5_多元线性回归模型统计检验.

§2.4多元线性回归模型的统计检验和区间估计StatisticalTestandIntervalEstimationofMultipleLinearRegressionModel拟合优度检验AIC和SC准则方程的显著性检验（F检验）变量的显著性检验（t检验）参数估计量的区间估计预测值的区间估计受约束回归参数稳定性检验说明由计量经济模型的数理统计理论要求的以多元线性模型为例包括拟合优度检验、总体显著性检验、变量显著性检验、偏回归系数约束检验、模型对时间的稳定性检验、参数估计量的区间估计、预测值的区间估计、受约束回归。一、拟合优度检验(TestingofSimulationLevel)1、概念检验模型对样本观测值的拟合程度通过构造一个可以表征拟合程度的统计量来实现。问题：采用普通最小二乘估计方法，已经保证了模型最好地拟合了样本观察值，为什么还要检验拟合程度？2、总体平方和、回归平方和、残差平方和定义2()iTSSYY总体平方和(TotalSumofSquares)2ˆ()iESSYY回归平方和(ExplainedSumofSquares)2ˆ()iiRSSYY残差平方和(ResidualSumofSquares)问题：既然RSS反映了样本观测值与估计值偏离的大小，可否直接用它来作为拟合优度检验的统计量？统计量必须是相对量。TSS、ESS、RSS之间的关系TSS=ESS+RSS3、一个有趣的现象：ˆˆiiiYYYYYY222ˆˆiiiYYYYYY222ˆˆiiiiYYYYYY=关键是在于TSS=ESS+RSS推导过程中用到的一组矩条件：ˆ00,1,...,jiiXYYjk矩条件在大样本下成立，只有一个样本时肯定不成立，在样本足够大时近似成立。理解教材中TSS=ESS+RSS的推导过程4、拟合优度检验统计量:可决系数r2和调整后的可决系数R2可决系数r221ESSRSSrTSSTSSr2越接近于1，模型的拟合优度越高。问题：如果在模型中增加一个解释变量，r2往往增大(?)是否越多的解释变量，模型拟合的越好？DependentVariable:CONSPMethod:LeastSquaresSample:19782000Includedobservations:23VariableCoefficientStd.Errort-StatisticProb.GDPP0.3861800.00722253.474710.0000C201.118914.8840213.512410.0000R-squared0.992710Meandependentvar905.3304AdjustedR-squared0.992363S.D.dependentvar380.6334S.E.ofregression33.26450Akaikeinfocriterion9.929800Sumsquaredresid23237.06Schwarzcriterion10.02854Loglikelihood-112.1927Hannan-Quinncriter.9.954632F-statistic2859.544Durbin-Watsonstat0.550636Prob(F-statistic)0.000000在消费模型中，Eviews软件估计结果DependentVariable:CONSPMethod:LeastSquaresSample(adjusted):19792000Includedobservations:22afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.GDPP0.2213590.0609733.6304620.0018CONSP(-1)0.4514080.1703182.6503800.0158C120.725336.513743.3062990.0037R-squared0.995403Meandependentvar928.4909AdjustedR-squared0.994919S.D.dependentvar372.6339S.E.ofregression26.56264Akaikeinfocriterion9.523012Sumsquaredresid13405.90Schwarzcriterion9.671791Loglikelihood-101.7531Hannan-Quinncriter.9.558060F-statistic2056.887Durbin-Watsonstat1.278902Prob(F-statistic)0.000000在消费模型中，Eviews软件估计结果调整后的可决系数R22111RSSnkRTSSn问题：•为什么以R2作为检验统计量避免了片面增加解释变量的问题？•R2多大才算通过拟合优度检验？二、AIC、SC准则(Akaikeinformationcriterion,AICSchwarzcriterion,SC)AIC、SC准则要求：在模型中增加解释变量的条件是能够减少AIC值或SC值。