计量经济学(英文)

Lecture OneMethodologyofEconometrics1立论？结论？立论？结论？v立论:要求给出求论的路径。v结论:要求说明结论的来源。v自以为是的东西并不见得是真v我们不是上帝！2我们的习惯是这样的吗？我们的习惯是这样的吗？v结论来自感觉（象上帝）v宏观思考（象战略家）v习惯地提出政策建议（象顾问）v得争取把一个个的大、小问题搞明白再说吧！3Mainstream Analysis ApproachesNormativeAnalysisPositiveAnalysis(empiricalanalysis)4The Writer D.N.GujarativProfessorofeconometricsattheMilitaryAcademyatWestPointvMasterofCommercevMBAvEditorialrefereevAuthorvVisitingProfessor5WhatisEconometricsvEmpiricalsupporttothemodelsvQuantitativeanalysisofactualeconomicphenomenavSocialscienceinwhichthetoolsofeconomictheory,mathematics,andstatisticalinferenceareappliedtotheanalysis.vPositivehelpvEconomictheory_measurements6MethodologyofEconometricsvStatementoftheoryorhypothesisvObtainingthedatavSpecificationofthemathematicalmodelvSpecificationoftheeconometricmodelvEstimationoftheparametersoftheeconometricmodelvHypothesistestingvForecastingorpredictionvUsingthemodelforcontrolorpolicypurposes7Statement of Theory or HypothesisvPostulate(givesomeexamples)vStatementvNote:hypothesisisnotthesameasanassumption8Obtaining the DatavNaturevSourcesvLimitations9Types of DataTimeseriesdata:quantitative,qualitative(dummyvariable)(SATIONARY)Cross-sectionaldata:(HETEROGENEITY)Pooleddata:(Paneldata)10Sources11Accuracy of Data vNon-experimentalinnaturevRound-offsandapproximationsvNon-responsevSelectivitybiasvAggregatelevelvConfidentialityvTheresultsofresearchareonlyasgoodthequalityofthedata.12Specification of the mathematical modelvYi=b1+b2*Xi0b2sampleparameter-estimate-estimatordistribution-populationparameter-populationcharacteristicsvConfirmationorrefutationofeconomictheoriesonthebasisofsampleevidencevThebasementisstatisticalinference(Hypothesistesting)20Forecasting or PredictionvHypothesisortheorybeconfirmedvKnownorpredictorvariableXvPredictthefuturevaluesofthedependent21Use of the Model for Control or Policy PurposesvControlvariableXvTargetvariableYvYi=b1+b2*XivManipulatethecontrolvariableXtoproducethedesiredlevelofthetargetvariableY22Anatomy of Classical Econometric ModelingvEconomictheoryvMathematicalmodeloftheoryvEconometricmodeloftheoryvDatavEstimationofeconometricmodelvHypothesistestingvForecastingorpredictionvUsingthemodelforcontrolorpolicypurposes23第第1章章计量量经济学研究的方法学研究的方法论4第第2-3章章基本基本统计概念，概率分布概念，概率分布4第第4章章估估计与假与假设4第第5章章双双变量模型的基本思想量模型的基本思想4第第6章章双双变量模型的假量模型的假设检验4第第7章章多元回多元回归：估：估计与假与假设检验4第第8章章回回归方程的函数形式方程的函数形式4第第9章章虚虚拟变量的回量的回归模型模型4第第10章章多重共多重共线性性4第第11章章异方差性异方差性4第第12章章自相关性自相关性4第第13章章模型模型选择：标准与准与检验4实验实验1-6624Please Give Some SuggestionsZ3-W163.COM027-62082852Thankyou.25A Review of Some Statistical ConceptsLecture Two26Sample space、Sample points、EventsvPopulationisthesetofallpossibleoutcomesofrandomexperiment(samplespace)vSample pointistheeachmemberofthissamplespacevEventisasubsetofthesamplespace27Probability and Random VariablesvP(A)probability(pr;p;pro)vXrandomvariable(rv)vXthevalueofarandomvariablev0=P(A)=0,29Cumulative Distribution FunctionCDFF(X)=P(X=x)(discrete)=(continuous)30CDFofDiscreteVRPDFCDFX times of face upf(x)(PDF)Value of X f(x)(CDF)01/16X=01/1614/16X=15/1626/16X=211/1634/16X=315/1641/16X=4131CDF11/165/1611/1615/161X234032CDFofContinuousVRPDFCDFValue of X timesf(x)PDF)Value of X timesf(x)(CDF)0=X11/16X=01/161=X24/16X=15/162=X36/16X=211/163=X44/16X=315/164=X51/16X0,symmetricalS=0orleftS2,itsvarianceis70An ExamplevGiven k1=10 and k2=8,what is