序列相關性的檢驗與修正

1、序列相關性的檢驗與修正案例：書本P115進口與國內生產總值的關系。一 檢驗準備工作：建立工作文件，導入數據。采用OLS方法建立進口方程。在命令框輸入： equati on EqO1 .Is m c gdpDependent Variable: IMMethod: Least Squares ate: 11/17/10 Time: 23:00Sample: 1978 2001 included observatio ns: 24CoefficientStd. Errort-StatisticProb.C152J05746.07B4E331B3760.0031GDP0.0203940.001014

2、20.116810.0000R-squar&d0.948440l-.lean dependent var826.9542Adjusted R-squared0.94609&S.D. dependent var667.4365S E of regression154.9500Akaike info criterion13.00387Sum squared res id52&277.4Schwarz critenon13.10204Log likelihood-1&4Q464Hannan-Quinn criter.13.02991F-statistic404.6960Durbin-Watson s

3、tat0.627922Pro b(F-si3ti stic)0.000000建立殘差序列精品文檔交流在命令框輸入:series e=resid建立殘差序列的滯后一期序列在命令框輸入：series e_lag仁resid(-1)方法1：利用兩個殘差序列畫圖、觀察。方法2：查看回歸方程的DW值=0.628，存在序列相關。方法3： LM檢驗在命令框輸入：equation Eq02.ls e c gdp e(-1) e(-2)Dep endent Variable: EMethod: Least SquaresDate: 11/17/10 Time: 23:35Sample adjusted): 19

4、S0 2001Included observations: 22 after adjustmentsCoeffi ci entStd Errort-Statis1icProb.G144403131.165810.4533&40,6487GDP-0.0004740 000711-0 6666300.5135E(-D1-1107930.1300306J70D460.0000E(-2)-0.8039980,213658-3.7630200,0014R-squared0 682108口ean dependent var3930846Rusted R-squared0.529126S.D depende

5、ntvar1552949S E of regression94.57362Akaike info criterion12.09960Sum squared resid160995.1Schwarz criterion12.29797Log likelihood-129.0956Hannan-Quinn criter.12.14633FStali stic12.87436Dur bin-Wats on stat1861943Protj(F-statistic)0.000099在命令框輸入：Scalar lm1= obs(e(-2)*eq02.r2可得 LM1= 15.006在命令框輸入：scal

6、ar chi qchisq(0.95,2)可得 chi1=5.99可以判定模型存在 2階序列相關。簡便方法：在方程 eq01 窗口中點擊 View/Residual Test/Series Correlation LM Test，并選擇滯 后期為2，則會得到如下圖所示的信息。注：LM計算結果與上面有差異，因為這里的輔助回歸所采用的resid(-1)、resid(-2)的缺失值用0補齊。Breusch-Godfrey Serial Correlation LM Test:F-statistic ObsR-squared19.5290515.87241Prob. F(2,20Prob. Chi-S

7、quare(20.00000.0004Test Equation:Dependent Variable: RESID Method: Least Squares Date: 11/18/10 Time: 00:04Sample: 1978 2001 Included observations: 24Presample missing value lagged residuals set to zero.CoefficientStd. Errort-StatisticProb.C6.55475023.561860.2303940.8197GDP-0.0003440.000633-0.504009

8、0.6190RESID(-11.0935900.1755246.2304590.0000RESID (-2)-0.7857760.212316-3.692270o.oouR-squared0.661350Mean dlependEntvar-1.52E-13Adjusted R-squared0.610553S.D. dependent var151.5538S.E. of regression94.57826Akaike info criterion12.08774Sum squared resid173900.9Schwarz criterion12.28409Log likelihood

9、-141.0529Hannan-Ouinn criter.12.13903F-statisticProb(F-statistic)13.019370.000061urbin-Watso n stat1.873142檢驗是否存在更高階的序列相關。繼續(xù)在命令框輸入：equation Eq03.ls e c gdp e(-1) e(-2) e(-3)Dependent Variable: EMethod: Least Squar&sDate 11/17/10 Time： 23:55Sample (adjusted): 1981 2001Included observations: 21 after

