放寬基本假定模型案例

第四章放寬基本假定模型案例一、異方差性1、中國農(nóng)村居民人均消費支出主要由人均純收入來決定。農(nóng)村人均純收入除從事農(nóng)業(yè)經(jīng)營的收入外，還包括從事其他產(chǎn)業(yè)的經(jīng)營性收入以及工資性收入、財產(chǎn)收入和轉(zhuǎn)移支出收入等。為了考察從事農(nóng)業(yè)經(jīng)營的收入和其他收入對中國農(nóng)村居民消費支出增長的影響，可使用如下雙對數(shù)模型：lnY=。+plnX+plnX+u01122其中Y表示農(nóng)村家庭人均消費支出，X］表示從事農(nóng)業(yè)經(jīng)營的收入，X2表示其他收入。表4.1列出了中國2001年各地區(qū)農(nóng)村居民家庭人均純收入及消費支出的相關(guān)數(shù)據(jù)。表4.1中國2001年各地區(qū)農(nóng)村居民家庭人均純收入與消費支出地區(qū)人均消費支出Y從事農(nóng)業(yè)經(jīng)營的收入天其他收入地區(qū)人均消費支出Y從事農(nóng)業(yè)經(jīng)營的收入天其他收入北京3552.1579.14446.4湖北2703.361242.92526.9天津2050.91314.62633.1湖南1550.621068.8875.6河北1429.8928.81674.8廣東1357.431386.7839.8山西1221.6609.81346.2廣西1475.16883.21088內(nèi)蒙古1554.61492.8480.5海南1497.52919.31067.7遼寧1786.31254.31303.6重慶1098.39764647.8吉林1661.71634.6547.6四川1336.25889.4644.3黑龍江1604.51684.1596.2貴州1123.71589.6814.4上海4753.2652.55218.4云南1331.03614.8876江蘇2374.71177.62607.2西藏1127.37621.6887浙江3479.2985.83596.6陜西1330.45803.8753.5安徽1412.41013.11006.9甘肅1388.79859.6963.4福建2503.110532327.7青海1350.231300.1410.3江西17201027.81203.8寧夏2703.361242.92526.9山東190512931511.6新疆1550.621068.8875.6河南1375.61083.81014.1用OLS法進行估計，結(jié)果如下:DependentVariable:LOG(Y)Method:LeastSquaresDate:07/03/08Time:16:31Sample:131Includedobsen/ations:31VariableCoefficientStd.Errort-StatisticProb.C1.6025200.0609701.8612080.0732LOG(X1)0.3254160.1037693.1359550.0040LOG(X2)0.5070700.04859910.433050.0000R-squared0.796506Meandependentvar7.448704AdjustedR-squared0.781971S.D.dependentvar0.364640S.E.ofregression0.170267Akaikeinfocriterion-0.611120Sumsquaredresid0.011747Schwarzcriterion-0.472355Loglikelihood12.47249F-statistic54.79806Durbin-Watsonstat1.964720Prob(F-statistic)0.000000對應的表達式為:InY=1.603+0.325lnX1+0.507lnX2(1.86)(3.14)(10.43)R2=0.7965,R=0.78,RSS=0.8117不同地區(qū)農(nóng)村人均消費支出的差別主要來源于非農(nóng)經(jīng)營收入及其他收入的差別，因此，如果存在異方差性，則可能是X2引起的。對異方差性的檢驗：做OLS回歸得到的殘差平方項與lnX2的散點圖：cnwchslijeLOG(X2)

