计量经济学第八章第5、8题答案

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1978~2007年中国货物进出口额的自然对数序列。年份LXLM年份LXLM19784.57994.690419936.82156.946619794.91715.054319947.09857.052819805.19965.299319957.30517.186019815.39415.394519967.32027.235819825.40815.262219977.51097.261019835.40405.365519987.51597.245919845.56615.613519997.57527.412819855.61136.046220007.82087.719119865.73466.061720017.88657.797919875.97746.068720028.08837.990119886.16376.314820038.38538.325519896.26426.382520048.68838.632719906.43126.279520058.93858.794719916.57806.458220069.17888.976519926.74456.692020079.40749.1653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：NullHypothesis:LXhasaunitrootExogenous:NoneLagLength:2(Fixed)t-Statistic

Prob.*AugmentedDickey-Fullerteststatistic

3.660835

0.9998Testcriticalvalues:1%level-2.6534015%level-1.95385810%level-1.609571*MacKinnon(1996)one-sidedp-values.AugmentedDickey-FullerTestEquationDependentVariable:D(LX)Method:LeastSquaresDate:05/28/11Time:12:13Sample(adjusted):19812007Includedobservations:27afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.

LX(-1)0.0221040.0060383.6608350.0012D(LX(-1))0.0893620.1982100.4508440.6561D(LX(-2))-0.0568220.179525-0.3165120.7544R-squared0.165948

Meandependentvar0.155844AdjustedR-squared0.096444

S.D.dependentvar0.094091S.E.ofregression0.089439

Akaikeinfocriterion-1.886086Sumsquaredresid0.191983

Schwarzcriterion-1.742104Loglikelihood28.46216

Hannan-Quinncriter.-1.843273Durbin-Watsonstat2.075404根据上表得”AugmentedDickey-Fullerteststatistics”的数值为

3.660835，大于5%criticalvalues: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：NullHypothesis:LMhasaunitrootExogenous:Constant,LinearTrendLagLength:1(Fixed)t-Statistic

Prob.*AugmentedDickey-Fullerteststatistic-2.391420

0.3756Testcriticalvalues:1%level-4.3239795%level-3.58062310%level-3.225334*MacKinnon(1996)one-sidedp-values.AugmentedDickey-FullerTestEquationDependentVariable:D(LM)Method:LeastSquaresDate:05/28/11Time:13:22Sample(adjusted):19802007Includedobservations:28afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.

LM(-1)-0.3664560.153238-2.3914200.0250D(LM(-1))0.4131680.1912162.1607460.0409C1.7679650.7124772.4814340.0205@TREND(1978)0.0524950.0208532.5174240.0189R-squared0.261758

Meandependentvar0.146821AdjustedR-squared0.169478

S.D.dependentvar0.131779S.E.ofregression0.120094

Akaikeinfocriterion-1.269523Sumsquaredresid0.346141

Schwarzcriterion-1.079208Loglikelihood21.77333

Hannan-Quinncriter.-1.211342F-statistic2.836560

Durbin-Watsonstat1.832088Prob(F-statistic)0.059412根据上表得”AugmentedDickey-Fullerteststatistics”的数值为

-2.391420，大于5%criticalvalues: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：NullHypothesis:D(LX)hasaunitrootExogenous:ConstantLagLength:1(Fixed)t-Statistic

Prob.*AugmentedDickey-Fullerteststatistic-3.339047

0.0229Testcriticalvalues:1%level-3.6998715%level-2.97626310%level-2.627420*MacKinnon(1996)one-sidedp-values.AugmentedDickey-FullerTestEquationDependentVariable:D(LX,2)Method:LeastSquaresDate:05/28/11Time:13:07Sample(adjusted):19812007Includedobservations:27afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.

