计量作业第2章-第4章

第二章一元线性回归模型1、最小二乘法对随机误差项u作了哪些假定？说明这些假定条件的意义。答：假定条件：(1)均值假设：E(ui)=0,i=1,2,…；(2)同方差假设：Var(ui)=E[ui-E(ui)]2=E(ui2)=σu2,i=1,2,…；(3)序列不相关假设：Cov(ui,uj)=E[ui-E(ui)][uj-E(uj)]=E(uiuj)=0,i≠j,i,j=1,2,…；(4)Cov(ui,Xi)=E[ui-E(ui)][Xi-E(Xi)]=E(uiXi)=0；(5)ui服从正态分布,ui～N(0,σu2)。意义：有了这些假定条件，就可以用普通最小二乘法估计回归模型的参数。2、阐述对样本回归模型拟合优度的检验及回归系数估计值显著性检验的步骤。答：样本回归模型拟合优度的检验：可通过总离差平方和的分解、样本可决系数、样本相关系数来检验。回归系数估计值显著性检验的步骤：(1)提出原假设H0：β1=0；(2)备择假设H1：β1≠0；(3)计算t=β1/Sβ1；(4)给出显著性水平α，查自由度v=n-2的t分布表，得临界值tα/2(n-2)；(5)作出判断。如果|t|tα/2(n-2)，接受H0:β1=0，表明X对Y无显著影响，一元线性回归模型无意义；如果|t|tα/2(n-2)，拒绝H0，接受H1:β1≠0，表明X对Y有显著影响。4、试说明为什么∑ei2的自由度等于n-2。答：在模型中，自由度指样本中可以自由变动的独立不相关的变量个数。当有约束条件时，自由度减少，其计算公式：自由度=样本个数-受约束条件的个数，即df=n-k。一元线性回归中SSE残差的平方和，其自由度为n-2,因为计算残差时用到回归方程，回归方程中有两个未知参数β0和β1，而这两个参数需要两个约束条件予以确定，由此减去2，也即其自由度为n-2。5、试说明样本可决系数与样本相关系数的关系及区别，以及样本相关系数与β^1的关系。答：样本相关系数r的数值等于样本可决系数的平方根，符号与β1相同。但样本相关系数与样本可决系数在概念上有明显的区别，r建立在相关分析的理论基础之上，研究两个随机变量X与Y之间的线性相关关系；样本可决系数r²建立在回归分析的理论基础之上，研究非随机变量X对随机变量Y的解释程度。6、已知某市的货物运输量Y(万吨)，国内生产总值GDP(亿元，1980年不变价)1985~1998年的样本观测值见下表(略)。DependentVariable:YMethod:LeastSquaresDate:10/28/13Time:10:25Sample:19851998Includedobservations:14VariableCoefficientStd.Errort-StatisticProb.GDP26.954154.1203006.5417920.0000C12596.271244.56710.121010.0000R-squared0.781002Meandependentvar20168.57AdjustedR-squared0.762752S.D.dependentvar3512.487S.E.ofregression1710.865Akaikeinfocriterion17.85895Sumsquaredresid35124719Schwarzcriterion17.95024Loglikelihood-123.0126Hannan-Quinncriter.17.85050F-statistic42.79505Durbin-Watsonstat0.859998Prob(F-statistic)0.000028(1)一元线性回归方程Yt=12596.27+26.95415GDPt(2)结构分析β^1=26.95425是样本回归方程的斜率，它表示某市货物运输量的情况，说明货物运输量每增加1亿元，将26,95425用于国内生产总值；β^0=12596.27是样本回归方程的截距，它表示不受货物运输量影响的国内生产总值。(3)统计检验r2=0.78说明总离差平方和的78％被样本回归直线解释，有22％没被解释，说明样本回归直线对样本点的拟合优度还是比较高的。显著性水平α=0.05，查自由度v=14-2=12的t分布表，得临界值t0.025(12)=2.18(4)预测区间1980~2000obsGDPRESIDYYFYFSE198019811982198319841985161.691294.518170471381824916954.481829528621837.8050429478071986171.071317.6882638304891852517207.311736169511827.8522585717681987184.07842.28434204673981840017557.715657953261815.3290745659511988194.75-1152.5859567725241669317845.585956772531806.1647435845771989197.86-2386.4133565223311554317929.413356522331803.6891930532051990208.55-2288.5531968198881592918217.553196819891795.8513778573231991221.06-246.74958616717411830818554.749586167181788.0138737937551992246.92-1729.783849038541752219251.783849038541776.4503159894641993276.81582.8262138154242164020057.173786184581770.9956488707011994316.382658.9810427230552378321124.018957276