计量经济学讲义(厦门大学 黄长全)Chap4

Chapter3多元线性回归模型(Multiplelinearregressionmodel)Youarerequiredtogetfamiliarwithmatrixalgebraformasteringthischapter!ClassicalMultiplelinearregressionmodel(CMLRM):1.model2.randomsample011kkYbbXbX1{;,}iikiYXX011(1,,)iikkiiYbbXbXinMatrixform:11101112212111,,,1kknnknknXXbYXXbYXXbYXbεYXbε3.Modelassumption:1.2.3.isnon-random.4.5.NormalityassumptionE()ε02E()(isaunitmatrix)nnεεIIXrank()=1knX2(0,)(1,,)iNinassumptions1and5implythattheerrorsareIndependent.Asinthecaseoftheunivariatelinearregressionmodels,wecanestimatetheregressioncoefficientsofthemultiplelinearregressionmodelsbyusingtheordinaryleastsquaresprocedure.Inmatrixform,theOLSEis1ˆ(bXX)XY4.OLSEfortheCMLRM5.PropertiesoftheOLSEfortheCMLRM1.2.3.TheGauss-Markovtheoremisstilltrue:TheOLSEfortheCMLRMistheBLUE.1ˆ(bXX)XYˆE()bb21ˆˆˆvar()E[()()]()bbbbbXX6.ResidualandEstimationofthepopulationvariance1.Residual1)Pisidempotent(幂等的)2)3)4)2ˆˆ[]()-1-1eYYYXbIX(XX)XYPYPIX(XX)XE()e02var()E()eeeP21ˆtr()niieYYbXYee2.Estimatorfor222ˆ(1)(1)ienknkee22ˆ()()1eeEEnk7.Goodness-of-fittesting1.1)Totalsumofsquares:2)Explainedsumofsquares:2.Coefficientofdetermination:22TSS()iYYnYYY222ˆESS()iiYYenYbXY2222ˆESS1TSS()inYRYYnYeebXYYY3.AdjustedR-squared:222(1)(1)(1)11()11inRnkRYYnknee8.Hypothesistesting1.Significancetestforthepopulationregressionequation1)Hypothesis:2)F-statistic:01:0vskHbb010:thereexsitssuchthat0kHkbESS(,1)RSS1kFFknknk3)TestingGivenasignificancelevel,pickupthecriticalvalue.if,reject.otherwise,acceptit.2.Significancetestforasingleparameter1)Hypothesis:2)t-statistic:FFF0H01:0vs:0jjHbHbˆˆˆˆ()var()jjjjbbtSbb9.Forecasting1.Pointforecast:Given,thenapredictor2.Intervalforecast:Letthen11,12,1,1(1,,,)nnnknXXXX11ˆˆnnYXb011ˆnneYY21111var()[1()]nnneXXXXLetThenHomework:1111ˆˆ()1()nnneXXXX11(1)ˆ()nnettnke对以下数据进行建模分析并做必要的假设检验（显著性水平=5%）Y：国家财政收入（不包括债务收入）（单位亿元）X1：国内生产总值（单位亿元）X2：进出口贸易总额（单位亿元）X3：国家外汇（单位亿美元）资料如下表：yx1x2x319781132.33624.1355.01.6719791146.44038.2454.08.419801159.94517.8570.0-12.9619811175.84862.4735.327.0819821212.35294.7771.369.8619831367.05934.5860.189.0119841642.97171.01201.082.219852004.88964.42066.726.4419862122.010202.22580.420.7219872199.411962.53084.229.2319882357.214928.33821.833.7219892664.916909.24155.955.519902937.118547.95560.1110.9319913149.521617.87225.8217.1219923483.426638.19119.6194.4319934349.034634.411271.0211.9919945218.146759.420381.9516.219956242.258478.123499.9735.9719967408.067884.624133.81050.2919978651.174462.626967.21398.919989876.078345.226896.31449.6199911444.182067.529896.31546.75200013395.289468.139274.21655.7200116386.097314.842183.62121.65200218913.9102397.951378.32864.07