D(LX(-1))-0.8026710.240389-3.3390470.0027D(LX(-1),2)0.0066990.1923270.0348300.9725C0.1247220.0425532.9310040.0073R-squared0.410284

Meandependentvar-0.001996AdjustedR-squared0.361141

S.D.dependentvar0.119873S.E.ofregression0.095813

Akaikeinfocriterion-1.748399Sumsquaredresid0.220323

Schwarzcriterion-1.604418Loglikelihood26.60339

Hannan-Quinncriter.-1.705586F-statistic8.348774

Durbin-Watsonstat1.994367Prob(F-statistic)0.001769根据上面表格得”AugmentedDickey-Fullerteststatistics”的数值为

-3.339047，小于5%criticalvalues: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阶单整序列。LM的单整性：NullHypothesis:D(LM)hasaunitrootExogenous:ConstantLagLength:1(Fixed)t-Statistic

Prob.*AugmentedDickey-Fullerteststatistic-5.027427

0.0004Testcriticalvalues:1%level-3.6998715%level-2.97626310%level-2.627420*MacKinnon(1996)one-sidedp-values.AugmentedDickey-FullerTestEquationDependentVariable:D(LM,2)Method:LeastSquaresDate:05/28/11Time:13:32Sample(adjusted):19812007Includedobservations:27afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.

D(LM(-1))-1.1292410.224616-5.0274270.0000D(LM(-1),2)0.4100440.1755472.3358080.0282C0.1647250.0408554.0319690.0005R-squared0.520213

Meandependentvar-0.002081AdjustedR-squared0.480230

S.D.dependentvar0.170029S.E.ofregression0.122583

Akaikeinfocriterion-1.255623Sumsquaredresid0.360636

Schwarzcriterion-1.111641Loglikelihood19.95091

Hannan-Quinncriter.-1.212810F-statistic13.01108

Durbin-Watsonstat2.003569Prob(F-statistic)0.000149根据上面表格得”AugmentedDickey-Fullerteststatistics”的数值为-5.027427，小于5%criticalvalues:的数值-2.976263，即

-5.027427<-2.976263，这就说明LM检验的序列（现在检验的是一次差分序列）是平稳的，所以LM序列为1阶单整序列。（3）.根据Eview软件操作得：检验形式1：Date:05/28/11Time:13:50Sample(adjusted):19802007Includedobservations:28afteradjustmentsTrendassumption:NodeterministictrendSeries:LXLM

Lagsinterval(infirstdifferences):1to1UnrestrictedCointegrationRankTest(Trace)HypothesizedTrace0.05No.ofCE(s)EigenvalueStatisticCriticalValueProb.**None*

0.385283

20.27831

12.32090

0.0019Atmost1*

0.211508

6.653716

4.129906

0.0117

Tracetestindicates2cointegratingeqn(s)atthe0.05level

*denotesrejectionofthehypothesisatthe0.05level

**MacKinnon-Haug-Michelis(1999)p-valuesUnrestrictedCointegrationRankTest(MaximumEigenvalue)HypothesizedMax-Eigen0.05No.ofCE(s)EigenvalueStatisticCriticalValueProb.**None*

0.385283

13.62459

11.22480

0.0186Atmost1*

0.211508

6.653716

4.129906

0.0117

Max-eigenvaluetestindicates2cointegratingeqn(s)atthe0.05level

*denotesrejectionofthehypothesisatthe0.05level

**MacKinnon-Haug-Michelis(1999)p-values

UnrestrictedCointegratingCoefficients(normalizedbyb'*S11*b=I):

LXLM

2.662861-2.984473

6.889796-6.895310

UnrestrictedAdjustmentCoefficients(alpha):

D(LX)-0.063612

0.000295D(LM)-0.036742

0.0564941CointegratingEquation(s):

Loglikelihood

49.93444Normalizedcointegratingcoefficients(standarderrorinparentheses)LXLM

1.000000-1.120777

(0.02827)Adjustmentcoefficients(standarderrorinparentheses)D(LX)-0.169391

(0.04279)D(LM)-0.097838

(0.06993)根据上面表格得：

20.27831>

12.32090和

13.62459>

11.22480,，这说明了至少有一个协整关系存在。

上述协整检验结果表明两个变量LX、LM之间存在协整关系。

检验形式2：Date:05/28/11Time:13:54Sample(adjusted):19802007Includedobservations:28afteradjustmentsTrendassumption:Nodeterministictrend(restrictedconstant)Series:LXLM