941776.9262940212641995363.521645.3625140395232404022394.637485960481803.3104801280861996415.51337.01636838282142413323795.983631617181855.6949869099331997465.78-60.968643007108762509025150.968643007111927.747214173007∧1998509.1-1813.622326981882450526318.622326981882004.9827372665981999200062029307.837321275562255.639096466328单个值预测区间Y2000∈[29307.84-2.10×2255.64,29307.84+2.10×2255.64]均值预测区间E(Y2000)∈[29307.84-2.10×2255.64,29307.84+2.10×2255.64]8、查中国统计年鉴，利用1978~2000的财政收入和GDP的统计资料，要求以手工和EViews软件。(1)散点图020,00040,00060,00080,000100,000010,00030,00050,00070,00090,000YGDPDependentVariable:YMethod:LeastSquaresDate:10/29/13Time:16:40Sample:19782000Includedobservations:23VariableCoefficientStd.Errort-StatisticProb.GDP0.9860970.001548637.03830.0000C174.417150.395893.4609390.0023R-squared0.999948Meandependentvar22634.30AdjustedR-squared0.999946S.D.dependentvar23455.82S.E.ofregression172.6972Akaikeinfocriterion13.22390Sumsquaredresid626310.6Schwarzcriterion13.32264Loglikelihood-150.0748Hannan-Quinncriter.13.24873F-statistic405817.8Durbin-Watsonstat0.984085Prob(F-statistic)0.000000一元线性回归方程Y=174.4174+0.98GDPt经济意义国名收入每增加1亿元，将有0.98亿元用于国内生产总值。(2)检验r²=99％，说明总离查平方和的99％被样本回归直线解释，仅有1％未被解释，所以说样本回归直线对样本点的拟合优度很高。显著性水平α=0.05，查自由度v=23-2=21的t分布表，得临界值t0.025(21)=2.08。(3)预测值及预测区间obsYYFYFSEGDP19783645.23768.939527560003178.87990788736163645.219794062.64180.536602486764178.77407772894174062.619804545.6000000000014656.821670023003178.65445312373664545.60000000000119814889.54998.011387140059178.57063446903184891.60000000000119824889.54998.011387140059178.57063446903184891.60000000000119835330.55423.808265322558178.46823011388035323.39999999999919845985.66054.220364030461178.32110832662425962.719857243.87282.306126162203178.04995048489017208.119869040.7000000000019065.07170297124177.69280630099319016198712050.612065.37179921504177.189939863891612058.6198810274.410306.76560988973177.469705227405810275.2198912050.612065.37179921504177.189939863891612058.6199015036.815008.08380447724176.817239439131815042.8199117000.916930.48077996771176.638587454027716992.3199218718.318582.68705461982176.526126442387818667.819933526035017.08573798564177.479184885403835333.9199421826.221653.09867943883176.418239372446321781.5199526937.326723.61175867555176.528268981976926923.519963526035017.08573798564177.479184885403835333.9199748108.547702.24331311228180.747077071159648197.9199859810.560122.92955260078185.968135704457960793.7199988479.288604.77659126783204.561247885819189677.1200070142.570361.48074871261191.66140