练习习题及参考解答(第四版)

1、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）19852066.719098.9200142183.62110863.119862580.410376.2200251378.15121717.419873084.212174.6200370483.45137422.019883821.815180.4200495539.09161840.219894155.917179.720

2、05116921.77187318.919905560.1218872.92006140974.74219438.519917225.7522005.62007166924.07270232.319929119.6227194.52008179921.47319515.5199311271.0235673.22009150648.06349081.4199420381.948637.52010201722.34413030.3199523499.9461339.92011236401.95489300.6199624133.8671813.62012244160.21540367.419971

3、9981999200026967.2426849.6829896.2339273.2579715.085195.590564.4100280.12013201420152016258168.89264241.77245502.93243386.46595244.4643974.0689052.1740598.7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lnYt=-2.714792+1.152178lnXt 0.316996

4、(0.027331) t=-8.5641 42.15675 R2=0.9834 F=1777.192 DW=0.3069自相关检验图示法 图1、2 et-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。DW检验由回归结果可知DW统计量为0.3069，同时n=32，k=1，在0.05的显著性水平下，dL=1.37，dU=1.50，因而模型中存在正相关。BG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，A

5、IC及SIC达到最小时，滞后阶数为2阶，此时nR2=22.57582，已知20.052=5.99，nR2=22.56454>5.99，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2 BG检验2阶回归结果自相关补救DW反算法求由DW=0.3069，可知=1-DW2=1-0.30692=0.84655，可得广义差分方程：lnYt-0.84655lnYt-1=11-0.84655+2lnXt-0.84655lnXt-1+ut 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，k=1，在0.05的显著性水平下，dL=1

6、.36，dU=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20.052=5.99，nR2=0.945667<5.99，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=-0.1531541-0.84655=-0.998071最终模型为：lnYt=-0.99

7、8071+1.009608lnXt残差过原点回归求Dependent Variable: EMethod: Least SquaresDate: 02/07/18 Time: 20:48Sample (adjusted): 1986 2016Included observations: 31 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  E(-1)0.9027060.1089908.2824420.0000R-squared0.695552    Mea

8、n dependent var0.004999Adjusted R-squared0.695552    S.D. dependent var0.206153S.E. of regression0.113749    Akaike info criterion-1.477927Sum squared resid0.388162    Schwarz criterion-1.431670Log likelihood23.90787    

9、Hannan-Quinn criter.-1.462848Durbin-Watson stat1.579983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=11-0.902706+2lnXt-0.902706lnXt-1+utDependent Variable: LNY-0.902706*LNY(-1)Method: Least SquaresDate: 02/07/18 Time: 20:51Sample (adjusted): 1986 2016Included observat

10、ions: 31 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  C-0.0057750.215666-0.0267780.9788LNX-0.902706*LNX(-1)0.9398970.1708675.5007560.0000R-squared0.510617    Mean dependent var1.175339Adjusted R-squared0.493742    S.D. dependent va

11、r0.158003S.E. of regression0.112422    Akaike info criterion-1.470776Sum squared resid0.366521    Schwarz criterion-1.378261Log likelihood24.79703    Hannan-Quinn criter.-1.440619F-statistic30.25831    Durbin-Watson stat

12、1.744560Prob(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，k=1，在0.05的显著性水平下，dL=1.36，dU=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20.052=5.99，nR2=0.464615<5

13、.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-Godfrey Serial Correlation LM Test:F-statistic0.205411    Prob. F(2,27)0.8156Obs*R-squared0.464615    Prob. Chi-Square(2)0.7927Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 02/07/18 Time: 2

14、1:30Sample: 1986 2016Included observations: 31Presample missing value lagged residuals set to zero.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)-0.0393490.20

15、1437-0.1953420.8466R-squared0.014988    Mean dependent var4.02E-16Adjusted R-squared-0.094458    S.D. dependent var0.110532S.E. of regression0.115635    Akaike info criterion-1.356845Sum squared resid0.361028    Schwarz

16、criterion-1.171814Log likelihood25.03110    Hannan-Quinn criter.-1.296530F-statistic0.136941    Durbin-Watson stat1.970873Prob(F-statistic)0.937095 广义差分BG检验2阶回归结果则可知，1=-0.0057751-0.902706=-0.059356最终模型为：lnYt=-0.059356+0.939897lnXt德宾两步法求构建模型 lnYt=11-+2lnXt-2lnX

