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: 20Variable Coefficient Std. Error t-Statistic Prob.CINCOMER-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)线性模型如下：CONSUMP = 5389 + *INCOME2）检验模型是否存在异方差性i) X Y -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 resid^2 或者genr ei2=resid^2； scat income 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 income Smpl 13 20； ls consump C income210.050.05615472.0126528.34.86(,)(81,81) 4.28118118111111RSS RSS F F F n k n k n k n k ===--=>=--------------，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。

22201122314251222012345220112122012:0,(),:0,(),ii i i i i i i ii i ie a a X a X a X a X a X X v H a a a a a nR q q ea a X a X v H a a nR q q χχ=++++++======+++==变量个数变量个数View ＼residual test ＼white heteroskedastcity(cross terms / no cross terms )White Heteroskedasticity Test: F-statistic Probability Obs*R-squaredProbabilityDependent Variable: RESID^2 Method: Least Squares Sample: 1 20Included observations: 20Variable CoefficientStd. Errort-StatisticProb.C INCOME INCOME^2R-squaredMean dependent var. of regression Akaike info criterion Sum squared resid +10 Schwarz criterionLog likelihood F-statistic Durbin-Watson statProb(F-statistic)22220.05(),212.65213(2) 5.99n R q q n R χχ===>，存在异方差。

还可以通过概率()220.0017890.05P n R χ>=<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: CONSUMP Method: Least Squares Sample: 1 20Included observations: 20 Weighting series: 1/EIJDZVariable CoefficientStd. Errort-StatisticProb.C INCOMEWeighted Statistics R-squaredMean dependent var. of regression Akaike info criterionSum squared resid Schwarz criterionLog likelihood F-statisticDurbin-Watson stat Prob(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_RESID^2Method: Least SquaresSample: 1 20Included observations: 20Variable Coefficient Std. Error t-Statistic Prob.CINCOME220.050.076420(2) 5.99n R q χ===<或()220.9625110.05P n R χ>=>,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)=+*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-statistic Probability Obs*R-squaredProbability模型取对数 F-statisticProbabilityObs*R-squared Probability模型不取对数，但加权F-statistic ProbabilityObs*R-squared Probability模型取对数，且加权F-statistic ProbabilityObs*R-squared Probability可见，各种方法都可以起到抑制异方差的效果。

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: 21Variable Coefficient Std. Error t-Statistic Prob.CLOG(INVEST)R-squared Mean dependent varAdjusted R-squared . dependent var. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson statProb(F-statistic)LOG(INDUSTRY) = 1. + *LOG(INVEST) i) 1,t t e e 散点图 ii) t e 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-Godfrey Serial Correlation LM Test:F-statistic Probability Obs*R-squaredProbabilityVariable CoefficientStd. Errort-StatisticProb.C LOG(INVEST) RESID(-1)R-squaredMean dependent var Adjusted R-squared . dependent var Log likelihood F-statistic Durbin-Watson statProb(F-statistic)一阶LM Test ：LM Test()()22220.05,(1),=(9.836218>) 3.84LM n p R p n p R p αααχχ==->=-=()()220.0017110.05P n p R χα>-=<=RESID(-1)的t 统计量显著（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：{}10111101111(1)()(1)..t t t t t t t t t t t t t OLS : ls y c y(-1) Y Y X X Y Y X X x x -)e (1i ββρβρρμρμβρρρβρνμμ---------⇒-+=++=+-+①--ls y c y(-1) x x(-1)y(-1)前面的系数：ˆρ，代回差分模型①，再次进行OLS 估计得到原模型的参数估计量，即 ***11000ˆˆˆˆˆˆˆ,,(1)OLS ρβββββρ−−−→==-→①。