计量经济学庞皓第二版第六章习题答案

练习题6.1参考解答:

（１）建立回归模型，回归结果如下：

Dependent Variable: Y Method: Least Squares Date: 05/06/10 Time: 22:58 Sample: 1960 1995 Included observations: 36

X C

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 0.935866 -9.428745

Std. Error 0.007467 2.504347

t-Statistic 125.3411 -3.7951

Prob. 0.0000 0.0006 2.9444 95.82125 5.907908 5.995881 5.938613 0.523428

0.997841 Mean dependent var 0.997777 S.D. dependent var 4.517862 Akaike info criterion 693.9767 Schwarz criterion -104.3423 Hannan-Quinn criter. 15710.39 Durbin-Watson stat 0.000000

估计结果如下

Se = (2.5043) (0.0075) t = (-3.7650) (125.3411)

R2 = 0.9978，F = 15710.39，d f = 34，DW = 0.5234

（２）对样本量为36、一个解释变量的模型、5%显著水平，查DW统计表可知，dL=1.411，dU= 1.525，模型中DWˆ9.42870.9359YXt tet0.72855et1

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：

Yt0.72855Yt11(10.72855)+2(Xt0.72855Xt1)ut0.72855ut1回归结果如下:

Dependent Variable: Y-0.72855*Y(-1) Method: Least Squares Date: 05/06/10 Time: 23:11 Sample (adjusted): 1961 1995

Included observations: 35 after adjustments

Coefficient

Std. Error

t-Statistic

Prob.

C

X-0.72855*X(-1) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

-3.783059 0.948406

1.8709 0.0105

-2.021984 50.16820

0.0513 0.0000 86.40203 26.56943 5.1317 5.224294 5.166097 2.097157

0.987058 Mean dependent var 0.986666 S.D. dependent var 3.068065 Akaike info criterion 310.6298 Schwarz criterion -87.86979 Hannan-Quinn criter. 2516.848 Durbin-Watson stat 0.000000

ˆ3.78310.9484XYtt (1.8710) (0.01) t= (-2.022) (50.1682) 查5%显著水平的DW统计表可知dL = 1.402，dU = 1.519，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：

R2=0.9871 R2=0.9867 F=2516.848 DW=2.097157

ˆˆ*/(1ˆ)00所以最终的消费模型为：

3.783113.936610.72855 *ˆˆ0.948411

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。

ˆ13.93660.9484XYtt

练习题6.2参考解答：

（1） 模型1中存在自相关，模型2中不存在自相关。

（2） 通过DW检验可以判定自相关的存在；在模型1中，DW=0.8252，查5%显著水平

的DW统计表可知dL1.106，dU1.371，DWdL，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，

dU1.539，dUDW4dU，因此模型2不存在自相关。

（3） 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）建立回归模型，回归结果如下：

Dependent Variable: Y Method: Least Squares

Date: 05/06/10 Time: 23:20 Sample: 2001 2019 Included observations: 19

X C

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 0.690488 79.93004

Std. Error 0.012877 12.39919

t-Statistic 53.62068 6.446390

Prob. 0.0000 0.0000 700.2747 246.4491 8.872095 8.971510 8.8820 0.574663

0.994122 Mean dependent var 0.993776 S.D. dependent var 19.44245 Akaike info criterion 26.149 Schwarz criterion -82.28490 Hannan-Quinn criter. 2875.178 Durbin-Watson stat 0.000000

估计结果如下

ˆ79.9300.690XYttSe(12.399)(0.013)t(6.446)(53.621)R20.994DW0.575(6.38)d1.18,dU1.40,DW1.18，说明

（２）DW＝0.575，取5%，查DW上下界L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

ˆet0.657et1Se(0.178)t(3.701)由上式可知=0.657352，对原模型进行广义差分，得到广义差分方程：

ˆ

Yt0.657352Yt11(10.657352)+2(Xt0.657352Xt1)ut0.657352ut1回归结果如下：

Dependent Variable: Y-0.657352*Y(-1) Method: Least Squares Date: 05/06/10 Time: 23:25 Sample (adjusted): 2002 2019

