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第六章练习题及参考解答6.1表6.5是中国1985-2016年货物进出口贸易总额（𝑌𝑡）与国内生产总值（𝑋𝑡）的数据。表6.5中国进出口贸易总额和国内生产总值单位：亿元年份货物进出口贸易总额（Y）国内生产总值（X）年份货物进出口贸易总额（Y）国内生产总值（X）19852066.719098.9200142183.62110863.119862580.410376.2200251378.15121717.419873084.212174.6200370483.45137422.019883821.815180.4200495539.09161840.219894155.917179.72005116921.77187318.919905560.1218872.92006140974.74219438.519917225.7522005.62007166924.07270232.319929119.6227194.52008179921.47319515.5199311271.0235673.22009150648.06349081.4199420381.948637.52010201722.34413030.3199523499.9461339.92011236401.95489300.6199624133.8671813.62012244160.21540367.4199719981999200026967.2426849.6829896.2339273.2579715.085195.590564.4100280.12013201420152016258168.89264241.77245502.93243386.46595244.4643974.0689052.1740598.7资料来源：《中国统计年鉴2017》（1）建立货物进出口贸易总额的对数ln𝑌𝑡对国内生产总值的对数ln𝑋𝑡的回归方程；（2）检测模型的自相关性；（3）采用广义差分法处理模型中的自相关问题。【练习题6.1参考解答】回归结果𝑙𝑛𝑌̂𝑡=−2.714792+1.152178𝑙𝑛𝑋𝑡(0.316996)(0.027331)𝑡=−8.564142.15675𝑅2=0.9834F=1777.192DW=0.3069自相关检验①图示法图1、2𝑒𝑡−1与𝑒𝑡的散点图以及模型残差图由上面两个图可以发现模型残差存在惯性表现，很可能存在正自相关。②DW检验由回归结果可知DW统计量为0.3069，同时n=32，=1，在0.05的显著性水平下，=1.37，=1.50，因而模型中存在正相关。③BG检验阶数5432AIC-1.275502-1.287655-1.276954-1.338140SIC-0.954873-1.012829-1.047933-1.154923滞后阶数从5阶减小到2阶，AIC及SIC达到最小时，滞后阶数为2阶，此时n𝑅2=22.57582，已知2.(2)=5.99，n𝑅2=22.564545.99，同时P值为0.0000，在0.05的显著性水平下拒绝原假设，即存在自相关。表2BG检验2阶回归结果自相关补救①DW反算法求由DW=0.3069，可知̂=1−2=1−.2=0.84655，可得广义差分方程：𝑙𝑛𝑌𝑡−0.84655𝑙𝑛𝑌𝑡−1=1(1−0.84655)+2(𝑙𝑛𝑋𝑡−0.84655𝑙𝑛𝑋𝑡−1)+𝑡表3广义差分结果-DW反算法DW检验：由回归结果可知DW统计量为1.6284，同时n=31，=1，在0.05的显著性水平下，=1.36，=1.50，即已消除自相关。BG检验：阶数5432AIC-1.260821-1.273263-1.337004-1.369938SIC-0.937017-0.995718-1.105716-1.184908滞后阶数从5阶减小到2阶，AIC及SIC达到最小时，滞后阶数为2阶，此时n𝑅2=0.945667，已知2.(2)=5.99，n𝑅2=0.9456675.99，同时P值为0.6232，在0.05的显著性水平下不拒绝原假设，即已消除自相关。表4广义差分BG检验2阶回归结果则可知，1=−.111−.=−0.998071最终模型为：ln𝑌̂𝑡=−0.998071+1.009608𝑙𝑛𝑋𝑡②残差过原点回归求DependentVariable:EMethod:LeastSquaresDate:02/07/18Time:20:48Sample(adjusted):19862016Includedobservations:31afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.E(-1)0.9027060.1089908.2824420.0000R-squared0.695552Meandependentvar0.004999AdjustedR-squared0.695552S.D.dependentvar0.206153S.E.ofregression0.113749Akaikeinfocriterion-1.477927Sumsquaredresid0.388162Schwarzcriterion-1.431670Loglikelihood23.90787Hannan-Quinncriter.-1.462848Durbin-Watsonstat1.579983表5残差序列过原点回归结果回归结果为：𝑒𝑡=0.902706𝑒𝑡−1，可知̂=0.902706。