计量经济学习题详解

第四章习题4.1没有进行t检验，并且调整的可决系数也没有写出来，也就是没有考虑自由度的影响，会使结果存在误差。4.3中国商品进口额、国内生产总值、居民消费价格指数年份商品进口额/亿元国内生产总值/亿元居民消费价格指数（以1985年为100）19851257.89016.0100.019861498.310275.2106.519871614.212058.6114.319882055.115042.8135.819892199.916992.3160.219902574.318667.8165.219913398.721781.5170.819924443.326923.5181.719935986.235333.9208.419949960.148197.9258.6199511048.160793.7302.8199611557.471176.6327.9199711806.578973.0337.1199811626.184402.3334.4199913736.489677.1329.7200018638.899214.6331.0200120159.2109655.2333.3200224430.3120332.7330.6200334195.6135822.8334.6200446435.8159878.3l347.7200554273.7183084.8353.9200663376.9211923.5359.2200773284.6249529.9376.5200879526.5314045.4398.7200968618.4340902.8395.9201094699.3401512.8408.92011113161.4472881.6431.0一研究的目的和要求我们知道，商品进口额与很多因素有关，了解其变化对进出口产品有很大帮助。为了探究和预测商品进口额的变化，需要定量地分析影响商品进口额变化的主要因素。二、模型的设定及其估计经分析，商品进口额可能与国内生产总值、居民消费价格指数有关。为此，考虑国内生产总值GDP、居民消费价格指数CPI为主要因素。各影响变量与商品进口额呈正相关。为此，设定如下形式的计量经济模型：叫M侑+SJnGDPt+SJnCP/t

式中，Z为第t年中国商品进口额(亿元)；InGDP为第t年国内生产总值(亿元)；InCPI为居民消费价格指数(以1985年为100)。各解释变量前的回归系数预期都大于零。为估计模型，根据上表的数据，利用EViews软件，生成Y、InGDP、InCPI等数据，采用OLS方法估计模型参数，得到的回归结果如下图所示：DependentVariable:LNYMethod:LeastSquar&sDate:04y15/17Time:19:54Sample:19352011Includedobservations:27VariableCoefficientStd.Errort-StatisticProbC-3.1114860.46J010 -6.7201260.0000LMGDP1.33S53-30.03S610 15.105820.0000LNCPI-0.4217910.2^295 -1.S079750.0832R-squared0.98805-1Meandependentvar9.4S4710AdjustedR-squared0.987055S.D.dependentyar1.455517S.E.口仃町曰33口门0.162189Akaikeinfocriterion-0.695670Sumsquared「e§id0.631326Schwarzcriterion-0.551689Loglikelihood12.3915-5Hannan-Quinncriter.-0.652857F-statisiic992.2532Durbin-Watsonstat0-522613Prob(F-statistic)O.OOQQOO模型方程为：lnY=-3.111486+1.338533lnGDP-0.421791lnCPI(0.463010)(0.088610) (0.233295)t=(-6.720126)(15.10582) (-1.807975)^2=0.988051R2=0.987055F=992.2582该模型氏2=0.988051，R2=0.987055，可决系数很高，F检验值为992.2582，明显显著。但是当a=0.05时，J(n-k)=%.O25(27-3)=2.064，不仅lnCPI的系数不显著，而且，lnCPI的符号与预期相反，这表明可能存在2严重的多重共线性。计算各解释变量的相关系数，选择lnGDRlnCPI数据，“view/correlation”得相关系数矩阵。CorrelationLMCPILNGDFLNCPILNGDPLNCPI1.0000000.9&37400&537401.0000001由相关系数矩阵可以看出，各解释变量相互之间的相关系数较高，证实确实存在一定的多重共线性。为了进一步了解多重共线性的性质，我们做辅助回归，即每个解释变量分别作为被解释变量都对剩余的解释变量进行回归。□epenoentvariaUte:lngdfMetnod:LeastsquaresDale04/15717TitHe：21：&9sample.13352011includedobservations:27DependentVariatIe:LNCPIMethod:LeastSquaresDate:04^15/17Time:21:10Sample.19852011incluclecJonser/allons27VariableCoefficient31dErrort-SlatisticProb.variablecoeffidemstd.Errort-siatisticProb.C-2.7963010.062790 -3.1B76340.0040c工5154020.256313 5.912301O.OODDLNOFI25110220.156302 15.B6227O.OODOLMGDP0.3B22510.022837 15.BB227O.OODOR-squared0.909621Fdeandependentvar11.16214R-squared0.9D9621Meandependentvar5.55B900AdjustedR-squared0.906005S.D.dependentvar1.194029.AdjustedR-squa.red0.9D6005S.D.dependentvar0.453513B.E.DfregressionO.J66072AkaikeinfocriterionQ.099213S.E.ofregression0.139042Akaikeinfocriterion-1.05BB94Sumsquaredresid3-350216Schwarzcriterion0.935201SumsquaredresiU0.463317Schwarzcriterion-0.94D907Loglikelihood-10.13933Hannan-Quinncriter.Q.92775&lccineiinood15.99SD8Hannan-ouinnenter-1WB352F-statistic251.6117Durbin-Watsonstat0.089623F-Etatstc251.6117Durtsin-Watsonslat□.114568ProKF-statiGtic]0.000000Proti(F-£tati£tic)O.DDOOOOInGDP与InCPI的相关系数很高，证明存在多重共线性。