Included observations: 18 after adjustments

C

Coefficient 35.97761

Std. Error 8.1036

t-Statistic 4.439737

Prob. 0.0004

X-0.657352*X(-1) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

0.668695 0.0202 32.39512 0.0000 278.1002 105.1781 8.115693 8.214623 8.129334 1.830746

0.984983 Mean dependent var 0.984044 S.D. dependent var 13.28570 Akaike info criterion 2824.158 Schwarz criterion -71.04124 Hannan-Quinn criter. 1049.444 Durbin-Watson stat 0.000000

估计结果如下

Y35.977610.668695X*tt(4.439737) (32.39512)R20.984983 DW=1.830746DW=1.830，已知L在广义差分模型中已无自相关。 由差分方程式可以得出：

^*td1.158 dU1.391，模型中dUDW1.834dU因此，

35.97761*ˆˆˆ)00/(1108.594 10.668695

ˆˆ*0.668695 11因此，修正后的回归模型应为

Yt108.5940.668695Xt

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。

6.4参

1．原题

（1）建立回归模型，回归结果如下：

Dependent Variable: Y Method: Least Squares Date: 11/26/10 Time: 19:47 Sample: 1970 1994 Included observations: 25

X C

R-squared

Coefficient 1.529712 -68.16026

Std. Error 0.050976 15.26513

t-Statistic 30.00846 -4.465096

Prob. 0.0000 0.0002 388.0000

0.975095 Mean dependent var

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

0.974012 S.D. dependent var 6.985763 Akaike info criterion 1122.420 Schwarz criterion -83.02805 Hannan-Quinn criter. 900.5078 Durbin-Watson stat 0.000000

43.33397 6.802244 6.97 6.8292 0.348288

ˆ68.060261.529712XYtt t= (-4.46509) (30.00846) R2=0.975 R2=0.974 F=900.5078 DW=0.348288' 给定n=25，k1,在0.05的显著水平下，查DW统计表可知，dL1.288,dU1.4。模型中DWdL，所以可以判断模型中存在正自相关。

（2）对模型的修正

1）采广义差分法修正自相关：

ˆ，得 使用普通最小二乘法估计的估计值et0.873772et1 t6.734519

ˆ=0.873772，对原模型进行广义差分，得到广义差分方程： 由上式可知Yt0.873772Yt11(10.873772)+2(Xt0.873772Xt1)ut0.873772ut1回归结果如下：

Dependent Variable: Y-0.873772*Y(-1) Method: Least Squares Date: 11/26/10 Time: 20:04 Sample (adjusted): 1971 1994

Included observations: 24 after adjustments

X-0.873772*X(-1)

C

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 1.252033 3.198065

Std. Error 0.187794 7.790739

t-Statistic 6.667059 0.410496

Prob. 0.0000 0.68 .86397 6.671848 5.652375 5.7507 5.678420 1.322343

0.6622 Mean dependent var 0.653873 S.D. dependent var 3.925217 Akaike info criterion 338.9612 Schwarz criterion -65.82850 Hannan-Quinn criter. 44.44968 Durbin-Watson stat 0.000001

ˆ*3.1980651.252033X*Ytt t= (0.410496) (6.667059) R2=0.669 R2=0.6 F=44.450 DW=1.322343'给定n=24，k1,在0.05的显著水平下，查DW统计表可知，

dL1.273,dU1.446。模型中dLDWdU，DW值落在了无法判断的区域。

ˆˆ*/(1ˆ)1.252033/(10.873772)9.91882 00ˆˆ*3.198065 11所以修正后的模型为：

ˆ9.918823.198065X

Ytt2）一阶差分法

对模型进行一阶差分，回归结果如下：

Dependent Variable: Y-Y(-1) Method: Least Squares Date: 11/26/10 Time: 20:37 Sample (adjusted): 1971 1994

Included observations: 24 after adjustments

X-X(-1)

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

'

Std. Error 0.131422

t-Statistic 10.143

Prob. 0.0000 6.208333 6.678839 5.619023 5.668109 5.632046

Coefficient 1.333333

0.652682 Mean dependent var 0.652682 S.D. dependent var 3.936084 Akaike info criterion 356.3333 Schwarz criterion -66.42828 Hannan-Quinn criter. 1.591830