Lagsinterval(infirstdifferences):1to1UnrestrictedCointegrationRankTest(Trace)HypothesizedTrace0.05No.ofCE(s)EigenvalueStatisticCriticalValueProb.**None*

0.478989

27.49647

20.26184

0.0042Atmost1*

0.281100

9.240922

9.164546

0.0484

Tracetestindicates2cointegratingeqn(s)atthe0.05level

*denotesrejectionofthehypothesisatthe0.05level

**MacKinnon-Haug-Michelis(1999)p-valuesUnrestrictedCointegrationRankTest(MaximumEigenvalue)HypothesizedMax-Eigen0.05No.ofCE(s)EigenvalueStatisticCriticalValueProb.**None*

0.478989

18.25555

15.89210

0.0209Atmost1*

0.281100

9.240922

9.164546

0.0484

Max-eigenvaluetestindicates2cointegratingeqn(s)atthe0.05level

*denotesrejectionofthehypothesisatthe0.05level

**MacKinnon-Haug-Michelis(1999)p-values

UnrestrictedCointegratingCoefficients(normalizedbyb'*S11*b=I):

LXLMC-6.499892

7.331780-3.773177

5.676638-5.917603

2.649028

UnrestrictedAdjustmentCoefficients(alpha):

D(LX)

0.063655

0.023968D(LM)-0.000144

0.0722951CointegratingEquation(s):

Loglikelihood

52.24991Normalizedcointegratingcoefficients(standarderrorinparentheses)LXLMC

1.000000-1.127985

0.580498

(0.02979)

(0.20010)Adjustmentcoefficients(standarderrorinparentheses)D(LX)-0.413754

(0.10441)D(LM)

0.000935

(0.17726)根据上面表格得：

27.49647>

20.26184和18.25555>

15.89210,，这说明了至少有一个协整关系存在。

上述协整检验结果表明两个变量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：DependentVariable:LMMethod:LeastSquaresDate:05/28/11Time:14:22Sample:19782007Includedobservations:30VariableCoefficientStd.Errort-StatisticProb.

C0.5037730.1260813.9956390.0004LX0.9226230.01807651.041700.0000R-squared0.989367

Meandependentvar6.824200AdjustedR-squared0.988987

S.D.dependentvar1.238187S.E.ofregression0.129939

Akaikeinfocriterion-1.179166Sumsquaredresid0.472755

Schwarzcriterion-1.085753Loglikelihood19.68749

Hannan-Quinncriter.-1.149282F-statistic2605.255

Durbin-Watsonstat0.966431Prob(F-statistic)0.000000对其进行加入LM滞后一期的最小二乘法回归分析：DependentVariable:LMMethod:LeastSquaresDate:05/28/11Sample(adjusted):19792007Includedobservations:29afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.

C0.3190270.1500022.1268110.0431LX0.5065740.1708162.9656210.0064LM(-1)0.4550770.1867032.4374430.0219R-squared0.990355

Meandependentvar6.897779AdjustedR-squared0.989613

S.D.dependentvar1.191488S.E.ofregression0.121434

Akaikeinfocriterion-1.281197Sumsquaredresid0.383401

Schwarzcriterion-1.139752Loglikelihood21.57735

Hannan-Quinncriter.-1.236898F-statistic1334.807

Durbin-Watsonstat1.227138Prob(F-statistic)0.000000

格兰杰因果关系检验：滞后一阶：PairwiseGrangerCausalityTestsDate:05/28/11Sample:19782007Lags:1

NullHypothesis:ObsF-StatisticProb.

LMdoesnotGrangerCauseLX

29

5.549100.0263

LXdoesnotGrangerCauseLM

0.991360.3286根据上面表格可得：LM是L