4 第四章 习习题 参考答案

1、第四章习题参考答案 P 1357. 1）用OLS法建立居民人均消费支出与可支配收入的线性模型。create u 20； data consump income；ls consump c incomeDependent Variable: CONSUMPMethod: Least SquaresSample: 1 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb. CINCOMER-squared Mean dependent varAdjusted R-squared . dependent var.

2、 of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)线性模型如下： CONSUMP = 5389 + *INCOME2）检验模型是否存在异方差性 i) 图：是否有明显的散点扩大/缩小/复杂型趋势 scat income consumpii)解释变量残差图：是否形成一条斜率为0的直线 scat income resid2 或者 genr ei2=resid2； scat income

3、ei2由两个图形，均可判定存在递增型异方差。 还可以用帕克检验，戈里瑟检验，戈德菲尔德-匡特检验，怀特检验等方法。iii) 戈德菲尔德-匡特检验：共有20个样本，去掉中间1/4个样本（4个），剩余大样本、小样本各8个。Sort income； smpl 1 8； ls consump C incomeSmpl 13 20； ls consump C income，存在异方差。iV)怀特检验：因为只有一个变量，故是否含有交叉项是一样的。 Viewresidual testwhite heteroskedastcity(cross terms / no cross terms )White Het

4、eroskedasticity Test:F-statistic ProbabilityObs*R-squared ProbabilityDependent Variable: RESID2Method: Least SquaresSample: 1 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb. CINCOMEINCOME2R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike in

5、fo criterionSum squared resid+10 Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)，存在异方差。还可以通过概率判定存在异方差。3）若存在异方差，用适当的方法估计模型对数(加权最小二乘法)ls consump C income； genr eijdz=abs(resid)ls(w=1/eijdz) consump C incomeDependent Variable: CONSUMPMethod: Least SquaresSample: 1 20Incl

6、uded observations: 20Weighting series: 1/EIJDZVariableCoefficientStd. Errort-StatisticProb. CINCOMEWeighted StatisticsR-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Pro

7、b(F-statistic)Unweighted StatisticsR-squared Mean dependent varAdjusted R-squared . dependent var. of regression Sum squared residDurbin-Watson statWhite Heteroskedasticity Test:F-statistic ProbabilityObs*R-squared ProbabilityTest Equation:Dependent Variable: STD_RESID2Method: Least SquaresSample: 1

8、 20Included observations: 20VariableCoefficientStd. Errort-StatisticProb. CINCOME或,均可判定加权处理后的模型不存在异方差。模型经取对数或加权处理都可以一定程度地消除异方差性。ls log(consump) C log(income)； genr eijdz=abs(resid)；ls(w=1/eijdz) log(Consump) C log(Income)普通最小二乘模型CONSUMP = 5389 + *INCOME加权最小二乘模型 CONSUMP = + *INCOME对数模型：LOG(CONSUMP)=+

9、*LOG(INCOME)加权对数模型：LOG(CONSUMP)=+ *LOG(INCOME)对各种模型的White检验结果，综合如下模型不取对数F-statisticProbabilityObs*R-squaredProbability模型取对数F-statisticProbabilityObs*R-squaredProbability模型不取对数，但加权F-statisticProbabilityObs*R-squaredProbability模型取对数，且加权F-statisticProbabilityObs*R-squaredProbability可见，各种方法都可以起到抑制异方差的效果

10、。8. 1）若采用对数模型，是否存在序列相关性ls log(industry) C log(invest)Dependent Variable: LOG(INDUSTRY)Method: Least SquaresSample: 1901 1921Included observations: 21VariableCoefficientStd. Errort-StatisticProb. CLOG(INVEST)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info cri

11、terionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)LOG(INDUSTRY) = 1. + *LOG(INVEST)i) 散点图ii) 随t变化的散点图 由两个图形，均可判定存在正序列相关。还可以利用回归检验法，D -W检验，拉格朗日乘数检验等方法。iii) D -W检验(DL(21, =, DU(21, =.= < DL(21, 2，=，至少存在一阶正自相关；但.只适用判别一阶自相关。iv) 拉格朗日乘数检验Breusch-Godf

12、rey Serial Correlation LM Test:F-statistic ProbabilityObs*R-squared ProbabilityVariableCoefficientStd. Errort-StatisticProb. CLOG(INVEST)RESID(-1)R-squared Mean dependent varAdjusted R-squared . dependent varLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)一阶LM Test：LM TestRESID(-1)的t统计