给定n=24，k1,在0.05的显著水平下，查DW统计表可知，dL1.273,dU1.446。模型中dUDW4dU，因此模型已不存在自相关。 3）德宾两步法

建立辅助回归方程Yt1(1)2Xt2Xt1Yt1vt，回归结果如下：

Dependent Variable: Y Method: Least Squares Date: 11/26/10 Time: 20:43 Sample (adjusted): 1971 1994

Included observations: 24 after adjustments

C X X(-1) Y(-1)

R-squared Adjusted R-squared

Coefficient -7.6331 1.172622 -1.006272 0.6255

Std. Error 12.84334 0.188527 0.2581 0.123909

t-Statistic -0.594366 6.219919 -3.952666 7.233172

Prob. 0.55 0.0000 0.0008 0.0000 391.6667 40.10927

0.992083 Mean dependent var 0.9906 S.D. dependent var

S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

3.827019 Akaike info criterion 292.9215 Schwarz criterion -.07673 Hannan-Quinn criter. 835.4552 Durbin-Watson stat 0.000000

5.673061 5.869403 5.725151 1.369050

ˆ看做的一个估计值，之后进行广义差分，回归模型为： 把Yt1的回归系数Yt0.6255Yt11(10.6255)+2(Xt0.6255Xt1)ut0.6255ut1回归结果如下：

Dependent Variable: Y-0.6255*Y(-1) Method: Least Squares Date: 11/26/10 Time: 20:47 Sample (adjusted): 1971 1994

Included observations: 24 after adjustments

X-0.6255*X(-1)

C

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

'

Std. Error 0.1305 6.595502

t-Statistic 6.344425 0.7066

Prob. 0.0000 0.4879 46.19771 6.352384 5.619501 5.717672 5.55 1.305817

Coefficient 1.201031 4.6529

0.6596 Mean dependent var 0.630532 S.D. dependent var 3.861224 Akaike info criterion 327.9990 Schwarz criterion -65.43401 Hannan-Quinn criter. 40.25173 Durbin-Watson stat 0.000002

给定n=24，k1,在0.05的显著水平下，查DW统计表可知，dL1.273,dU1.446。模型中dLDWdU，DW值落在了无法判断的区域。

2．调换X和Y之后

（1）建立回归模型，回归结果如下：

Dependent Variable: Y Method: Least Squares Date: 12/04/10 Time: 11:21 Sample: 1970 1994 Included observations: 25

X C

Coefficient 0.637437 50.874

Std. Error 0.021242 8.291058

t-Statistic 30.00846 6.136073

Prob. 0.0000 0.0000

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

0.975095 Mean dependent var 0.974012 S.D. dependent var 4.509491 Akaike info criterion 467.7167 Schwarz criterion -72.08580 Hannan-Quinn criter. 900.5078 Durbin-Watson stat 0.000000

298.2000 27.97320 5.9268 6.024374 5.953909 0.352762

ˆ50.8740.637437XYtt t= (6.1361) (30.00846) R2=0.975 R2=0.974 F=900.5078 DW=0.352762' 给定n=25，k1,在0.05的显著水平下，查DW统计表可知，dL1.288,dU1.4。模型中DWdL，所以可以判断模型中存在正自相关。

（2）对模型的修正

1）采广义差分法修正自相关：

ˆ，得 使用普通最小二乘法估计的估计值et0.850961et1 t6.682710

ˆ=0.850961，对原模型进行广义差分，得到广义差分方程： 由上式可知Yt0.850961Yt11(10.850961)+2(Xt0.850961Xt1)ut0.850961ut1回归结果如下：

Dependent Variable: Y-0.850961*Y(-1) Method: Least Squares Date: 12/04/10 Time: 11:17 Sample (adjusted): 1971 1994

Included observations: 24 after adjustments

X-0.850961*X(-1)

C

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 0.535125 13.97334

Std. Error 0.074793 4.7436

t-Statistic 7.1796 2.917533

Prob. 0.0000 0.0080 48.03762 4.550930 4.790616 4.888787 4.816661 2.377660

0.699417 Mean dependent var 0.6857 S.D. dependent var 2.551144 Akaike info criterion 143.1833 Schwarz criterion -55.48739 Hannan-Quinn criter. 51.19110 Durbin-Watson stat 0.000000