10、adjustmentsCoefficientStd Error(-StatisticProb.C25.1811234.779320.7240260.4795GDP-0.0007140 000738-0 9050790,3783E(-1：1.1078830.2663774.4 S 59 3 90.0004E(-2)-1,0597900.50B427-2 0824600.0537E(-3)0.1971500.4140190 4761660.6404R-squared0.594482Mean dependent var1149151Adjusted R-squared0.518103S.D depe

11、ndentvar158 6532S.E of regressionS8.04427Akaike info criterion12.21297Sum squared resid153002.9Schwarz criterion1246167Log likelihood-123.2362Hannan-Quinn criter.1226655F-statlstic9 092526Durbin-Watson stat1.996014ProDF-statistic)0.D0049S在命令框輸入：Scalar lm2= obs(e(-3)*eq03.r2可得 LM2=14.58在命令框輸入：scalar

12、chi2=qchisq(0.95,3)可得 chi2=7.185仍然存在序列相關性，但由于e(-3)的參數不顯著，可認為不存在3階序列相關。在方程 eq01 窗口中點擊 View/Residual Test/Series Correlation LM Test，并選擇滯后期為3，則會得到如下圖所示的信息。Brecsch-Goctfre/ Serial Correlation LM TestF-staflstic Obs*R-squared12375761S.87561Prob F(3.19)Prob Chi-Square(3)0Q001U0012Test Equation.Depen dent

13、 Vari able: RESIDMethod: Least SquaresDate; 11/18/10 Time: 00:11Sample: 1978 2001Included obsen/atians: 24Presample missing value lagged residuals set to zero.CoefficientStd. ErrorStatisticPfOtiC6.69153S29319490.2282320.821 gGDP-0(3003490.000703-04967170fi251RESID(-1)1 1073330.2439554.5411510.0002RE

14、3ID(-2-0.B192S50.444735-1 842210Q.0811RESlD(-30.03229703733510 0S65070.9320R-equared0.661464Mean dependent var-1.52E-13Adjusted R-squared0.590217S D dependent var151.5539S.E of regression97.01S14Aka ike info criterion12.17066Burn squared resid173630.5Schwarz 匚 riterion12.41511Log likelihood-141.0482

15、Hannan-Quinn criter.12 2350QF-statistic9.231322Tbin-Watson stat1 SSS505Prcb(F-statistic)0.000247顯然，LM檢驗的結果拒絕原假設(無序列相關)，表明存在序列相關性。二序列相關性的修正與補救廣義差分法就是廣義最小二乘法(GLS)，但損失了部分樣本觀測值，損失的數量依賴于序列相關性的階數(如一階序列相關，至少損失1個樣本值)。在實際操作中，往往基于廣義差分法完成估計。根據隨機擾動項相關系數估計方法的不同， 可以分為C-O迭代法和Durbin兩步法。(1) Durbin兩步法第一步，估計隨機擾動項的相關系數

16、根據前面檢驗可知存在二階序列相關，因此設定方程為Mt =氏 +RMt_L + P2Mt +0；GDR +P；GDR_l + P；GDR/ +野在命令框輸入 Is m c m(-1) m(-2) gdp gdp(-1) gdp(-2)即得到隨機擾動項相關系數的估計值，結果記為eq04Dependent Variable: MMethod Least SquaresDate: 11/22/10 Time: 17:03Sample (adjusted): 1980 2001Included observations: 22 after adjutmenteCoefficientStd Errort-

17、StatisticProb.C70 0857544 462561.7562130.0982M(-1)0.9382490.1412506.64245&Q.OOQOM(-2)-0.4686520.265786-1 76326B0.0969GDP0 05477700093115 882762o.ooooJGDP(-1)-0.0963950 018567-5.1S18770.0001GDP(-2)0.0535830.0101065.3020790.0001Rsquared0991335(dean dependentvar890.0591Adjusted R-squared0,988627S.D. de