三、其他分析1.进行下面的回归①In%=4+4联吗+?DependentVariable:LNYMethod:LeastSquaresDate:04y27/17Time:09:10Sample:19852011Includedobservations:27VariableQoefTiGientStd.Errort-StatisticProbC-3,7&06700.312255-12.011560.0000LNGDP1.1357390.027S2242,615330.0000R-squared0.986423Meandependentvar9.484710Adjusi&fiR-squaretl0.98S3SOS.D.dependsntvar1.425517&.E.ofregression0.1&33S9Akaikeinfocriterion-0.642056Suinsquaredresid0.717312Schwarzcriterion-0.546068Loglikelihood10.66776Hannan-Quinncriter.-0.613514F-statistic1S16.407□urbin-Watsonstat0471111ProbfF-statistic）0.000000模型的估计结果为：叫=-3.750670+1.185739InGDPt（0.312255） （0.027822）t=（-12.01156） （42.61933）^2=0.986423 丘2=0.985880 F=1816.407②叫②1+断叫+%DependentVariable:LNYMethod:LeastSquares□ate:04722/17Time:09:12Sample:19852011Includedobsenations:27VariableCoefficientStd.Errort-3tatisticProb.G-6.8545351.242243-5.517871a.ooaoLNCPI2.9392950.22275613,19511a.ooooR-squared0.874-442Meandependentvar3.434710AdjustedR-squared0.869419S.D.dependsntwar1425517S.E.ofregression0.&1&124Akaikeinfocriterion1.5S2363Sumsquaredresid6.633S10Schwarzcriterion1.673356Lo-glikelihood-1936196Hannan-Quinncriter.1.610910F-statistic174.1108□urbin-Watsonstat0.137042Prob[F-statistic）0.000000模型的估计结果为：叫=-6.854535+2.939295lnCPZt（1.242243） （0.222756）t=（-5.517871） （13.19511）

^2=0.874442 R2=0.869419F=174.1108③lnGDf+qlnCP"」DependentVariable:LNGDPMethod:LeastSquares□ate:04J22/17Tim-e:09:14Sample:19852011Includedobservations:27VariableCaefficientStdErrort-StatisticProb.C-2.7963S-10.8S2793 -5.167634O.QO+OLNCPI25110220.1&B302 15.862270.0000R-squared0.909621Meandependentvar11,16214AdjustedR-squared0.906005S.D.dependentvar1194029S.E.ofregression0366072Akaikeinfocriterion0899213Sumsquaredresid33&0216Schwarzcriterion0.995201Loglikelihood-10.13938Hannan-Quinncriter.0.927755F-statistic251.6117□urbin-Watsonstat0.099623-Prob(F-statistic)0.000000模型的估计结果为：[1]lnGDPt=-2.796381+2.511022lnCP，t(0.882798) (0.158302)t=(-3.167634) (15.86227)R2=0.909621 R2=0.906005 F=251.6117由此对多重共线性的认识：由上面的几组拟合效果可知，单方程拟合效果都很好，可决系数分别为：0.986423和0.874442,可决系数较高，说明GDP和CPI单个对商品进口额有显著的影响。但是，当这两个变量同时引进模型时，影响方向发生了改变，这只有通过相关系数的检验才能发现，第三个回归结果也说明了，它们间有很强的线性相关关系。建议：如果仅仅是做预测，可以不用在意这些多重共线性，如果是进行结构分析，就需要注意了4.41985~2011年财政收入及其影响因素数据年份财政收入CZSR/亿元财政支出CZZC/亿元国内生产总值GDP(现价)/亿元税收总额SSZE/亿元19852004.802004.259016.042040.7919862122.002204.9110275.182090.7319872199.402262.1812058.622140.3619882357.202491.2115042.822390.4719892664.902823.7816992.322727.4019902937.103083.5918667.822821.8619913149.483386.6221781.502990.1719923483.373742.2026923.483296.9119934348.954642.3035333.924255.3019945218.105792.6248197.865126.88