ˆ*13.973340.535125X*Ytt t= (2.91753) (7.1796) R2=0.699 R2=0.685 F=51.191 DW=2.37766'给定n=24，k1,在0.05的显著水平下，查DW统计表可知，dL1.273,dU1.446。模型中dUDW4dU，因此可以判断模型不存在自相关。

ˆˆ*/(1ˆ)13.97334/(10.850961)93.756265 00ˆˆ*0.535125 11所以修正后的模型为：

ˆ93.7562560.535125X

Ytt

6.5参考解答：

（1）建立回归模型，回归结果如下：

Dependent Variable: LOG(Y) Method: Least Squares Date: 05/07/10 Time: 00:17 Sample: 1980 2000 Included observations: 21

C LOG(X)

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 2.171041 0.951090

Std. Error 0.241025 0.0387

t-Statistic 9.007529 24.45123

Prob. 0.0000 0.0000 8.039307 0.5686 -1.0785 -1.1307 -1.619196 1.159788

0.969199 Mean dependent var 0.967578 S.D. dependent var 0.101822 Akaike info criterion 0.196987 Schwarz criterion 19.22825 Hannan-Quinn criter. 597.8626 Durbin-Watson stat 0.000000

ˆ2.1710410.95109lnXlnYii t= (9.007529) (30.00846) 给定n=21，k1,在0.05的显著水平下，查DW统计表可知，

R2=0.969199 R2=0.967578 F=597.8626 DW=1.159788 dL1.221 dU1.42。DW1.159788dL，

模型中所以可以判断模型中存在正自相关。

（2）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，得

ˆet0.400234et1t1.722522

由上式可知=0.400234，对原模型进行广义差分，得到广义差分方程：

ˆlnYt0.400234lnYt11(10.400234)+2(lnXt0.400234lnXt1)ut0.400234ut1回归结果如下：

Dependent Variable: LOG(Y)-0.400234*LOG(Y(-1)) Method: Least Squares Date: 05/07/10 Time: 00:21 Sample (adjusted): 1981 2000

Included observations: 20 after adjustments

C

LOG(X)-0.400234*LOG(X(-1)) R-squared

Coefficient 1.477095 0.9059

Std. Error 0.225636 0.059767

t-Statistic 6.6372 15.15871

Prob. 0.0000 0.0000 4.882162 0.344052 -1.769534 -1.669961 -1.750096 1.4413

0.927357 Mean dependent var 0.923321 S.D. dependent var 0.095271 Akaike info criterion 0.163380 Schwarz criterion 19.69534 Hannan-Quinn criter. 229.78 Durbin-Watson stat 0.000000

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

ˆ*1.4770950.9059lnX*lnYtt t6.6372 15.15871R20.927357 R20.923321 F=229.78 DW=1.4413 模型中不存在自相关。

由差分方程式可以得出：

给定n=20，k1,在0.05的显著水平下，查DW统计表可知，

dL1.201 dU1.411。模型中dUDW1.44134dU，所以可以判断广义差分

ˆˆ*/(1ˆ)1.477095/(10.400234)2.46278500

ˆˆ*0.905911所以修正后的模型为：

ˆ2.4627850.9059lnXlnYtt

（3）变换数据后的回归结果如下：

Dependent Variable: LOG(Y/Y(-1)) Method: Least Squares Date: 05/07/10 Time: 00:23 Sample (adjusted): 1981 2000

Included observations: 20 after adjustments

C LOG(X/X(-1)) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 0.0047 0.442224

Std. Error 0.013322 0.066024

t-Statistic 4.0566 6.697901

Prob. 0.0007 0.0000 0.091592 0.098311 -2.903219 -2.8036 -2.883781 1.590363 0.713658 Mean dependent var 0.697750 S.D. dependent var 0.0049 Akaike info criterion 0.052583 Schwarz criterion 31.03219 Hannan-Quinn criter. 44.86188 Durbin-Watson stat 0.000003

ˆ*0.00470.442224lnX*lnYtt t4.0566 6.697901R20.713658 R20.69775 F=44.86188 DW=1.590363 d1.201 dU1.411。k1,在0.05的显著水平下，给定n=20，查DW统计表可知，L模型中

dUDW1.5903634dU，所以可以判断变化数据后的模型中不存在自相关。