序列相关性的检验与修正

1、序列相关性的检验与修正案例：书本P115进口与国内生产总值的关系。一检验准备工作：建立工作文件，导入数据。采用OLS方法建立进口方程。在命令框输入：equationEqOl.lsmcgdpDependentVariable:MMethod:LeastSquaresDate:11/17/10Time:23:00Sample:19782001Includedobservations:24CoefficienlStd.Errort-StatiSticProb.c152.905716.07S4S3.31S37600031GDP0J2Q3-9+Q.00101420.116S1O.OQQOR.-squar

2、ed0.943440臨朗dependentvarS26.9&42AdjustedR-squared0.946096S.D.dependsntvar667.4365S.E.ofregression154.9600Akaikeinfocriterion13.00J37Sumsquaredresid52S2774Schwarzcriterion13.10204Loglikelihood-154.0464Hannan-Quinncriter.13.02991F-statistic404.6860Durbin-Watsonstat0.627922Prob(F-statistic0.000000建立残差序

3、列在命令框输入：seriese=resid建立残差序列的滞后一期序列在命令框输入：serieseag1=resid(-1)方法1：利用两个残差序列画图、观察。方法2：查看回归方程的DW值=0.628,存在序列相关。方法3：LM检验在命令框输入：equationEq02.lsecgdpe(-1)e(-2)在命令框输入：Scalarlm1=obs(e(-2)*eq02.r2可得LMl=15.006在命令框输入：scalarchi1=qchisq(0.95,2)可得chi1=5.99可以判定模型存在2阶序列相关。简便方法：在方程eq01窗口中点击View/ResidualTest/SeriesCor

4、relationLMTest,并选择滞后期为2,则会得到如下图所示的信息。注:LM计算结果与上面有差异，因为这里的辅助回归所采用的resid(-1)、resid(-2)的缺失值用0补齐。Breusch-GodfteySerialCorrelationLMTestF-statislic19.52905Prob.FC2.20)0.0000ObsR-squared15.37241Prob.Chi-Square(20.0004TestEqua:ion:DependentVariable:RE3IDMetnoa：LeastsquaresDate.11/1B/10Time:00.04Sample:197S

5、2001Includedobsrvatiang：24Presamplemissingvalulaggedreidualset1ozero.CoefficientStd.Errort-StalisticProbC6.59475028561860.2303940.3197GDP-0.00D3440.300633-0.5040090.6190RESID(-1)1.095930.1755246.2304590.0000RESID(-2)-07857760.212016-3.692270O.OOUR-squared0.0G1350Meandependentvar-1.52E-13AdjustedR-sq

6、uared0.61D553S.D.dlependlentvar151.5539S.E.ofregression94.57026AkaikeinfDcriterion12.00774Sumsquaredresid179900.9Schwaizcritarion12.29409Loglikslihood-1J1.D529H甘nnan-Quinncriter.12.13983F-statis1ic12.01937urbin-Watsanstat1.S73142Prob(F-statistic)0.000061检验是否存在更高阶的序列相关。继续在命令框输入:equationEq03.lsecgdpe(

7、-1)e(-2)e(-3)在命令框输入：Scalarlm2=obs(e(-3)*eq03.r2可得LM2=14.58在命令框输入：scalarchi2=qchisq(0.95,3)可得chi2=7.185仍然存在序列相关性，但由于e(-3)的参数不显著，可认为不存在3阶序列相关。在方程eq01窗口中点击View/ResidualTest/SeriesCorrelationLMTest，并选择滞后期为3,则会得到如下图所示的信息。Breusch-GodfreySerialCorrelationLMTest:F-statistic1237576Prob.F(3,19)0.0001Obs*R-squ

8、ared15.87561Prob.Chi-Square(30.0012TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:11/13/10Time:00:11Sample:19782001Includedobservations:24Presamplemissingvaluelaggedresidualssettozero.CoefficientStd.Errort-StatisticProb.C6.69163829.319490.22S2320.3219GDP-0.0003490.000703-0.4967170.6251R

9、ESID(-1)1.1078380.24S95545411510.0002RESID(-2)-0.8192950444735-1.3422100.0811RESID(-3)0.0322970.3733510.0065070.9320R-squared0.661404Meandepe-1.52E-13AdjustedR-squared0.590217S.D.dependentvar151.5539S.E.ofregression97.01614Akaikeinfocriterion12.17068Sumisquaredresid178830.5Schwarzcriterion12.4-1611L