19956242.206823.7260793.736038.0419967407.997937.5571176.596909.8219978651.149233.5678973.038234.0419989875.9510798.1884402.289262.80199911444.0813187.6789677.0510682.58200013395.2315886.5099214.5512581.51200116386.0418902.58109655.1715301.38200218903.6422053.15120332.6917636.45200321715.2524649.95135822.7620017.31200426396.4728486.89159878.3424165.68200531649.2933930.28184937.3728778.54200638760.2040422.73216314.4334804.35200751321.7849781.35265810.3145621.97200861330.3562592.66314045.4354223.79200968518.3076299.93340902.8159521.59201083101.5189874.16401512.8073210.792011103874.43109247.79472881.5689738.39、研究的目的和要求国家财政收入的高低是政府有效实施其各项职能的重要保障。国家财政收入主要来源于各项税收收入，只有经济持续而健康地增长，才能提供持续的税收来源，因而经济增长是其重要的影响因素；另外，财政收入需要满足日益增长的财政支出的需要。为此，需要定量地分析影响国家财政收入的主要因素。二、模型设定及其估计为了分析各主要因素对国家财政收入的影响，建立财政收入（亿元）（CZSR）为被解释变量，财政支出（亿元）（CZZC）、国内生产总值（亿元）（GDP）、税收总额（亿元）（SSZE）等为解释变量的计量模型。为此，设定如下形式的计量经济模型：CZSRi=B0+B1cZZCi+B2GDPi+B3sSZEi+i式中，CZSRi为第i年财政收入（亿元）；CZZCi为第i年财政支出（亿元;；GD4为第i年国内生产总值GDP（现价）（亿元）；SSZEi为第i年税收总额（亿元）。各解释变量的系数预期都大于零。利用EViews软件，生成CZSR、CZZC、GDP、SSZE等数据，采用OLS方法估计模型参数，得到回归结果如下图所示：DependentVariable:CZSRMethod:Leas.tSquaresDate:04/16J17Time:14:21Sample:1985.2011includedobsedations:27VariableCoefficientStd.Errort-StatisticProb.“修-221.8540130.6522-1.69803^80.1030pZZQ0.0901140.0443672.03'11<290.0540;GDP-0.02^340005069-4.9980360.0000SSZE1.176S940.062162.18.93-2710.0000R-squared0.999857Meandependentvar22572.56AdjustedR-squared0.999838S.D.dependentvar27739.498：.E.ofregregion353.0540Akaikeinfocriterion1470707Sumsquaredre.sid■2866884.Schwarzcriterion14,89905Loglikelihood-194.5455Hannan-Quinncriter.1476416F-statistic...53493.93Durbin-Watsonw怕t1.45S128PmbFstating'0.000000回归方程可写为： - i(130.6532)i(130.6532)t=(-1.698038)R2=0.999857(0.044367)(2.031129)(0.044367)(2.031129)R2=0.999838(-4.998036) (18.93271)F=53493.93—该模型区2=0.999857,R2=0.999838,可决系数很高，F检验值为53493.93,明显显著。但是当a=0.05时，td(n-k)=时，td(n-k)=t0.02K27-4)=2.069，不仅CZZC的系数不显著，并且，GDP的系数与预期相反，这表明可能存在严2重的多重共线性。计算各解释变量的相关系数，选择CZZC、GDP、SSZE数据，点“view/correlation〃得相关系数矩阵，如下图所示：由各相关系数矩阵可知，各解释变量之间的相关系数较高，证实确实存在一定的多重共线性。为了进一步了解多重共线性的性质，我们做辅助回归，即将每个解释变量分别作为被解释变量都对其余的解释变量进行回归。DependentVariable:CZZtMethod:LeastSquaresDate:04/16/17Time:14:49Sample:19852011Includedobservations:27VariableCoefficientStd.Errort-StatisticProb.C-285.2687598.2876-0.4768090.6378GDP-0.0084360.023257-0.36275107200SSZE1.2600840.12479110.102390.0000R-squared0.997168Meandependentvar24168.23AdjustedR-squared0.996932S.D.dependentvar29327.97S.E.ofregression1624.346Akaikeinfocriterion1772804Sumsquaredresid63323999Schwarzcriterion17,87202Loglikelihood-236.3286Harnan-Quiriricriter.1777086F-statistic4225.S95Durbin-Watsonstat1.379907Prob(F-statistic)0.000000Dependentvariable:GDPMethod:LeastSquaresDate:04n6>17Time:14:51Sample:19S52011Includedobservations:27VariableCoefficientStd.Errort-StatisticProb.C18385.223687.876 4.9853Ua.oooaCZZC-0.64636217B1835 -0.362751a.72oaSSZE6.1213642.169215 