18、pendent var661.6499S E. of regression70.56198Aka ike info criterion11.57786Sum squared resid79663.8SSchwarz criterion11.87542Log likelihood-121.3565Hannan-Quinn criter.11.64796F-statistic3660872Durbin-Watson stat2.306915Prob(F-statistic)0 000000第二步，進行差分變換，然后對應書本（4220 ）公式進行回歸。M : = Mt(0.938Mt0.469Mt/

19、)GDPt* =GDPt -(0.983GDPtJL -0.469GDPt在命令框中輸入Series m_star=m-(0.938*m(-1)-0.469*m(-2)Series gdp_star=gdp-(0.938*gdp(-1)-0.469*gdp(-2)將Mt*對GDPt*回歸。在命令框輸入Ls m_star c gdp_star結果記為eq05Dependent Variable: M_STARMethod: Least SquaresDate: 11/22/10 Time: 17:13Sample: 197B 2001Included observations: 24Coeffi

20、cientStd Errort-StatisticProb.C78.1624727.633352.8285560.0098GDP_STAR0.0206240.00114418.023990.0000R-squared0.936614Mean dependent 4427313Adjusted R-sqjared0.933733S.D. dependentvar358 4401S.E. of regression92.27101Akaike info criterion11.96699Sum squared resid1873066Schwarz criterion12.06516Log lik

21、elihood-141.6039Hannan-Quinn criter11.99304F-statistic325.0807Durbin-Watson stat1.549633Prob(F-statistic)0.000000第三步，還原?0（參考書本4220式）78.161 -0.938 0.469= 147.19這一結果與eq01有差別。(2) C-O迭代法原理：參見教材113頁。操作：非常簡單。根據前面檢驗可知存在二階序列相關，因此設定方程為Mt =Po +RGDR +RAR(1)+ F2AR(2) +薊在命令框中輸入Equation eq06.Is m c gdp ar(1) ar(2

22、)Dependent Variable. UMettiod: Least Squar&sDate: 11/22/10 Time: 1729Sample (adjusted). 1980 2001Included observations: 22 after adjustmentsConvergence achieved after 5 iterationsCoefficientStd Errorb StatisticProbC1&9 321044.390073.8143890.0013GDP00197920.00107316.452500.0000AR1 10817701812465.1142

23、270 0QO0AR2)-0 8011940.221892-3.6107360 0020R-squared0 9S2325LI色白n depentignt ar890 0591Adjusted R-sqjar&d0 979379S.D. dependentvar661 6499S E of regression95.01304Aka ike info crit&ricn12.10037Sum squared resid162494.6Sctiwaiz criterion12.30724Lag likelihoodd29 1976Hannan-Quinn enter.12.155S0F-stat

24、istic333.4596Durbin-Watson stat1 853364ProbfF-stafisiic)0.000000Inverted AR Roots.55+70155-70i從DW值來看，已經不存在序列相關性，并且每項AR的回歸系數都具有統(tǒng)計顯著性。假設采用1階廣義差分，在命令框中輸入Equatio n eq07 .ls m c gdp ar(1)Dependent Variable: MMethod: Least SquaresDate： 11/22/1Q Time： 1737Sample (adjusted): 1979 2001IndJded observations: 2

25、3 after adjustments Convergence achieved after 7 iteratiansCoefficient Std Error ( Statistic Prob.C163.5484142316711491860.2640GDP0.02142000025088.54044400000AR(1)0.7280930 1812864.0162800.0007R-squared0.970150Mean dependent var853 1739Adjusted R-squared0.967165S.D dependent var664.2776S.E ofregression120.3699Akaike info criterion12,54012Sum squared resid2897784Schwarz criterion1266823Log likelihood-141 2114HannanQuinn criter.12.57737F-statistic325.0086Durbin-Watson stat1.122150Prob(F-statistic)0.000000Inverted AR Roots.73從DW來看，序列相關性并未消除。假設采用3階廣義差分，在命令框中輸