计量经济学（庞浩）第五章练习题参考解答

Method: Least Squares Date: 08/05/05 Time: 13:17 Sample: 1 60

Included observations: 60 Weighting series: W1

Variable Coefficient Std. Error t-Statistic Prob. C 10.37051 2.629716 3.943587 0.0002 X 0.630950 0.018532 34.04667 0.0000 Weighted Statistics R-squared

0.211441 Mean dependent var 106.2101 Adjusted R-squared 0.197845 S.D. dependent var 8.685376 S.E. of regression 7.778892 Akaike info criterion 6.973470 Sum squared resid 3509.647 Schwarz criterion 7.043282 Log likelihood -207.2041 F-statistic 1159.176 Durbin-Watson stat 0.958467 Prob(F-statistic) 0.000000 Unweighted Statistics R-squared

0.946335 Mean dependent var 119.6667 Adjusted R-squared 0.945410 S.D. dependent var 38.68984 S.E. of regression 9.039689 Sum squared resid 4739.526

Durbin-Watson stat 0.800564

其估计的书写形式为

Y??10.3705?0.6310X(3.9436)(34.0467)

R2?0.2114,s.e.?7.7789,F?1159.18

练习题5.5参考解答

（1）建立样本回归模型。

Y??192.9944?0.0319X

(0.1948)(3.83)

R2?0.4783,s.e.?2759.15,F?14.6692（2）利用White检验判断模型是否存在异方差。

White Heteroskedasticity Test: F-statistic 3.057161 Probability 0.076976 Obs*R-squared 5.212471 Probability 0.073812

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 08/08/05 Time: 15:38 Sample: 1 18

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Included observations: 18 Variable C X X^2 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -6219633. 229.3496 -0.000537 Std. Error 6459811. 126.2197 0.000449 t-Statistic -0.962820 1.817066 -1.194942 Prob. 0.3509 0.0892 0.2507

35.77968 35.92808 3.057161 0.076976 0.289582 Mean dependent var 6767029. 0.194859 S.D. dependent var

Akaike info criterion

2.61E+15 Schwarz criterion -319.0171 F-statistic 1.694572 Prob(F-statistic)

给定??0.05和自由度为2下，查卡方分布表，得临界值??5.9915，而White统计量

2(2)，则不拒绝原假设，说明模型中不存在异方差。 nR2?5.2125，有nR2??0.052（3）有Glejser检验判断模型是否存在异方差。经过试算，取如下函数形式 e??2X?? 得样本估计式

??6.4435Xe

(4.5658) R2?0.2482由此，可以看出模型中随机误差项有可能存在异方差。

（4）对异方差的修正。取权数为w?1/X，得如下估计结果

???243.4910?0.0367XY

(?1.7997)(5.5255)

R2?0.1684,s.e.?694.2181,F?30.5309

练习题5.7参考解答 （1）求回归估计式。

??4.6103?0.7574XY

(4.2495)(5.0516)R2?0.5864,s.e.?3.3910,F?25.5183

作残差的平方对解释变量的散点图

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504030E220100051015X202530

由图形可以看出，模型有可能存在异方差。

（2）去掉智利的数据后，回归得到如下模型

??6.7381?0.2215XY

(2.8254)(0.3987)R2?0.0093,s.e.?3.3906,F?0.1589

作残差平方对解释变量的散点图

4030E220100051015X202530

从图形看出，异方差的程度降低了。

（3）比较情况（1）和情况（2），实际上根据所给的数据，我们发现情况（1）的异方差性比情况（2）的异方差性要低。

练习题5.9参考解答

（1）建立样本回归函数。

??43.8967?0.8104XY

(2.1891)(37.7771)R2?0.9854,s.e.?60.4920,F?1427.112

从估计的结果看，各项检验指标均显著，但从残差平方对解释变量散点图可以看出，模型很

可能存在异方差。

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2000015000E2100005000005001000X15002000

（2）用White检验判断是否存在异方差。

White Heteroskedasticity Test: F-statistic Obs*R-squared

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 08/08/05 Time: 17:04 Sample: 1978 2000 Included observations: 23 Variable C X X^2 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2319.690 10.85979 -0.002560 Std. Error 2268.373 6.644388 0.003247 t-Statistic -1.022623 1.634430 -0.788315 Prob. 0.3187 0.1178 0.4398 5013.402 19.42572 19.57383 9.509463 0.001252 9.509463 Probability 11.21085 Probability

0.001252 0.003678

0.487428 Mean dependent var 3337.769 0.436171 S.D. dependent var 3764.490 Akaike info criterion 2.83E+08 Schwarz criterion -220.3958 F-statistic 1.552514 Prob(F-statistic)

由上表可知，nR?11.2109，给定??0.05，在自由度为2下，查卡方分布表，得临界值

2为??5.9915，显然，nR?11.2109>??5.9915，则拒绝原假设，说明模型存在异

222方差。

进一步，用ARCH检验判断模型是否存在异方差。经试算选滞后阶数为1，则ARCH检验结果见下表

ARCH Test: F-statistic

9.394796 Probability

0.006109

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Obs*R-squared

Test Equation:

7.031364 Probability

0.008009

Dependent Variable: RESID^2 Method: Least Squares Date: 08/08/05 Time: 17:11 Sample(adjusted): 1979 2000

Included observations: 22 after adjusting endpoints Variable C RESID^2(-1) R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient 1676.876 0.588797 Std. Error 1086.874 0.192098 t-Statistic 1.542843 3.065093 Prob. 0.1385 0.0061 5097.707 19.66118 19.76037 9.394796 0.006109 0.319607 Mean dependent var 3457.332 0.285588 S.D. dependent var 4308.730 Akaike info criterion 3.71E+08 Schwarz criterion -214.2730 F-statistic 1.874793 Prob(F-statistic)

由上表可知，(n?p)R?7.0314，在??0.05和自由度为1下，查卡方分布表，得临界值为?0.05(1)?3.8415，显然，(n?p)R?7.0314>?0.05(1)?3.8415，则说明模型中随机误差项存在异方差。

（3）修正异方差。取权数为W?1/X，得如下估计结果

22222??8.3065?0.8558XY

(1.8563)(34.1172)R2?0.9941,s.e.?13.4795,F?1163.99

经检验异方差的表现有明显的降低。

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