13、量显著（P=<），至少存在一阶自相关。2）按照一阶自相关，用杜宾两步法和广义最小二乘法估计原模型。杜宾两步法：ls y c y(-1) x x(-1)y(-1)前面的系数：，代回差分模型，再次进行OLS估计得到原模型的参数估计量，即 。genr y = log(industry)； genr x = log(invest)；Step 1: ls y c y(-1) x x(-1)Dependent Variable: YMethod: Least SquaresSample(adjusted): 1981 2000Included observations: 20 after adjus

14、ting endpointsVariableCoefficientStd. Errort-StatisticProb. CY(-1)XX(-1)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)Step 2: ls y - * y(-1) c x - * x

15、(-1)Dependent Variable: *Y(-1)Method: Least SquaresSample(adjusted): 1981 2000Included observations: 20 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. C*X(-1)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid S

16、chwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic).= 介于 DL(21-1, 2，=与DU(21-1, 2，=之间，不能判别是否存在一阶正自相关，但可由拉格朗日乘数法判断，此时不存在序列相关性。Breusch-Godfrey Serial Correlation LM Test:F-statistic ProbabilityObs*R-squared ProbabilityTest Equation:Dependent Variable: RESIDMethod: Least Squar

17、esVariableCoefficientStd. Errort-StatisticProb. C*X(-1)RESID(-1)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)拉格朗日乘数检验：D-W stat： > ，不存在序列相关性。所以 矫正后

18、的模型：LOG(INDUSTRY) = + *LOG(INVEST)原模型：LOG(INDUSTRY) = 1. + *LOG(INVEST)广义差分法ls y c x ar(1) （不能判定是否存在一阶自相关）Dependent Variable: YMethod: Least SquaresSample(adjusted): 1981 2000Included observations: 20 after adjusting endpointsConvergence achieved after 15 iterationsVariableCoefficientStd. Errort-Sta

19、tisticProb. CXAR(1)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)但由LM检验：概率为>，故此时不存在序列相关性。因此模型只存在一阶自相关性。Breusch-Godfrey Serial Correlation LM Test:F

20、-statisticProbabilityObs*R-squared ProbabilityDependent Variable: RESIDVariableCoefficientStd. Errort-StatisticProb. CXAR(1)RESID(-1)Durbin-Watson stat Prob(F-statistic)模型为 Y = + *X + * AR(1) 与杜宾两步法矫正的模型：LOG(INDUSTRY) = + *LOG(INVEST) 非常接近。广义最小二乘法若仅存在一阶自相关ls log(industry) C log(invest) genr resid_co

21、rr = residls resid_corr resid_corr(-1) 注：resid是内置变量；Dependent Variable: RESID_CORRMethod: Least SquaresVariableCoefficientStd. Errort-StatisticProb. CRESID_CORR(-1)R-squared Mean dependent varDurbin-Watson stat Prob(F-statistic)直接计算 模型为LOG(INDUSTRY)=+*LOG(INVEST)，误差偏大。3）采用差分形式，估计原模型。ls D(industry) C

22、 D(invest)ls industryindustry(-1) C investinvest(-1)Dependent Variable: D(INDUSTRY)Method: Least SquaresSample(adjusted): 1981 2000Included observations: 20 after adjusting endpointsVariableCoefficientStd. Errort-StatisticProb. CD(INVEST)R-squared Mean dependent varAdjusted R-squared . dependent var

23、. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)Breusch-Godfrey Serial Correlation LM Test:F-statistic ProbabilityObs*R-squared ProbabilityTest Equation:Dependent Variable: RESIDMethod: Least SquaresVariableCoeffic

24、ientStd. Errort-StatisticProb. CD(INVEST)RESID(-1)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(F-statistic)原模型存在一阶正自相关，但经过一阶自相关差分处理后不存在序列相关性（.= > 或=>）。模型为：

25、D(INDUSTRY) = + *D(INVEST)说明：在有的方法不能判别自相关性时，可以用其他方法测试。9. 说明下述回归模型是否可靠Ls CONSUMP C INCOME WEALTHDependent Variable: CONSUMPMethod: Least SquaresSample: 1 10Included observations: 10VariableCoefficientStd. Errort-StatisticProb. CINCOMEWEALTHR-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterionSum squared resid Schw