4 第四章 習習題 參考答案

1、第四章習題參考答案 P 1357. 1）用OLS法建立居民人均消費支出與可支配收入的線性模型。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)線性模型如下： CONSUMP = 5389 + *INCOME2）檢驗模型是否存在異方差性 i) 圖：是否有明顯的散點擴大/縮小/復雜型趨勢 scat income consumpii)解釋變量殘差圖：是否形成一條斜率為0的直線 scat income resid2 或者 genr ei2=resid2； scat income

3、ei2由兩個圖形，均可判定存在遞增型異方差。 還可以用帕克檢驗，戈里瑟檢驗，戈德菲爾德-匡特檢驗，懷特檢驗等方法。iii) 戈德菲爾德-匡特檢驗：共有20個樣本，去掉中間1/4個樣本（4個），剩余大樣本、小樣本各8個。Sort income； smpl 1 8； ls consump C incomeSmpl 13 20； ls consump C income，存在異方差。iV)懷特檢驗：因為只有一個變量，故是否含有交叉項是一樣的。 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)，存在異方差。還可以通過概率判定存在異方差。3）若存在異方差，用適當?shù)姆椒ü烙嬆Ｐ蛯?shù)(加權最小二乘法)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,均可判定加權處理后的模型不存在異方差。模型經(jīng)取對數(shù)或加權處理都可以一定程度地消除異方差性。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加權最小二乘模型 CONSUMP = + *INCOME對數(shù)模型：LOG(CONSUMP)=+

9、*LOG(INCOME)加權對數(shù)模型：LOG(CONSUMP)=+ *LOG(INCOME)對各種模型的White檢驗結果，綜合如下模型不取對數(shù)F-statisticProbabilityObs*R-squaredProbability模型取對數(shù)F-statisticProbabilityObs*R-squaredProbability模型不取對數(shù)，但加權F-statisticProbabilityObs*R-squaredProbability模型取對數(shù)，且加權F-statisticProbabilityObs*R-squaredProbability可見，各種方法都可以起到抑制異方差的效果

10、。8. 1）若采用對數(shù)模型，是否存在序列相關性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) 散點圖ii) 隨t變化的散點圖 由兩個圖形，均可判定存在正序列相關。還可以利用回歸檢驗法，D -W檢驗，拉格朗日乘數(shù)檢驗等方法。iii) D -W檢驗(DL(21, =, DU(21, =.= < DL(21, 2，=，至少存在一階正自相關；但.只適用判別一階自相關。iv) 拉格朗日乘數(shù)檢驗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)一階LM Test：LM TestRESID(-1)的t統(tǒng)計

13、量顯著（P=<），至少存在一階自相關。2）按照一階自相關，用杜賓兩步法和廣義最小二乘法估計原模型。杜賓兩步法：ls y c y(-1) x x(-1)y(-1)前面的系數(shù)：，代回差分模型，再次進行OLS估計得到原模型的參數(shù)估計量，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，=與DU(21-1, 2，=之間，不能判別是否存在一階正自相關，但可由拉格朗日乘數(shù)法判斷，此時不存在序列相關性。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)拉格朗日乘數(shù)檢驗：D-W stat： > ，不存在序列相關性。所以 矯正后

18、的模型：LOG(INDUSTRY) = + *LOG(INVEST)原模型：LOG(INDUSTRY) = 1. + *LOG(INVEST)廣義差分法ls y c x ar(1) （不能判定是否存在一階自相關）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檢驗：概率為>，故此時不存在序列相關性。因此模型只存在一階自相關性。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)模型為 Y = + *X + * AR(1) 與杜賓兩步法矯正的模型：LOG(INDUSTRY) = + *LOG(INVEST) 非常接近。廣義最小二乘法若僅存在一階自相關ls log(industry) C log(invest) genr resid_co

21、rr = residls resid_corr resid_corr(-1) 注：resid是內置變量；Dependent Variable: RESID_CORRMethod: Least SquaresVariableCoefficientStd. Errort-StatisticProb. CRESID_CORR(-1)R-squared Mean dependent varDurbin-Watson stat Prob(F-statistic)直接計算 模型為LOG(INDUSTRY)=+*LOG(INVEST)，誤差偏大。3）采用差分形式，估計原模型。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)原模型存在一階正自相關，但經(jīng)過一階自相關差分處理后不存在序列相關性（.= > 或=>）。模型為：

25、D(INDUSTRY) = + *D(INVEST)說明：在有的方法不能判別自相關性時，可以用其他方法測試。9. 說明下述回歸模型是否可靠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