计量经济学第5、8题答案

1、第8章练习5证明:对方程两边同时减去，得：然后对该式等号右边加上再减去一个，得： 将第二个方程代入，得： 其中，第8章练习8下表给出了19782007年中国货物进出口额的自然对数序列。年份LXLM年份LXLM19784.57994.690419936.82156.946619794.91715.054319947.09857.052819805.19965.299319957.30517.186019815.39415.394519967.32027.235819825.40815.262219977.51097.261019835.40405.365519987.51597.24591984

2、5.56615.613519997.57527.412819855.61136.046220007.82087.719119865.73466.061720017.88657.797919875.97746.068720028.08837.990119886.16376.314820038.38538.325519896.26426.382520048.68838.632719906.43126.279520058.93858.794719916.57806.458220069.17888.976519926.74456.692020079.40749.1653解：根据Eview软件操作得：对

3、1978-2007年中国货物进、出口额的自然对数系列LX， LM的单位根检验分别如下： 对LX的单位根检验：Null Hypothesis: LX has a unit rootExogenous: NoneLag Length: 2 (Fixed)t-StatisticProb.*Augmented Dickey-Fuller test statistic3.6608350.9998Test critical values:1% level-2.6534015% level-1.95385810% level-1.609571*MacKinnon (1996) one-sided p-val

4、ues.Augmented Dickey-Fuller Test EquationDependent Variable: D(LX)Method: Least SquaresDate: 05/28/11 Time: 12:13Sample (adjusted): 1981 2007Included observations: 27 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.LX(-1)0.0221040.0060383.6608350.0012D(LX(-1)0.0893620.1982100.4508440.6

5、561D(LX(-2)-0.0568220.179525-0.3165120.7544R-squared0.165948Mean dependent var0.155844Adjusted R-squared0.096444S.D. dependent var0.094091S.E. of regression0.089439Akaike info criterion-1.886086Sum squared resid0.191983Schwarz criterion-1.742104Log likelihood28.46216Hannan-Quinn criter.-1.843273Durb

6、in-Watson stat2.075404根据上表得Augmented Dickey-Fuller test statistics的数值为3.660835，大于5% critical values:的数值-1.953858，即3.660835-1.953858。所以说明LX检验的序列现在检验的是原序列是不平稳的，至少有一个单位根。对LM的单位根检验：Null Hypothesis: LM has a unit rootExogenous: Constant, Linear TrendLag Length: 1 (Fixed)t-StatisticProb.*Augmented Dickey-

7、Fuller test statistic-2.3914200.3756Test critical values:1% level-4.3239795% level-3.58062310% level-3.225334*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(LM)Method: Least SquaresDate: 05/28/11 Time: 13:22Sample (adjusted): 1980 2007Included observat

8、ions: 28 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.LM(-1)-0.3664560.153238-2.3914200.0250D(LM(-1)0.4131680.1912162.1607460.0409C1.7679650.7124772.4814340.0205TREND(1978)0.0524950.0208532.5174240.0189R-squared0.261758Mean dependent var0.146821Adjusted R-squared0.169478S.D. depende

9、nt var0.131779S.E. of regression0.120214Akaike info criterion-1.269523Sum squared resid0.346141Schwarz criterion-1.079208Log likelihood21.77333Hannan-Quinn criter.-1.211342F-statistic2.836560Durbin-Watson stat1.832088Prob(F-statistic)0.059412根据上表得Augmented Dickey-Fuller test statistics的数值为-2.391420，

10、大于5% critical values:的数值-3.580623，即-2.391420-3.580623。所以说明LM检验的序列现在检验的是原序列是不平稳的，至少有一个单位根。附：LX,LM的时间序列图如下：2根据Eview软件操作得：LX的单整性：Null Hypothesis: D(LX) has a unit rootExogenous: ConstantLag Length: 1 (Fixed)t-StatisticProb.*Augmented Dickey-Fuller test statistic-3.3390470.0229Test critical values:1% le

11、vel-3.6998715% level-2.97626310% level-2.627420*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test EquationDependent Variable: D(LX,2)Method: Least SquaresDate: 05/28/11 Time: 13:07Sample (adjusted): 1981 2007Included observations: 27 after adjustmentsVariableCoefficientStd. Errort-Sta

12、tisticProb.D(LX(-1)-0.8026710.240389-3.3390470.0027D(LX(-1),2)0.0066990.1923270.0348300.9725C0.1247220.0425532.9310040.0073R-squared0.410284Mean dependent var-0.001996Adjusted R-squared0.361141S.D. dependent var0.119873S.E. of regression0.095813Akaike info criterion-1.748399Sum squared resid0.220323

13、Schwarz criterion-1.604418Log likelihood26.60339Hannan-Quinn criter.-1.705586F-statistic8.348774Durbin-Watson stat1.994367Prob(F-statistic)0.001769根据上面表格得Augmented Dickey-Fuller test statistics的数值为-3.339047，小于5% critical values:的数值-2.976263，即-3.339047-2.976263，这就说明LX检验的序列现在检验的是一次差分序列是平稳的，所以LX序列为1阶单整

14、序列。LM的单整性：Null Hypothesis: D(LM) has a unit rootExogenous: ConstantLag Length: 1 (Fixed)t-StatisticProb.*Augmented Dickey-Fuller test statistic-5.0274270.0004Test critical values:1% level-3.6998715% level-2.97626310% level-2.627420*MacKinnon (1996) one-sided p-values.Augmented Dickey-Fuller Test Equ