222(1)lnlnlniiekAICnnekSCnnnDependentVariable:CONSPMethod:LeastSquaresSample:19782000Includedobservations:23VariableCoefficientStd.Errort-StatisticProb.C201.118914.8840213.512410.0000GDPP0.3861800.00722253.474710.0000R-squared0.992710Meandependentvar905.3304AdjustedR-squared0.992363S.D.dependentvar380.6334S.E.ofregression33.26450Akaikeinfocriterion9.929800Sumsquaredresid23237.06Schwarzcriterion10.02854Loglikelihood-112.1927Hannan-Quinncriter.9.954632F-statistic2859.544Durbin-Watsonstat0.550636Prob(F-statistic)0.000000在消费模型中,用AIC、SC准则判断是否新增解释变量DependentVariable:CONSPMethod:LeastSquaresSample(adjusted):19792000Includedobservations:22afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.GDPP0.2213590.0609733.6304620.0018CONSP(-1)0.4514080.1703182.6503800.0158C120.725336.513743.3062990.0037R-squared0.995403Meandependentvar928.4909AdjustedR-squared0.994919S.D.dependentvar372.6339S.E.ofregression26.56264Akaikeinfocriterion9.523012Sumsquaredresid13405.90Schwarzcriterion9.671791Loglikelihood-101.7531Hannan-Quinncriter.9.558060F-statistic2056.887Durbin-Watsonstat1.278902Prob(F-statistic)0.000000在消费模型中,用AIC、SC准则判断是否新增解释变量例：测度教育的回报问题wage:小时工资(元)，educ:受教育的年数，exper:以年数计的工作经历。其他非观测因素：天生能力、职业道德等。•E(μ|educ,exper)=0，影响wage的其它因素与educ和exper无关。比如，如果μ是天生能力，这个假定就是要求，工人总体中受教育和工作经历的各种组合，其平均能力都相同。•Var(μ|educ,exper)=σ2，Var(wage|educ,exper)=σ2，如果这个方差随着两个解释变量中的任何一个变化，就出现了异方差。012eduwcexgeuperaDependentVariable:WAGEMethod:LeastSquaresSample:1526Includedobservations:526VariableCoefficientStd.Errort-StatisticProb.C-3.3905400.766566-4.4230230.0000EDUC0.6442720.05380611.973970.0000EXPER0.0700950.0109786.3852910.0000R-squared0.225162Meandependentvar5.896103AdjustedR-squared0.222199S.D.dependentvar3.693086S.E.ofregression3.257044Akaikeinfocriterion5.205204Sumsquaredresid5548.160Schwarzcriterion5.229531Loglikelihood-1365.969Hannan-Quinncriter.5.214729F-statistic75.98998Durbin-Watsonstat1.820274Prob(F-statistic)0.000000在教育回报模型中，Eviews估计结果:012ln()wageeducexperuDependentVariable:LOG(WAGE)Method:LeastSquaresSample:1526Includedobservations:526VariableCoefficientStd.Errort-StatisticProb.C0.2168540.1085951.9969090.0464EDUC0.0979360.00762212.848390.0000EXPER0.0103470.0015556.6533930.0000R-squared0.249343Meandependentvar1.623268AdjustedR-squared0.246473S.D.dependentvar0.531538S.E.ofregression0.461407Akaikeinfocriterion1.296614Sumsquaredresid111.3447Schwarzcriterion1.320940Loglikelihood-338.0094Hannan-Quinncriter.1.306139F-statistic86.86167Durbin-Watsonstat1.789452Prob(F-statistic)0.000000在教育回报对数模型中，Eviews估计结果:DependentVariable:LOG(WAGE)Method:LeastSquaresSample:1526Includedobservations:526VariableCoefficientStd.Errort-StatisticProb.C0.2843600.1041902.7292300.0066EDUC0.0920290.00733012.555250.0000EXPER0.0041210.0017232.3914370.0171TENURE0.0220670.0030947.1330700.