经济学第六章习题讲解计量经济学

第六章自相关习题参考答案练习题6.1参考解答:（１）建立回归模型，回归结果如下：DependentVariable:YMethod:LeastSquaresDate:05/06/10 Time:22:58Sample:19601995Includedobservations:36CoefficientStd.Errort-StatisticProb.X 0.9358660.007467125.34110.0000C -9.4287452.504347-3.7649510.0006R-squared0.997841Meandependentvar289.9444AdjustedR-squared0.997777S.D.dependentvar95.82125S.E.ofregression4.517862Akaikeinfocriterion5.907908Sumsquaredresid693.9767Schwarzcriterion5.995881Loglikelihood-104.3423Hannan-Quinncriter.5.938613F-statistic15710.39Durbin-Watsonstat0.523428Prob(F-statistic)0.000000估计结果如下

Y9.42870.9359Xt tSe=(2.5043) (0.0075)t=(-3.7650) (125.3411)R2=0.9978，F=15710.39，df=34，DW=0.523436DWdU=1.525，模型中DW<dL，显然消费模型中有自相关。（３）采用广义差分法e0.72855et t由上式可知0.728550，对原模型进行广义差分，得到广义差分方程：Y0.72855Yt

t

(10.72855)+(X1 2

0.72855

t

)ut

0.72855

t1回归结果如下:DependentVariable:Y-0.72855*Y(-1)Method:LeastSquaresDate:05/06/10 Time:23:11Sample(adjusted):1961Includedobservations:35afteradjustmentsCoefficient Std.Error t-Statistic Prob.C-3.7830591.870964-2.0219840.0513X-0.72855*X(-1)0.9484060.01890550.168200.0000R-squared0.987058Meandependentvar86.40203AdjustedR-squared0.986666S.D.dependentvar26.56943S.E.ofregression3.068065Akaikeinfocriterion5.135417Sumsquaredresid310.6298Schwarzcriterion5.224294Loglikelihood-87.86979Hannan-Quinncriter.5.166097F-statistic2516.848Durbin-Watsonstat2.097157Prob(F-statistic)0.0000003.78310.9484Xt t(1.8710)(0.0189)t=(-2.022)(50.1682)R2=0.9871R2=0.9867 F=2516.848DW=2.0971571355%DW统计表可知dL==模型中DW=2.0972>dU，说明广义差分模型中已无自相关。同时，判定系数R2、t、F统计量均达到理想水平。由差分方程式可以得出：ˆˆ*/(ˆ)0 0

3.7831 10.72855所以最终的消费模型为：

ˆˆ1 1

0.948413.93660.9484Xt t由上述模型可知，美国个人实际可支配收入每增加1元，个人实际消费支出平均增加0.9484元。练习题6.2参考解答：12中不存在自相关。DW1中，DW=0.8252显著水平的DW统计表可知dL

1.106，dU

1.371，DWdL

，因此模型1存在正自相关；而在模型2中，DW=1.82,查5%显著水平的DW统计表可知dL

0.982，d 1.539，dU

DW4dU

，因此模型2不存在自相关。虚假自相关是由模型设定失误所造成的自相关，主要包括遗漏某些重要的解释变量或者模型函数形式不正确，因此在区分虚假自相关和真正自相关是主要从这两个方面来判断，即根据经济意义检查解释变量是否遗漏了重要的变量，或者根据数据的数字特征检验模型形式的设定是否恰当。练习题6.3参考解答：（１）先对数据进行处理，收入-消费模型（个人实际收入与个人实际消费支出）个人实际消费支出=人均生活消费支出/商品零售物价指数*100建立回归模型，回归结果如下：DependentVariable:YMethod:LeastSquaresDate:05/06/10 Time:23:20Sample:20012019Includedobservations:19CoefficientStd.Errort-StatisticProb.X 0.6904880.01287753.620680.0000C 79.9300412.399196.4463900.0000R-squared0.994122Meandependentvar700.2747AdjustedR-squared0.993776S.D.dependentvar246.4491S.E.ofregression19.44245Akaikeinfocriterion8.872095Sumsquaredresid6426.149Schwarzcriterion8.971510Loglikelihood-82.28490Hannan-Quinncriter.8.888920F-statistic2875.178Durbin-Watsonstat0.574663Prob(F-statistic)0.000000估计结果如下