17、t-1+lnYt-1+utDependent Variable: LNYMethod: Least SquaresDate: 02/07/18 Time: 21:43Sample (adjusted): 1986 2016Included observations: 31 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  C-0.0689790.496455-0.1389430.8905LNX1.3740570.4219163.2567030.0030LNX(-1)-1.2756850.36133

18、4-3.5304850.0015LNY(-1)0.8952960.1274917.0224420.0000R-squared0.995027    Mean dependent var10.65304Adjusted R-squared0.994475    S.D. dependent var1.518884S.E. of regression0.112903    Akaike info criterion-1.404662Sum squared resid0.34417

19、1    Schwarz criterion-1.219631Log likelihood25.77226    Hannan-Quinn criter.-1.344347F-statistic1800.834    Durbin-Watson stat1.685337Prob(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=11-0.895296+2lnXt-0.89529

20、6lnXt-1+utDependent Variable: LNY-0.895296*LNY(-1)Method: Least SquaresDate: 02/07/18 Time: 22:03Sample (adjusted): 1986 2016Included observations: 31 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  C-0.0219770.214312-0.1025490.9190LNX-0.895296*LNX(-1)0.9504930.1590425.9763

21、580.0000R-squared0.551894    Mean dependent var1.253138Adjusted R-squared0.536442    S.D. dependent var0.164971S.E. of regression0.112320    Akaike info criterion-1.472580Sum squared resid0.365861    Schwarz criterion-1.

22、380065Log likelihood24.82500    Hannan-Quinn criter.-1.442423F-statistic35.71686    Durbin-Watson stat1.731160Prob(F-statistic)0.000002 广义差分-德宾两步法回归结果DW检验：由回归结果可知DW统计量为1.731160，同时n=31，k=1，在0.05的显著性水平下，dL=1.36，dU=1.50，因而模型已不存在自相关。BG检验：阶数5432AIC-1.251149-1.25857

23、9-1.322263-1.359935SIC-0.927345-0.981033-1.090975-1.174904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=0.503846，已知20.052=5.99，nR2=0.503846<5.99，同时P值为0.7773，在0.05的显著性水平下不拒绝原假设，即已消除自相关。Breusch-Godfrey Serial Correlation LM Test:F-statistic0.223042    Prob. F(2,27)0.8015Obs*R-squared0

24、.503846    Prob. Chi-Square(2)0.7773Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 02/07/18 Time: 22:16Sample: 1986 2016Included observations: 31Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C0.004

25、6100.2220210.0207620.9836LNX-0.895296*LNX(-1)-0.0035380.164928-0.0214500.9830RESID(-1)0.1278020.1945140.6570330.5167RESID(-2)-0.0346800.201586-0.1720380.8647R-squared0.016253    Mean dependent var-1.88E-16Adjusted R-squared-0.093052    S.D. dependent var0.1104

26、33S.E. of regression0.115456    Akaike info criterion-1.359935Sum squared resid0.359914    Schwarz criterion-1.174904Log likelihood25.07899    Hannan-Quinn criter.-1.299620F-statistic0.148695    Durbin-Watson stat1.97256

27、0Prob(F-statistic)0.929624 广义差分BG检验2阶回归结果则可知，1=-0.0219771-0.895296=-0.209896最终模型为：Yt=-0.209896+0.950493Xt科克兰·奥科特迭代法Dependent Variable: LNYMethod: Least SquaresDate: 02/07/18 Time: 22:38Sample (adjusted): 1986 2016Included observations: 31 after adjustmentsConvergence achieved after 16 iteration

28、sVariableCoefficientStd. Errort-StatisticProb.  C-0.4817443.749680-0.1284760.8987LNX0.9702560.2861873.3902890.0021AR(1)0.8807660.1284896.8548010.0000R-squared0.994721    Mean dependent var10.65304Adjusted R-squared0.994344    S.D. dependent var1.5188

29、84S.E. of regression0.114233    Akaike info criterion-1.409380Sum squared resid0.365380    Schwarz criterion-1.270607Log likelihood24.84538    Hannan-Quinn criter.-1.364143F-statistic2637.882    Durbin-Watson stat1.70195