theprobabilityofobtaininganFvalue(a)of3.4orgreater;(b)of5.8orgreater?vTheseprobabilitiesare(a)approximately0.05;(b)approximately0.01.71Relationships1.If the denominator df,k2,is fairly large,thefollowingrelationshipholds:2.3.Largedf,thet,chisquare,andFdistributionsapproachthenormaldistribution,thesedistributions are known as the distributionsrelatedtothenormaldistribution.72 An ExamplevLet k1=20 and k2=120.The 5 percentcritical F value for these df is 1.48Therefore,k1F=(20)*(1.48)=29.6.vFrom the chi-square distribution for 20 df,the 5 percent critical chi-square value isabout31.41.73Lecture 3(2)Estimation and Inference74ESTIMATIONvAssumethatarandomvariableXfollowsaparticularprobabilitydistributionbutdonotknowthevalue(s)oftheparameter(s)ofthedistribution.vifXfollowsthenormaldistribution,wemaywanttoknowthevalueofitstwoparameters,namely,themeanandthevariance.75Estimate the UnknownsvWehavearandomsampleofsizenfromtheknownprobabilitydistribution;vUsethesampledatatoestimatetheunknownprobabilitydistribution;(non-pa)vUsethesampledatatoestimatetheunknownparameters.(pa)76Two CategoriesPointestimationIntervalestimation.77Point EstimationvLetXbearvwithPDFf(x;),istheparameterofthedistribution(forsimplicityonlyoneunknownparameter).vAssume that we know the theoreticalPDF,suchasthetdistributiondonotknowthevalueof.wedrawarandomsample of size n from this known thisPDFandthendevelopafunctionofthesamplevalues78Estimator or Estimatevprovidesusanestimateofthetrue.is known as a statistic,or an estimator,vA particular numerical value taken by the estimator is known as an estimate.can be treated as a random variable.provides us with a rule,or formula,that tells us how we may estimate the true.79An ExamplevSamplemeanisanestimatorofthetruemeanvalue,.Ifinaspecificcase=50,thisprovidesanestimateof.Theestimator obtainedisknownasapointestimatorbecauseitprovidesonlyasingle(point)estimateof.80Interval EstimationDefofintervalestimation:vweobtaintwoestimatesof,byconstructingtwoestimators1(x1,x2,xn)and2(x1,x2,xn),andsaywithsomeconfidence(i.e.,probability)thattheintervalbetween1and2includesthetrue.vweprovidearangeofpossiblevalueswithinwhichthetrue maylie.81Key conepts vSampling,Probability distribution,An estimator.vX is normally distributed,then thesamplemeanisalsonormallydistributedwithmean=(thetruemean)andvariance=2/n,N(,2).vprobabilityis95%82IntervalEstimationvMoregenerally,inintervalestimationweconstructtwoestimatorsand,bothfunctionsofthesampleXvalues,suchthatvTheintervalisknownasa confidence interval ofsize1for,v1beingknownastheconfidence coefficient,v isknownasthelevel of significance.83An ExamplevSupposethatthedistributionofheightofmeninapopulationisnormallydistributedwithmean=and=2.5.vAsampleof100mendrawnrandomlyfromthispopulationhadanaverageheightof67.Establisha95%confidenceintervalforthemeanheight(=)inthepopulationasawhole.84SolutionvAsnoted,N(,2/n),whichinthiscasebecomesN(,2.52/100).v95%confidenceintervalas66.5167.4985OLS and MLvThereareseveralmethodsofobtainingpointestimators,thebestknownbeingthemethod of Ordinary leastsquares andthe method of maximumlikelihood(ML).vThedesirablestatisticalpropertiesfallintotwocategories:small-sample,andlargesample,orasymptotic.86SmallSample PropertiesvUnbiasedness.Anestimatorissaidtobean unbiased estimator of if the expectedvalueofisequaltothetrue;thatis,vE()=0vIf this equality does not hold,then theestimatorissaidtobebiased,andthebiasiscalculatedasvBias()=E()87Minimum VariancevMinimum variance.1issaidtobeaminimum-varianceestimatorofifthevarianceof1 issmallerthanoratmostequaltothevarianceof2，whichisanyotherestimatorof。