进而得广义差分方程：ln𝑌𝑡−0.902706𝑙𝑛𝑌𝑡−1=1(1−0.902706)+2(𝑙𝑛𝑋𝑡−0.902706𝑙𝑛𝑋𝑡−1)+𝑡DependentVariable:LNY-0.902706*LNY(-1)Method:LeastSquaresDate:02/07/18Time:20:51Sample(adjusted):19862016Includedobservations:31afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C-0.0057750.215666-0.0267780.9788LNX-0.902706*LNX(-1)0.9398970.1708675.5007560.0000R-squared0.510617Meandependentvar1.175339AdjustedR-squared0.493742S.D.dependentvar0.158003S.E.ofregression0.112422Akaikeinfocriterion-1.470776Sumsquaredresid0.366521Schwarzcriterion-1.378261Loglikelihood24.79703Hannan-Quinncriter.-1.440619F-statistic30.25831Durbin-Watsonstat1.744560Prob(F-statistic)0.000006表6广义差分-残差序列过原点回归结果DW检验：由回归结果可知DW统计量为1.744560，同时n=31，=1，在0.05的显著性水平下，=1.36，=1.50，因而模型已不存在自相关。BG检验：阶数5432AIC-1.248335-1.254886-1.318563-1.356845SIC-0.924532-0.977340-1.087275-1.171814滞后阶数从5阶减小到2阶，AIC及SIC达到最小时，滞后阶数为2阶，此时n𝑅2=0.464615，已知2.(2)=5.99，n𝑅2=0.4646155.99，同时P值为0.7927，在0.05的显著性水平下不拒绝原假设，即已消除自相关。Breusch-GodfreySerialCorrelationLMTest:F-statistic0.205411Prob.F(2,27)0.8156Obs*R-squared0.464615Prob.Chi-Square(2)0.7927TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:02/07/18Time:21:30Sample:19862016Includedobservations:31Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C0.0036930.2234790.0165250.9869LNX-0.902706*LNX(-1)-0.0030000.177227-0.0169260.9866RESID(-1)0.1211960.1944720.6232050.5384RESID(-2)-0.0393490.201437-0.1953420.8466R-squared0.014988Meandependentvar4.02E-16AdjustedR-squared-0.094458S.D.dependentvar0.110532S.E.ofregression0.115635Akaikeinfocriterion-1.356845Sumsquaredresid0.361028Schwarzcriterion-1.171814Loglikelihood25.03110Hannan-Quinncriter.-1.296530F-statistic0.136941Durbin-Watsonstat1.970873Prob(F-statistic)0.937095广义差分BG检验2阶回归结果则可知，1=−.1−.2=−0.059356最终模型为：ln𝑌̂𝑡=−0.059356+0.939897𝑙𝑛𝑋𝑡③德宾两步法求̂构建模型ln𝑌𝑡=1(1−)+2𝑙𝑛𝑋𝑡−2𝑙𝑛𝑋𝑡−1+𝑙𝑛𝑌𝑡−1+𝑡DependentVariable:LNYMethod:LeastSquaresDate:02/07/18Time:21:43Sample(adjusted):19862016Includedobservations:31afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C-0.0689790.496455-0.1389430.8905LNX1.3740570.4219163.2567030.0030LNX(-1)-1.2756850.361334-3.5304850.0015LNY(-1)0.8952960.1274917.0224420.0000R-squared0.995027Meandependentvar10.65304AdjustedR-squared0.994475S.D.dependentvar1.518884S.E.ofregression0.112903Akaikeinfocriterion-1.404662Sumsquaredresid0.344171Schwarzcriterion-1.219631Loglikelihood25.77226Hannan-Quinncriter.-1.344347F-statistic1800.834Durbin-Watsonstat1.685337Prob(F-statistic)0.000000德宾两步法回归结果由此可知，̂=0.895296，进而得广义差分方程：ln𝑌𝑡−0.895296𝑙𝑛𝑌𝑡−1=1(1−0.895296)+2(𝑙𝑛𝑋𝑡−0.895296𝑙𝑛𝑋𝑡−1)+𝑡DependentVariable:LN