30、3Prob(F-statistic)0.000000Inverted AR Roots      .88 科克兰·奥科特迭代法回归结果DW检验：由回归结果可知DW统计量为1.701953，同时n=31，k=1，在0.05的显著性水平下，dL=1.36，dU=1.50，因而模型已不存在自相关。BG检验：阶数5432AIC-1.196683-1.227634-1.291215-1.308816SIC-0.826622-0.903830-1.013670-1.077528滞后阶数从5阶减小到2阶，AIC及SIC达到最小时，滞后阶数

31、为2阶，此时nR2=0.870091，已知20.052=5.99，nR2=0.870091<5.99，同时P值为0.6472，在0.05的显著性水平下不拒绝原假设，即已消除自相关。Breusch-Godfrey Serial Correlation LM Test:F-statistic0.375414    Prob. F(2,26)0.6907Obs*R-squared0.870091    Prob. Chi-Square(2)0.6472Test Equation:Dependent Variable

32、: RESIDMethod: Least SquaresDate: 02/07/18 Time: 22:42Sample: 1986 2016Included observations: 31Presample missing value lagged residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C-2.0865514.831163-0.4318940.6694LNX0.1565470.3660070.4277150.6724AR(1)-0.0877980.182587-0.480

33、8570.6346RESID(-1)0.2100670.2430450.8643130.3953RESID(-2)0.0317620.2362270.1344580.8941R-squared0.028067    Mean dependent var2.13E-08Adjusted R-squared-0.121461    S.D. dependent var0.110360S.E. of regression0.116870    Akaike info criteri

34、on-1.308816Sum squared resid0.355125    Schwarz criterion-1.077528Log likelihood25.28665    Hannan-Quinn criter.-1.233422F-statistic0.187707    Durbin-Watson stat1.921934Prob(F-statistic)0.942677 BG检验2阶回归结果最终模型为：lnYt=-0.481744+0.970256lnXt6

35、.2表6.6是中国1985-2016年国家财政一般公共预算收入、各项税收、经济活动人口（劳动力）以及国民总收入的数据。 表6.6 中国财政收入等数据年份一般公共预算收入（亿元）Y各项税收合计（亿元）X2经济活动人口（万人）X3国民总收入（亿元）X419852004.822040.79501129123.6019862122.012090.735154610375.4019872199.352140.365306012166.6019882357.242390.475463015174.4019892664.902727.405570717188.4019902937.102821.866532

36、318923.3019913149.482990.176609122050.3019923483.373296.916678227208.2019934348.954255.306746835599.2019945218.105126.886813548548.2019956242.206038.046885560356.6019967407.996909.826976570779.6019978651.148234.047080078802.9019989875.959262.807208783817.60199911444.0810682.587279189366.50200013395.

37、2312581.517399299066.10200116386.0415301.3873,884109276.20200218903.6417636.4574492120480.40200321715.2520017.3174911136576.30200426396.4724165.6875290161415.40200531649.2928778.5476120185998.90200638760.2034804.3576315219028.50200751321.7845621.9776531270844.00200861330.3554223.7977046321500.502009

38、68518.3059521.5977510348498.50201083101.5173210.7978388411265.202011103874.4389738.3978579484753.202012117253.52100614.2878894539116.52013129209.64110530.7079300590422.42014140370.03119175.3179690644791.12015152269.23124922.2080091686449.62016159604.97130360.7380694741140.4资料来源：中国统计年鉴2017（1）建立国家财政一般

39、公共预算收入与各项税收、经济活动人口及国民总收入的回归方程。（2）检测模型是否存在自相关性，并修正模型。【练习题6.2参考解答】回归结果Yt=12815.73+0.802396X2t-0.240367X3t+0080063X4t 3446.508 0.137414 0.057824 (0.026809) t=3.718469 5.839260 -4.156877 2.986379 R2=0.999309 F=13497.65 DW=0.572280自相关检验图示法图1、2 et-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。DW检验由回归结果可知D

40、W统计量为0.572280，同时n=32，k=3，在0.05的显著性水平下，dL=1.24，dU=1.65，因而模型中存在正相关。BG检验阶数5432AIC16.7978316.9050316.8425616.78012SIC17.2100717.2714617.1631917.05494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=17.38280，已知20.052=5.99，nR2=17.38280>5.99，同时P值为0.0002，在0.05的显著性水平下拒绝原假设，即存在自相关。Breusch-Godfrey Serial Correlation

41、 LM Test:F-statistic15.45963    Prob. F(2,26)0.0000Obs*R-squared17.38280    Prob. Chi-Square(2)0.0002Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 02/08/18 Time: 01:19Sample: 1985 2016Included observations: 32Presample missing value lagged