2.E219200.0094R-squared0.988633Meandependentfar126659.6AdjustedR-squared0.987902S.D.dependentvar129265.4S.E.ofregression14218.02Akaikeinfocriterion22,06685Sumsquaredresid4.85E40QSchwarzcriterion22,21083Loglikelihood-294.9024Hannan-Quinncriter.22,10966F-statistic1062.558Durbin-Watsonstat0.180550Prob(F-statiStic)0.000000DependentVariable:SSZEMeihod:LeastSquaresDate:04>16J17Time:14:52Sample:19852011Includedabsen/ations:27VariableCoefficientStdErrort-StallSticProb.C-432.2846419.8597-1.0295930.3135CZZC0.6422010.06356910,10239a.oooaGDP0.0407000.0144232.821920a.0094R-squared0.997862Meandependentvar20244.81AdjustedR-squared0.997684S.D.dependentvar24oga.5BS.E.ofregression1159.340Akaikeinfocriterion17,05353Sumsquaredresid32257673Schwarzcriterion17,19751Loglikelihood-237.2236Hannan-Quinncriter.17,09634F-statistic5601.263Durbin-Watsonstat1.262S11Prob(F-statiStic)0.000000下表是所得到的可决系数和方差扩大因子的数值，如下表所示:被解释变量可决系数R2的值方差扩大因子VIF.=j1-Rj2CZZC0.997168353GDP0.98883390SSZE0.997862468由上表可知，辅助回归的可决系数很高，经验表明，方差扩大因子VIFj210时，通常说明该解释变量与其余解释变量之间有严重的多重共线性。三、对多重共线性的处理运用逐步回归法，逐步选择和剔除引起多重共线性的变量，具体步骤如下：.先用被解释变量对每一个所考虑的解释变量作简单回归，结果如下所示：aCZSR与CZZC的一元回归结果DependentVariable:CZSRMethod:LeastSquares口日怡：口4门印Time:15:16Sample:19852011Includedobservations:27VariableCoefTicientStd.Errort-StatisticProb.c-257.5992350.3645-0.7100190.4709CZZC0.944635a.00953699,063660.0000R-squared0.997459Meandependentvar22572.56AdjustedR-sqjared0.997357S.D.dependentvar27739.49S.E.ofregression1425.996Akaikeinfocriterion17,43432Sumsquaredresid50B36616Schwarzcriterion17.53Q30Loglikelihood-233.3633Hannan-Quinncriter.17,46236F-staiistic9013.609Durbin-Watsonstat1.259741Prob(F-statistic}0.000000R2=0.997459R2=0.997357F=9813.609bCZSR与GDP的一元回归的结果DependentVariable:CZSRMethod:LeastSquaresDate:04；16/17Time:15:17Sample:19852011Includedobsetvaiions:27VariableCoefficientStd.Errort-StatisticProb.C-4419.476919.2456-4.8077200.0001□DP0.2130560.00512741,552030.0000R-squared0.985727Meandependentvar22572.56AdjustedR-squared0.985156S.D.dependentvar27739.49S.E.ofregression3379.646Akaikeinfocriterion19,16012Sumsquaredresid2,%E+口NSchwarzcriterion19,2561CLoglikelihood-25G,eG1GHannan-Quinncriter.19,1086CF-statistic1726.571Durbin-Watsonstat0.178877Prob(F-statistic)0.000000R2=0.985727R2=0.985156F=1726.571c.CZSR与SSZE的一元回归结果DependentVariable:CZSRMethod:LeastSquaresDate:04/16/17Time:15:18Sample:19852011Includedotiservations:27VariableCoefficientStd.Errort-StatisticProb.C-734.7667131.1949-5.6005590.0000SSZE1.1512740.004215273.1237a.ooaoR-squared0.999665Mearidependertvar22572.56AdjustedR-squared0.999652S.D.dependentvar27739.49S.E.ofregression517.7892Akaikeinfocriterion15,40820Sumsquaredresid6703642.Schwa(2criterion15,50419Loglikelihood-206.0107Hannan-Quinncriter.15,43674F-statistic74596.56Durbin-Watsonstat0.899852Prob(F-statistic)0.000000R2=0.999665R2=0.999652F=74596.56.对以被解释变量贡献最大的解释变量所对应的回归方程为基础，518D逐个引入其余的解释变量。由上面的回归结果可知，SSZE对CZSR的回归结果可决系数最大，再此基础上，逐个引入剩下的解释变量CZZC和GDP在c的基础上引入解释变量CZZC,得到如下的回归结果：

DependentVariable:CZSRMethod:LeastSquaresDate:04/16/17Time:15:41Sample:19852011Includedobservations:27VariableCoefficientSid.Error t-StatisticProb.C-687.6167129.4806 -5.3105790.