79.9300.690Xt tSe(12.399)(0.013)t(6.446)(53.621)R20.994 DW0.575

(6.38)（２）DW＝0.575，对样本量为36、一个解释变量的模型、5%显著水平的DW统计表可知d 1.18,d 1.40,DW1.18L U ，说明误差项存在正自相关。（３）采用广义差分法使用普通最小二乘法估计

的估计值，得et

t1由上式可知

Se(0.178t(3.701)=0.657352，对原模型进行广义差分，得到广义差分方程：Y0.657352Yt t

(10.657352)+(X1 2

t

)ut

0.657352ut1回归结果如下：DependentVariable:Y-0.657352*Y(-1)Method:LeastSquaresDate:05/06/10 Time:23:25Sample(adjusted):2002Includedobservations:18afteradjustmentsCoefficient

Std.

t-Statistic

Prob.CX-0.657352*X(-1)

35.977610.668695

8.1035460.020642

4.43973732.39512

0.00040.0000R-squared0.984983Meandependentvar278.1002AdjustedR-squared0.984044S.D.dependentvar105.1781S.E.ofregression13.28570Akaikeinfocriterion8.115693Sumsquaredresid2824.158Schwarzcriterion8.214623Loglikelihood-71.04124Hannan-Quinncriter.8.129334F-statistic1049.444Durbin-Watsonstat1.830746Prob(F-statistic)0.000000估计结果如下

^Y*35.97761+0.668695X*^t tt(4.439737) (32.39512)R20.984983 DW=1.830746d 1.158

d DW1.834dDW=1.830，已知L U在广义差分模型中已无自相关。由差分方程式可以得出：

，模型中U

U因此，ˆˆ0 0

/(ˆ)

35.9776110.668695

（错误）ˆ ˆ0 0

/(1ˆ)

35.9776110.657352

（正确）ˆˆ*1 1

0.668695因此，修正后的回归模型应为Y108.5940.668695Xt t由上述模型可知，个人实际收入每增加1元，个人实际支出平均增加0.668695元。参考答案原题建立回归模型，回归结果如下：DependentVariable:YMethod:LeastSquaresDate:11/26/10 Time:19:47Sample:19701994Includedobservations:25CoefficientStd.Errort-StatisticProb.X 1.5297120.05097630.008460.0000C -68.1602615.26513-4.4650960.0002R-squared0.975095Meandependentvar388.0000AdjustedR-squared0.974012S.D.dependentvar43.33397S.E.ofregression6.985763Akaikeinfocriterion6.802244Sumsquaredresid1122.420Schwarzcriterion6.899754Loglikelihood-83.02805Hannan-Quinncriter.6.829289F-statistic900.5078Durbin-Watsonstat0.348288Prob(F-statistic)0.00000068.060261.529712Xt tt=(-4.46509)(30.00846)R2=0.975R2=0.974 F=900.5078DW=0.348288给定n=25，k'1,在0.05的显著水平下，查DW 统计表可知，d 1.288,dL

1.454。模型中DWdL

，所以可以判断模型中存在正自相关。对模型的修正采广义差分法修正自相关：使用普通最小二乘法估计，得e0.873772et tt6.734519由上式可知ˆ=0.873772，对原模型进行广义差分，得到广义差分方程：Y0.873772Yt t

(10.873772)+(X1 2

t

)ut

0.873772ut1回归结果如下：DependentVariable:Y-0.873772*Y(-1)Method:LeastSquaresDate:11/26/10 Time:20:04Sample(adjusted):1971Includedobservations:24afteradjustmentsCoefficient

Std.