42、residuals set to zero.VariableCoefficientStd. Errort-StatisticProb.  C-909.86962473.584-0.3678350.7160X20.1291290.0992501.3010510.2047X30.0179380.0416710.4304720.6704X4-0.0231710.019330-1.1987280.2414RESID(-1)0.6647000.1944813.4178130.0021RESID(-2)0.2841830.2679281.0606690.2986R-squared0.5

43、43213    Mean dependent var-4.46E-12Adjusted R-squared0.455369    S.D. dependent var1327.778S.E. of regression979.8889    Akaike info criterion16.78012Sum squared resid    Schwarz criterion17.05494Log likelihood-262.4819

44、    Hannan-Quinn criter.16.87121F-statistic6.183852    Durbin-Watson stat1.984539Prob(F-statistic)0.000667表2 BG检验2阶回归结果自相关补救DW反算法求由DW=0.306575，可知=1-DW2=1-0.5722802=0.71386，可得广义差分方程：Yt-0.71386Yt-1=11-0.71386+2X2t-0.71386X2t-1+3X3t-0.71386X3t-1+4X4t-0.71386X4t-1

45、+utDependent Variable: Y-0.71386*Y(-1)Method: Least SquaresDate: 02/08/18 Time: 01:41Sample (adjusted): 1986 2016Included observations: 31 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.  C1937.5992008.7540.9645770.3433X2-0.71386*X2(-1)0.8675340.1399266.1999450.0000X3-0.7138

46、6*X3(-1)-0.1487350.102036-1.4576680.1565X4-0.71386*X4(-1)0.0674100.0264442.5492040.0168R-squared0.997291    Mean dependent var15685.47Adjusted R-squared0.996990    S.D. dependent var17950.24S.E. of regression984.8333    Akaike info criterio

47、n16.74274Sum squared resid    Schwarz criterion16.92777Log likelihood-255.5124    Hannan-Quinn criter.16.80305F-statistic3313.118    Durbin-Watson stat1.897337Prob(F-statistic)0.000000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1.897337，同时n=31，k=3，在0.

48、05的显著性水平下，dL=1.23，dU=1.65，即已消除自相关。BG检验：阶数5432AIC16.4996916.6846516.7902816.78215SIC16.9160117.0547117.1140917.05969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5阶，此时nR2=13.39193，已知20.015=15.09，nR2=13.39193<15.09，同时P值为0.0200，在0.01的显著性水平下不拒绝原假设，即已消除自相关。Breusch-Godfrey Serial Correlation LM Test:F-statistic3.3464

49、50    Prob. F(5,22)0.0213Obs*R-squared13.39193    Prob. Chi-Square(5)0.0200Test Equation:Dependent Variable: RESIDMethod: Least SquaresDate: 02/08/18 Time: 01:47Sample: 1986 2016Included observations: 31Presample missing value lagged residuals set to zero.Vari

50、ableCoefficientStd. Errort-StatisticProb.  C-501.36691740.841-0.2880030.7760X2-0.71386*X2(-1)-0.0182870.168376-0.1086100.9145X3-0.71386*X3(-1)0.0203630.0888370.2292230.8208X4-0.71386*X4(-1)0.0049530.0303320.1633090.8718RESID(-1)0.0488990.1934300.2527990.8028RESID(-2)0.3860850.2998871.28743

51、40.2113RESID(-3)0.6204360.3089392.0082810.0570RESID(-4)0.4576780.3575401.2800750.2139RESID(-5)-0.9277830.371591-2.4967880.0205R-squared0.431998    Mean dependent var9.09E-13Adjusted R-squared0.225452    S.D. dependent var934.2949S.E. of regression822.2583 

52、;   Akaike info criterion16.49969Sum squared resid    Schwarz criterion16.91601Log likelihood-246.7451    Hannan-Quinn criter.16.63540F-statistic2.091531    Durbin-Watson stat1.811014Prob(F-statistic)0.081653表4 广义差分BG检验2阶回归结果则可知，1=1937.5991-0.71386=6771.506955最终模型为：Yt=6771.506955+0.867534X2t-0.148735X3t+0.067410X4t残差过原点回归求Dependent Variable: EMethod: Least SquaresDate: 02/08/18 Time: 03:49Sample (adjusted): 1986 2016Included observations: 31 after