15、ationDependent Variable: D(LM,2)Method: Least SquaresDate: 05/28/11 Time: 13:32Sample (adjusted): 1981 2007Included observations: 27 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.D(LM(-1)-1.1292410.224616-5.0274270.0000D(LM(-1),2)0.4100440.1755472.3358080.0282C0.1647250.0408554.03196

16、90.0005R-squared0.520213Mean dependent var-0.002081Adjusted R-squared0.480230S.D. dependent var0.170029S.E. of regression0.122583Akaike info criterion-1.255623Sum squared resid0.360636Schwarz criterion-1.111641Log likelihood19.95091Hannan-Quinn criter.-1.212810F-statistic13.01108Durbin-Watson stat2.

17、003569Prob(F-statistic)0.000149根据上面表格得Augmented Dickey-Fuller test statistics的数值为-5.027427，小于5% critical values:的数值-2.976263，即-5.02742712.32090和13.6245911.22480,，这说明了至少有一个协整关系存在。上述协整检验结果说明两个变量LX、LM之间存在协整关系。检验形式2：Date: 05/28/11 Time: 13:54Sample (adjusted): 1980 2007Included observations: 28 after ad

18、justmentsTrend assumption: No deterministic trend (restricted constant)Series: LX LMLags interval (in first differences): 1 to 1Unrestricted Cointegration Rank Test (Trace)HypothesizedTrace0.05No. of CE(s)EigenvalueStatisticCritical ValueProb.*None *0.47898927.4964720.261840.0042At most 1 *0.2811009

19、.2409229.1645460.0484Trace test indicates 2 cointegrating eqn(s) at the 0.05 level* denotes rejection of the hypothesis at the 0.05 level*MacKinnon-Haug-Michelis (1999) p-valuesUnrestricted Cointegration Rank Test (Maximum Eigenvalue)HypothesizedMax-Eigen0.05No. of CE(s)EigenvalueStatisticCritical V

20、alueProb.*None *0.47898918.2555515.892100.0209At most 1 *0.2811009.2409229.1645460.0484Max-eigenvalue test indicates 2 cointegrating eqn(s) at the 0.05 level* denotes rejection of the hypothesis at the 0.05 level*MacKinnon-Haug-Michelis (1999) p-valuesUnrestricted Cointegrating Coefficients (normali

21、zed by b*S11*b=I):LXLMC-6.4998927.331780-3.7731775.676638-5.9176032.649028Unrestricted Adjustment Coefficients (alpha):D(LX)0.0636550.023968D(LM)-0.0001440.0722951 Cointegrating Equation(s):Log likelihood52.24991Normalized cointegrating coefficients (standard error in parentheses)LXLMC1.000000-1.127

22、9850.580498(0.02979)(0.20010)Adjustment coefficients (standard error in parentheses)D(LX)-0.413754(0.10441)D(LM)0.000935(0.17726)根据上面表格得：27.4964720.26184和18.2555515.89210,，这说明了至少有一个协整关系存在。上述协整检验结果说明两个变量LX、LM之间存在协整关系。注：上面两种检验形式不同，但得到的结论是一样的。4根据Eview软件操作得：对其进行最小二乘法回归分析：Dependent Variable: LMMethod: Le

23、ast SquaresDate: 05/28/11 Time: 14:22Sample: 1978 2007Included observations: 30VariableCoefficientStd. Errort-StatisticProb.C0.5037730.1260813.9956390.0004LX0.9226230.01807651.041700.0000R-squared0.989367Mean dependent var6.824200Adjusted R-squared0.988987S.D. dependent var1.238187S.E. of regression

24、0.129939Akaike info criterion-1.179166Sum squared resid0.472755Schwarz criterion-1.085753Log likelihood19.68749Hannan-Quinn criter.-1.149282F-statistic2605.255Durbin-Watson stat0.966431Prob(F-statistic)0.000000对其进行参加LM滞后一期的最小二乘法回归分析：Dependent Variable: LMMethod: Least SquaresDate: 05/28/11 Time: 14:

25、25Sample (adjusted): 1979 2007Included observations: 29 after adjustmentsVariableCoefficientStd. Errort-StatisticProb.C0.3190270.1500022.1268110.0431LX0.5065740.1708162.9656210.0064LM(-1)0.4550770.1867032.4374430.0219R-squared0.990355Mean dependent var6.897779Adjusted R-squared0.989613S.D. dependent

26、 var1.191488S.E. of regression0.121434Akaike info criterion-1.281197Sum squared resid0.383401Schwarz criterion-1.139752Log likelihood21.57735Hannan-Quinn criter.-1.236898F-statistic1334.807Durbin-Watson stat1.227138Prob(F-statistic)0.000000格兰杰因果关系检验：滞后一阶：Pairwise Granger Causality TestsDate: 05/28/11 Time: 14:27Sample: 1978 2007Lags: 1Null Hypothesis:ObsF-StatisticProb.LM does not Granger Cause LX295.549100.0263LX does not Granger Cause LM0.991360.3286根据上面表格可得：LM是LX的格兰杰原因的接受率是0.0263，LX不是LM的格兰杰原因的接受率是0.3286滞后二阶：Pairwise Granger Causality TestsDate