88EfficientvBest unbiased，or efficient，estimator.If1and2 are twounbiased estimators of,and are twounbiased estimators of 1，and thevarianceof1issmallerthanoratmostequaltothevarianceof2，then1isa minimumvariance unbiased，orbest unbiased，orefficient，estimator.89Linearity.vAn estimatoris said to be a linearestimatorofifitisalinearfunctionofthesample observations。Thus，the samplemeandefinedas90BLUEvBest linear unbiased estimator（BLUE）。If is linear，is unbiased，and hasminimumvarianceintheclassofalllinearunbiasedestimatorsof，thenitiscalledabestlinearunbiasedestimator，orBLUEforshort。91Hypothesis TestingvEstimationandhypothesis testingconstitutethetwinbranchesofclassicalstatisticalinference.vAssumethatwehaveanrvXwithaknowPDFf(;),where is the parameter of the distribution.Having obtained a random sample of size n,we obtain the point estimator.The true is rarely known,Is the estimator “compatible”with some hypothesized value of,say,=*,?92HypothesisvIn the language of hypothesis testing=*is called the null null hypothesishypothesis and is generally denoted by Ho.The null hypothesis is tested against an alternative hypothesisalternative hypothesis,denoted by H1,which,for example,may state that*.93Simple and CompositevA simplesimple hypothesis:it specifies the value(s)of the parameter(s)of the distribution;otherwise it is called a composite composite hypothesis.vThus,if XN(,2)and we state that v Ho:=15 and=2vit is a simple hypothesis,whereas v Ho:=15 and 2vis a composite hypothesis because here the value of is not specified.94Test the Null HypothesisvTo test the null hypothesis(i.e.to test its validity),we use the sample information to obtain what is known as the test statistic.The point estimator of the unknown parameter.Then we try to find out the sampling,or probability,distribution of the test statistic and use the confidence interval or test of significance approach to test the null hypothesis.95AnExample The height(X)of men in a population.We are told that Xi N(,2)=N(,2.52)vAverage height=67 =67 n=100vLet us assume that Ho:=*=69 H1:6996vCouldthesamplewith=67,theteststatistic,havecomefromthepopulationwiththemeanvalueof69?vwemaynotrejectthenullhypothesisif is“sufficientlyclose”to*;otherwisewemayrejectitinfavorofthealternativehypothesis.howdowedecidethat is“sufficientlyclose”to*?97Two Approaches(1)(1)Confidence interval Confidence interval(2)(2)Test of significanceTest of significance(3)Both leading to identicalidentical conclusions in any specific application.98Confidence Interval Confidence Interval ApproachApproachvSinceXiN(,2),weknowthattheteststatisticisdistributedasv100(1)confidenceintervalforbasedonandseewhetherthisconfidenceintervalincludes=*?99vTheactualmechanicsareasfollows:sinceN(,2/n),itfollowsthatvPr(1.96Zi 1.96)=0.95100Turningtoourexample,wehavealreadyestablisheda95%confidenceintervalfor,whichis66.51 67.49Thisintervalobviouslydoesnotinclude=69.Therefore,wecanrejectthenullhypothesisthatthetrueis69witha95%confidencecoefficient.101Region and Critical ValuesvTheconfidenceintervalthatwehaveestablishediscalledtheacceptance region.vThearea(s)outsidetheacceptanceregionis(are)calledtheregion(s)of rejectionofthenullhypothesis.