t-Statistic

Prob.X-0.873772*X(-1)C

1.2520333.198065

0.1877947.790739

6.6670590.410496

0.00000.6854R-squared0.668922Meandependentvar54.86397AdjustedR-squared0.653873S.D.dependentvar6.671848S.E.ofregression3.925217Akaikeinfocriterion5.652375Sumsquaredresid338.9612Schwarzcriterion5.750547Loglikelihood-65.82850Hannan-Quinncriter.5.678420F-statistic44.44968Durbin-Watsonstat1.322343Prob(F-statistic)0.000001ˆˆ*3.1980651.252033X*t tt=(0.410496)(6.667059)R2=0.669R2=0.654 F=44.450DW=1.322343给定n=24，k'1,在0.05的显著水平下，查DW 统计表可知，d 1.273,dˆˆˆ*/(ˆ)1.252033/(0.873772)9.918820 0ˆˆ*3.198065

1.446。模型中dL

DWdU

，DW值落在了无法判断的区域。1 19.91882t t一阶差分法对模型进行一阶差分，回归结果如下：DependentVariable:Y-Y(-1)Method:LeastSquaresDate:11/26/10 Time:20:37Sample(adjusted):1971Includedobservations:24afteradjustmentsCoefficient

Std.

t-Statistic

Prob.X-X(-1)

1.333333

0.131422

10.14543

0.0000R-squared0.652682Meandependentvar6.208333AdjustedR-squared0.652682S.D.dependentvar6.678839S.E.ofregression3.936084Akaikeinfocriterion5.619023Sumsquaredresid356.3333Schwarzcriterion5.668109Loglikelihood-66.42828Hannan-Quinncriter.5.632046Durbin-Watsonstat1.591830给定n=24，k'1,在0.05的显著水平下，查DW 统计表可知，d 1.273,dL

1.446。模型中dU

DW4dU

，因此模型已不存在自相关。德宾两步法建立辅助回归方程Yt

(1)X1 2

2

Xt

vt

，回归结果如下：DependentVariable:YMethod:LeastSquaresDate:11/26/10 Time:20:43Sample(adjusted):1971Includedobservations:24afteradjustmentsCoefficientStd.Errort-StatisticProb.C-7.63364112.84334-0.5943660.5589X1.1726220.1885276.2199190.0000X(-1)-1.0062720.254581-3.9526660.0008Y(-1)0.8962550.1239097.2331720.0000R-squared0.992083Meandependentvar391.6667AdjustedR-squared0.990896S.D.dependentvar40.10927S.E.ofregression3.827019Akaikeinfocriterion5.673061Sumsquaredresid292.9215Schwarzcriterion5.869403Loglikelihood-64.07673Hannan-Quinncriter.5.725151F-statistic835.4552Durbin-Watsonstat1.369050Prob(F-statistic)0.000000把Yt1

的回归系数

看做的一个估计值，之后进行广义差分，回归模型为：Y0.896255Yt t

(10.896255)+(X1 2

t

)ut

0.896255ut1回归结果如下：DependentVariable:Y-0.896255*Y(-1)Method:LeastSquaresDate:11/26/10 Time:20:47Sample(adjusted):1971Includedobservations:24afteradjustmentsCoefficient

Std.

t-Statistic

Prob.X-0.896255*X(-1)C

1.2010314.652899

0.1893056.595502

6.3444250.705466

0.00000.4879R-squared0.646596Meandependentvar46.19771AdjustedR-squared0.630532S.D.dependentvar6.352384S.E.ofregression3.861224Akaikeinfocriterion5.619501Sumsquaredresid327.9990Schwarzcriterion5.717672Loglikelihood-65.43401Hannan-Quinncriter.5.645545F-statistic40.25173Durbin-Watsonstat1.305817Prob(F-statistic)0.000002给定n=24，k'1,在0.05的显著水平下，查DW 统计表可知，d 1.273,dL

1.446。模型中dL

DWdU

，DW值落在了无法判断的区域。XY之后建立回归模型，回归结果如下：DependentVariable:YMethod:LeastSquaresDate:12/04/10 Time:11:21Sample:19701994Includedobservations:25CoefficientStd.Errort-StatisticProb.X 0.6374370.02124230.008460.0000C 50.874548.2910586.1360730.0000R-squared0.975095Meandependentvar298.2000AdjustedR-squared0.974012S.D.dependentvar27.97320S.E.ofregression4.509491Akaikeinfocriterion5.926864Sumsquaredresid467.7167Schwarzcriterion6.024374Loglikelihood-72.08580Hannan-Quinncriter.5.953909F-statistic900.5078Durbin-Watsonstat0.352762Prob(F-statistic)0.00000050.874540.637437Xt tt=(6.1361)(30.00846)R2=0.975R2=0.974 F=900.5078DW=0.352762给定n=25，k'1,在0.05的显著水平下，查DW 统计表可知，d 1.288,dL