10、oglikelihood-141.0482Hannan-Quinncriter.12.23500F-statistic9.281822urbin-Watsonstat1.380605Prob(F-statistic0.000247显然，LM检验的结果拒绝原假设(无序列相关)，表明存在序列相关性。二序列相关性的修正与补救广义差分法就是广义最小二乘法(GLS)，但损失了部分样本观测值，损失的数量依赖于序列相关性的阶数(如一阶序列相关，至少损失1个样本值)。在实际操作中，往往基于广义差分法完成估计。根据随机扰动项相关系数估计方法的不同,可以分为C-0迭代法和Durbin两步法。(1)Durbin两步

11、法第一步，估计随机扰动项的相关系数根据前面检验可知存在二阶序列相关，因此设定方程为M*+pM+pM+*GDP+*GDP+*GDP,t01t12t21t2t13t2t在命令框输入lsmcm(-1)m(-2)gdpgdp(-1)gdp(-2)即得到随机扰动项相关系数的估计值，结果记为eq04DependentVariable:MMethod:LeastSquaresate:11/22/10Time:17:03Sample(adjusled):19802001Includedobservations:22alteradjustmentsCoefficientStdErrort-StaiisticPr

12、ob.C78.0857544.4625617562130.09S20.932490.14125066424660.0000-0.46S6520.265736-1.76326S0.0969GDP0.0547770.0093115.SQ2762ci.oooaGDP(-1)-0.0963950.013567-5.1913770.0001GDP(-2)0.0535830.0101065.3020790.0001R-squared0.991335Meandependentvar890.0591AdjustedR-squared0.98B627S.D.dependentvar661.6499S.E.ofr

13、egression70.56198Akaikeinfocriterion11.57736Sumsquaredreaid79663.88Schwarzcriterion11.37542Loglivelihood-121.3565Hannan-Quinncriter.11.64796F-statistic366.0372Djrbin-Watsonstat2.306915Prob(F-statistic)0.000000第二步，进行差分变换，然后对应书本（4.2.20）公式进行回归。M*二M(0.938M0.469M)ttt1t2GDP*=GDP(0.983GDP0.469GDP)ttt1t2在命令

14、框中输入Seriesm_star=m-(0.938*m(-1)-0.469*m(-2)Seriesgdp_star=gdp-(0.938*gdp(-1)-0.469*gdp(-2)将M*对GDP*回归。在命令框输入ttLsm_starcgdp_star结果记为eq05第三步，还原（参考书本4.2.20式）*1垐-p1278.161-0.9380.469，147.19这一结果与eq01有差别。(2)C-O迭代法原理：参见教材113页。操作：非常简单。根据前面检验可知存在二阶序列相关，因此设定方程为M，GDPpAR(1)pAR(2)t01t12t在命令框中输入Equationeq06.lsmcgd

15、par(1)ar(2)DependentVariable:LIMethod:LeastSquaresate:11/22/10Time:17:29Sample沮djusted:19302001Includedobservations:22afteradjustmentsConvergenceachievedafter5iterationsCoefficientStd.Errort-StatisticProb.c169.321044.390073.8143090.0013GDP0.0197920.00107318.45-2500.0000AR(1)1.1031770.1-31246&1142270

16、.0000AR(2)-0.3011940.221892-3.6107360.0020R-squared0.932325Meandependsntvmr890.0591AdjustedR-squared0.979379S.D.dependentvar661.6499S.E.ofregression95.01304Akaikzeinfocriterion12.10387Sumsqua.redresid162494.6Schwarzcriterion12.30724Loglikelihood-129.1976Hannan-Quinncriter.12.15560F-statistic333.4596

17、urbin-Watsonstat1.353364Prob(F-statistic0.000000InvertedARRoots.55+.70i.55.7Oi从DW值来看，已经不存在序列相关性，并且每项AR的回归系数都具有统计显著性。假设采用1阶广义差分，在命令框中输入Equationeq07.1smcgdpar(1)DependentVariable:MMethod:LeastSquaresDate:11/22/10Time:T7:37Sample(adjusted):19792Q01Includedobservations:23afteradjustmentsConvergenceachievedafte1iterationsCoefficientStd.Errort-StatisticProb.C163.&43斗142.31671.1491560.2640GDP0.0214200.0025080.0000ARC1)072-80930.1B12864.0162B00.0007R-squared0.970150Meandependentvar858.1739AdjustedR.-squared0.967165S.D.dependentvar664.2776S.E.ofregression120.699Aaikeinfocriterion12.&4Q12Sums