(Criticalregions)vThelowerandupperlimitsoftheacceptanceregionarecalledthe critical values.102Criterion for RejectionvLanguageofhypothesistesting,ifthehypothesizedvaluefallsinsidetheacceptanceregion,onemaynotrejectthenullhypothesis;votherwiseonemayrejectit.103Two Types of ErrorvWearelikelytocommittwotypesoferrors:(1)wemayrejectH0whenitis,infact,true;thisiscalledatypeerror.v(2)wemaynotrejectH0whenitis,infact,false;thisiscalledatypeerror.104vIdeally,wewouldliketominimizebothtypeandtypeerror.vButunfortunately,foranygivensamplesize,itisnotpossibletominimizeboththeerrorssimultaneously.105Level of SignificancevIntheliteraturetheprobabilityoftypeerrorisdesignatedasandiscalledthelevel of significance.vTheprobabilityoftypeerrorisdesignatedas.vTheprobabilityofnotcommittingatypeerror,1,iscalledthepowerof the test.106Theclassicalapproachtohypothesistestingistofixatlevelsuchas0.01or0.05andthentrytomaximizethepowerofthetest;thatisminimize.107Confidence CoefficientvConfidencecoefficient(1-)issimplyoneminustheprobabilityofcommittingatypeIerror.vA95%confidencecoefficientmeansthatwearepreparedtoacceptatthemosta5%probabilityofcommittingatypeIerror.vwedonotwanttorejectthetruehypothesisbymorethan5outof100times.108P valuevThepvalue,orexactlevelofsignificance.Insteadofpreselectingatarbitrarylevels,suchas1,5,or10percent.onecanobtainthep(probability)value,orexactlevelofsignificanceofateststatistic.vThep valueisdefinedasthelowestsignificancelevelatwhichanullhypothesiscanberejected.109The Test of Significance Approachvatest(statistic)issignificant,wegenerallymeanthatwecanrejectthenullhypothesis.vtheteststatisticisregardedassignificantiftheprobabilityofourobtainingitisequaltoorlessthan,theprobabilityofcommittingatypeIerror.110SummarizetheStepsInvolvedinTestingaStatisticalHypothesisvStep1.StatethenullhypothesisHoandthealternativehypothesisH1(e.g.Ho:=69andH1:69).vStep2.Selecttheteststatistic(e.g,).vStep3.Determinetheprobabilitydistributionoftheteststatistic(e.g.,N(,2/n).111vStep4.Choosethelevelofsignificance(i.e.,theprobabilityofcommittingatypeIerror).vStep5.Usingtheprobabilitydistributionoftheteststatistic,establisha100(1-)%confidenceinterval.112vIfthevalueofparameterunderthenullhypothesis(e.g,=*=69)liesinthisconfidenceregion,theregionofacceptance,donotrejectthenullhypothesis.Butifitfallsoutsidethisinterval(i.e,itfallsintotheregionofrejection),youmayrejectthenullhypothesis.vKeepinmindthatinnotrejectingorrejectinganullhypothesisyouaretakingachanceofbeingwrongpercentofthetime113Examplevif=*=69,vif=0.05,theprobabilityofobtainingaZvalueof1.96or1.96is5percent(or2.5percentineachtailofthestandardizednormaldistribution).vInourillustrativeexampleZwas8.114vZ=-8isstatisticallysignificant;thatis,werejectthenullhypothesisthatthetrue*is69.vOfcourse,wereachedthesameconclusionusingtheconfidenceintervalapproachtohypothesistesting.115Lecture FourTwovariable Regression Analysis(A)Some Basic Ideas(B)Estimation(point and interval)(C)Hypothesis Testing116Some Basic IdeasvConceptofRegressionvPopulationRegressionFunction（PRF）vSampleRegressionFunction（SRF）117Two and Multiple Regression AnalysisThemoregeneralmultipleregressionanalysisisinmanywaysalogicalextensionofthetwo-variablecase.Sowefirstintroducethetwovariableregressionanalysis.118Regression