1.454DWdL

，所以可以判断模型中存在正自相关。对模型的修正1）采广义差分法修正自相关：使用普通最小二乘法估计，得e0.850961et tt6.682710=0.850961，对原模型进行广义差分，得到广义差分方程：Y0.850961Yt t

(10.850961)+(X1 2

t

)ut

0.850961ut1回归结果如下：DependentVariable:Y-0.850961*Y(-1)Method:LeastSquaresDate:12/04/10 Time:11:17Sample(adjusted):1971Includedobservations:24afteradjustmentsCoefficient

Std.

t-Statistic

Prob.X-0.850961*X(-1)C

0.53512513.97334

0.0747934.789436

7.1547962.917533

0.00000.0080R-squared0.699417Meandependentvar48.03762AdjustedR-squared0.685754S.D.dependentvar4.550930S.E.ofregression2.551144Akaikeinfocriterion4.790616Sumsquaredresid143.1833Schwarzcriterion4.888787Loglikelihood-55.48739Hannan-Quinncriter.4.816661F-statistic51.19110Durbin-Watsonstat2.377660Prob(F-statistic)0.000000ˆ*13.973340.535125X*t tt=(2.91753)(7.154796)R2=0.699R2=0.685 F=51.191DW=2.37766给定n=24，k'1,在0.05的显著水平下，查DW 统计表可知，d 1.273,dˆˆˆ*/(ˆ)13.97334/(0.850961)93.7562650 0ˆˆ*0.535125

1.446。模型中dU

DW4dU

，因此可以判断模型不存在自相关。1 193.756256t t参考解答：DependentVariable:LOG(Y)Method:LeastSquaresDate:05/07/10 Time:00:17Sample:19802000Includedobservations:21CoefficientStd.Errort-StatisticProb.C 2.1710410.2410259.0075290.0000LOG(X) 0.9510900.03889724.451230.0000R-squared0.969199Meandependentvar8.039307AdjustedR-squared0.967578S.D.dependentvar0.565486S.E.ofregression0.101822Akaikeinfocriterion-1.640785Sumsquaredresid0.196987Schwarzcriterion-1.541307Loglikelihood19.22825Hannan-Quinncriter.-1.619196F-statistic597.8626Durbin-Watsonstat1.159788Prob(F-statistic)0.000000ln2.1710410.95109lnXit=(9.007529)(30.00846)R2=0.969199R2=0.967578iF=597.8626DW=1.159788给定n=21，k1,在0.05的显著水平下ln2.1710410.95109lnXit=(9.007529)(30.00846)R2=0.969199R2=0.967578iF=597.8626DW=1.159788d 1.221 dL

DW1.159788

L，所以可以判断模型中存在正自相关。采用广义差分法修正自相关： 使用普通最小二乘法估计

的估计值et

，得0.400234et1由上式可知

t1.722522=0.400234，对原模型进行广义差分，得到广义差分方程：lnYt

0.400234lnYt1

(10.400234)+1

(lnXt

t

)ut

0.400234ut1回归结果如下：DependentVariable:LOG(Y)-0.400234*LOG(Y(-1))Method:LeastSquaresDate:05/07/10 Time:00:21Sample(adjusted):1981Includedobservations:20afteradjustmentsCoefficient

Std.Error

t-Statistic

Prob.CLOG(X)-0.400234*LOG(X(-1))

1.4770950.905989

0.2256360.059767

6.54637215.15871

0.00000.0000R-squared0.927357Meandependentvar4.882162AdjustedR-squared0.923321S.D.dependentvar0.344052S.E.ofregression0.095271Akaikeinfocriterion-1.769534Sumsquaredresid0.163380Schwarzcriterion-1.669961Loglikelihood19.69534Hannan-Quinncriter.-1.750096F-statistic