishardlysurprising,forwereallydonothavetwoindependentvariablesinthemodel.不能估计解释变量X2,X3各自对应变量Y的影响，没什么好奇怪的，因为在模型中确实没有两个独立的变量。OLSprinciple最小二乘法TheOLSprinciplechoosesthevalueoftheunknownparametersinsuchawaythattheresidualsumofsquares(RSS)Assmallaspossible.BLUE：UnderassumedconditionstheOLSestimatorsarebestlinearunbiasedestimators在古典线性回归模型的基本假定下，双变量模型的OLS估计量是最优无偏估计量EachregressioncoefficientestimatedbyOLSislinearandunbiased.每一个回归系数都是线性的和无偏的Ontheaverageitcoincideswiththetruevalue.平均而言，他与真实值一致Amongallsuchlinearunbiasedestimators,theOLSestimatorshavetheleastpossiblevariancesothatthetrueparametercanbeestimatedmoreaccuratelythanbycompetinglinearunbiasedestimators.在所有线性无偏估计量中，OLS估计量具有最小方差性，所以，OLS估计量比其他线性无偏估计量更准确地估计了真实的参数值。Inshort,theOLSestimatorsareefficient.简言之，OLS是最有效的Intwo-variablecasewesawthatr^2measuresthegoodnessoffitofthefittedsampleregressionline(SRL)r^2度量了样本回归直线（SRL）的拟合优度Inthree-variablecase，WewouldliketoknowtheproportionofthetotalvariationinY(yt2)explainedbyX2andX3jointly.在三变量模型中，我们用多元判定系数度量X2和X3对应变量Y变动的联合解释比例Inmultipleregressionmodel,RcanbeinterpretedasthedegreeoflinearassociationbetweenYandalltheXvariablesjointly.Antiqueclockauctionrevision（Eviews）LetY=auctionprice,X2=ageofclock,X3=numberofbiddersInterpretationoftheresults回归结果的解释：Theinterpretationoftheslopecoefficientofabout12.74(b2)meansthatholdingothervariablesconstant,iftheageoftheclockgoesupbyayear,theaveragepriceoftheclockwillgoupbyabout12.74$.Thetestofsignificanceapproach显著性检验法wedevelopateststatisticfindoutitssamplingdistributionchoosealevelofsignificanceαdeterminethecriticalvalue(s)oftheteststatisticatthechosenlevelofsignificancecomparethevalueoftheteststatisticobtainedfromthesampleathandwiththecriticalvalue(s)rejectthenullhypothesisifthecomputedvalueoftheteststatisticexceedsthecriticalvalue(s)iftheteststatistichasanegativevalue,weconsideritsabsolutevalueandsaythatiftheabsolutevalueoftheteststatisticexceedsthecriticalvalue,werejectthenullhypothesis.Wecanfindthepvalueoftheteststatisticandrejectthenullhypothesisifthepvalueissmallerthanthechosenαvalue求得统计量的p值，如果p值小于显著水平α，则拒绝零假设TestingthejointhypothesisthatB2=B3=0orR2=0：检验联合假设NullhypothesisThisnullhypothesisisajointhypothesisthatB2andB3arejointlyorsimultaneouslyequaltozero.这个零假设成为联合假设，即B2,B3联合或同时为令（而不是单独为零）ThishypothesisstatesthatthetwoexplanatoryvariablestogetherhavenoinfluenceonY.这个假设表明两个解释变量联合对应变量Y无影响。Thisisthesameassayingthat等同于TemptationThetemptationhereistostatethatsinceindividuallyb2andb3arestatisticallydifferentfromzerointhepresentexample,thenjointlyorcollectivelytheyalsomustbestatisticallydifferentfromzero,thatwerejectthenullhypothesis.这里潜在的逻辑是，既然b2，b3各自均显著不为零，那么它们一定也联合或集体显著不为零，即拒绝这个零假设Inotherwords,sinceageoftheantiqueclockandthenumberofbiddersattheauction,eachhasasignificanteffectontheauctionprice,togethertheyalsomusthaveasignificanteffectontheauctionprice.既然钟表年代和竞标人数各自都对拍卖价格有显著影响，那么它们一起也一定会对拍卖价格有显著影响Whenmulticollineratiyexists,inamultipleregressiononeoremorevariablesindividuallyhavenoeffectonthedependentvariablebutcollectivelytheyhaveasignificantimpactonit.在多元回归模型中，一个或多个解释变量各自对应变量没有影响，但却联合对应变量有影响。Thismeansthatthet-testingprocedurediscussedpreviously,althoughvalidfortestingthestatisticalsignificanceofanindividualregressioncoefficient,isnotvalidfortestingthejointhypothesi