Analysis(conditional)vIs largely concerned with estimating and/or predicting the population mean or average value of the dependent variable on the basis of the known or fixed values of the explanatory variables.vConditional distribution(probabilities,expectation,mean)119PRF and SRFvPopulationRegressionFunctionisakeyconceptunderlyingregressionanalysis.vThedisturbancetermplaysacriticalroleinestimatingthePRF.vPRFisanidealizedconcept.vSampleRegressionFunctionisusedtoestimatethePRF120Income XConsumption 80.100 120 140 160 180 200Weekly family consumption expenditure Y,$5560657075Total 325121Income Xp(Y|X=Xi)80.100 120 140 160 180 200p(Y|X=Xi)1/51/51/51/51/5Conditional means of Y65122Conditional Distribution8010012014016018020050100150200Consumption expenditureWeekly income123Population Regression Line(Curve)E(Y|X=Xi)8010012014016018020050101150200Conditional MeanWeekly income65149124Linear Population Regression FunctionWhereandareunknownbutfixedparametersknownastheregressioncoefficients.andarealsoknownastheinterceptandslopecoefficient.125Stochastic Specification of P R F126Sample Regression FunctionvOurtaskistoestimatethePRFonthebasisofthesampleinformation.vSampleisfluctuation.vNdifferentsampleshaveNdifferentSRFs,buttheyarenotthesame.vEstimatororstatisticissimplyaruleorformulathatisusedtoestimatethetotalparameterfromsampleinformation.127IsreadasYhatorY-capEstimatorofE(Y|Xi)Denotesthesampleresidualterm128Consumption expenditureWeekly income129The Problem of Estimation130Method of Ordinary Least Squares(OLS)vNowgivennpairsofobservationsonYandX,wewouldliketodeterminetheSRFinsuchamannerthatitisascloseaspossibletotheactualY.vWemayadopttheleast-squarescriterion,whichstatesthattheSRFcanbefixedinsuchwaythat131Normal Equations and the Solution of OLS132Numerical Properties of Estimators From OLSvTheOLSestimatorsareexpressedsolelyintermsoftheobservablequantities.vTheyarepointestimators.vOncetheOLSestimatesareobtainedfromthesampledata,thesampleregressionlinecanbeeasilyobtained.vTheresidualsareuncorrelatedwithYi.vTheresidualsareuncorrelatedwithXi.133Properties of SRLvSRF passes through the sample means of Y and X.vThe mean value of the estimated Y is equal to the mean value of the actual Y.vThe mean value of the residuals is zero.134XSRF135CLRMvOurobjectiveisnotonlytoobtainthesampleparametersbutalsotodrawinferencesaboutthetruetotalparameters.vCLRM:TheGaussian,standard,orClassicalLinearRegressionModel.vItisthecornerstoneofmosteconometrictheory.136Assumptions on CLRM(1)1.Theregressionmodelislinearintheparameters.2.Xvaluesarefixedinrepeatedsampling,theregressionanalysisisconditionalregression.3.Zeromeanvalueofdisturbanceui.4.Homoscedasticityorequalvarianceofui.5.notheteroscedasticity.1375.Noautocorrelationbetweenthedisturbances.Fordifferentobservations,covarianceiszero.ornoserialcorrelation.6.ZerocovariancebetweenuiandXi.7.Thenumberofobservationsnmustbegreaterthanthenumberofparameterstobeestimated.Assumptions on CLRM(2)138Assumptions on CLRM(3)8.VariabilityinXvalues.9.Theregressionmodeliscorrectlyspecified.10.Thereisnoperfectmulti-collinearity.139Precision or Standard Errors of Leastsquares EstimationStandarderroroftheestimaterepresentsthevarianceofthebothuiandYi.140Properties of Leastsquares EstimationvTheGauss_Markov:BLUEvItisLinear.ThatisalinearfunctionofarandomvariableYintheregressionmodel.vItisUnbiased.Itsaverageorexpectedvalueisequaltothetruevalue.vEfficientestimator.Ithasminimumvariableintheclassofallsuchlinearunbiasedestimators.141The Coefficient of Determination r2vTotal sum of squares(TSS)vExplained sum of squares(ESS)vResidual sum of squares(RSS)vTSS=ESS+RSS 142The Coefficient of Determination r2143YXiYi =due to residualYi =total =due to regression144Lecture Five1.Review(General concepts)2.Interval Estimation 3.Hypothesis Testing145ReviewvProbability DistributionsvProperties of Normality Assumption vPoint EstimationvInterval EstimationvHypothesis TestingvConfidence Interval ApproachvTesting of Significance Approach146MethodologyofEconometricsvStatementoftheoryorhypothesisvObtainingthedatavSpecificationofthemathematicalmodelvSpecificationoftheeconometricmodelvEstimationoftheparametersoftheeconometricmodelvHypothesistestingvForecastingorpredictionvUsingthemodelforcontrolorpolicypurposes147Normal DistributionChisquare DistributionT DistributionF Distribution148Normal Distribution vA(continuous)random variable X is said to benormally distributed if its PDF has the followingform:whereand2,knownastheparametersofthedistribution,are,respectively,the mean and thevarianceofthedistribution.X N (,2)149The Properties of This Distribution vItissymmetricalarounditsmeanvalue.v68%oftheareaunderthenormalcurveliesbetweenthevaluesof;v95%95%of the area lies between 22;v99.7%99.7%of the area lies between 33150Standardized Normal VariablevweconvertthegivennormallydistributedvariableXwithmeanand2intoastandardized normal variable ZbythefollowingtransformationvvvAnimportantpropertyofanystandardizedvariableisthatitsmeanvalueiszeroanditsvarianceisunity.X N(0,1)151Properties of NDvLet X1 N(1,21)and X2 N(2,22)andtheyareindependent.thelinearcombination Y=aX1+bX2,a,bareconstants.vThenitcanbeshownthat Y N(a1 1+b+b2 2),(a),(a2 221+b222)152Central Limit TheoremvLet X1,X2,Xn.denote n independentrandom variables,all of which have thesamePDFwithmean=and variance=2.Let=Xi/n(i.e.,the sample mean).(i.e.,n),153vThat is,approaches the normaldistributionwithmeanand variance2/n.Notice that this resultholdstrueregardlessoftheformofthePDF.Asaresult,itfollowsthatvZ is a standardized normal variable.154Momentsv The third and fourth moments of the normal distribution around the mean value are as of follows:E(X-)3=0 E(X-)4=34 Note:All odd-powered moments about the mean value of a normally distributed variable are zero.155Skewness and KurtosisvFor a normal PDF,For a normal PDF,vSkewness=0 and Kurtosis=3Skewness=0 and Kurtosis=3;vA A normal normal distribution distribution is is symmetric symmetric and and mesokurtic.mesokurtic.vA A simple simple test test of of normality normality is is Jarque-Jarque-Bera(JB)test of normalityBera(JB)test of normality156The The(Chi-square)(Chi-square)DistributionDistributionvLet Z1,Z2,ZK be independent standardized normal variable.v distribution with k degrees of freedom(df),vdf means the number of independent quantities in the previous sum,vA chi-square-distributed variable is denoted by ,where the subscript k indicates the df.157Properties of the Distributionvthedistributionisskewed,thedegreeoftheskewnessdependingonthedf.vForcomparativelyfewdf,thedistributionishighlyskewedtotheright;vAs the number of df increases,thedistributionbecomesincreasinglysymmetrical.158vTh