從散點圖可以看出，兩者存在異方差性。下面進行統(tǒng)計檢驗。cnwchslije采用White異方差檢驗：EViews提供了包含交叉項和沒有交叉項兩個選擇。本例選擇沒有包含交叉項。SEViews-[Equation:UWTITLEDTorkfile:UNTITLED::Untit1e□FileEditObjectViewFi的巨卿的口匚］［口北艷匚t.PrindWrnnEFrevERejireserLtatiuileEstimationLlutputActual.,Fitted,Reeidual*AFX1AStructufH...Gradientsand.Ilerivatives*Cuy：±fi:±tli：eMatrix_ocQuick0^+ioilsWindowHelp司［e如印1弓!:司作口佗匚2］［51:北］怛藥山itStd.Errort-StatisticProb.Correli：igi-：iiTi一Q_EtatisticeCorrelugi-的SguaredReeidu：dleHistugi-：diTi一Nurm：±lityTestSeri：ilCorrelaticmLMTest...AJICHLMTest...Cue￡￡icientTests卜ResidualThe+e*StabilityTests?LabelR-squared0.79651AdjustedR-squared0.7819S.E.ofregression0.17021Sumsquaredresid0.8117-Loglikelihood12.4724Durbin-Watsonstat1.96472j￥hiteHetHroEkedasticity(jiocroeeterms)eHeteruskedasticity(_croeeterms〕9F-statistic54.79806□Prob(F-statistic)0.000000得到如下結(jié)果：WhiteHeteroskedasticityTest:F-statistic4.920995Prob.F(4,26)0.004339Obs*R-squared13.35705Prob.Chi-Square(4)D.DCl疵TTestEquation:DependentVariable:RESIDA2Method:LeastSquaresDate:07/03/08Time:16:51Sample:131Includedobservations:31VariableCoefficientStd.Errort-StatisticProb.C3.9821372.8828511.3813190.1789L0G(X1J-0.5792890.916069-0.6323640.5327(L0G(X1J^20.0418390.0668660.6257100.5370L0G(X2J-0.5636560.203228-2.7735140.0101(LOG(X2J^20.0402800.0138792.9021730.0075R-squared0.430873Meandependentvar0.026185AdjustedR-squared0.343315S.D.dependentvar0.038823S.E.ofregression0.031460Akaikeinfocriterion-3.933482Sumsquaredresid0.025734Schwarzcriterion-3.702194Loglikelihood65.96890F-statistic4.920995Durbin-Watsonstat1.526222Prob(F-statistic)0.004339所以輔助回歸結(jié)果為:e2=3.982-0.579lnX1+0.042(lnX1)2-0.563lnX2+0.04(lnX2)2(1.38)(-0.63)(0.63)(-2.77)(2.9)其他收入X2與X2的平方項的參數(shù)的t檢驗是顯著的，且White統(tǒng)計量為13.36,在5%的顯著性水平下，拒絕同方差性這一原假設(shè)，方程確實存在異方差性。用加權(quán)最小二乘法對異方差性進行修正，重新進行回歸估計，過程如下：在EViews工作窗口輸入如下命令，定義加權(quán)數(shù)：估計過程如下:得到加權(quán)后消除異方差性的估計結(jié)果:

DependentVariable:LOG(Y)Method:LeastSquaresDate:03/30/09Time:11:56Sample:131Includedobservations:31Weightingseries:WVariableCoefficientStd.Errort-StatisticProb.C1.2279290.2972684.1307080.0003LOG(X1)0.3757480.0568306.6117340.0000LOG(>2)0.5101200.01778128.688470.0000WeightedStatisticsR-squared0.999990Meandependentvar7.558578AdjustedR-squared0.999989S.D.dependentvar12.31758S.E.ofregression0.041062Akaikeinfocriterion-3.455703Sumsquaredresid0.047210Schwarzcriterion-3.316930Loglikelihood56.56339F-statistic1960.131Durbin-Watsonstat2.487309Prob(F-statistic)0.000000UnweightedStatisticsR-squared0.794514Meandependentvar7.448704AdjustedR-squared0.779836S.D.dependentvar0.364648S.E.ofregression0.171099Sumsquaredresid0.819694Durbin-Watsonstat2.007122回歸表達式為:InY=1.228+0.376lnX+0.51lnX12(4.13)(6.61)(28.69)R2=0.999,R=0.999,RSS=0.047從上面的結(jié)果看出，運用加權(quán)最小二乘法估計的結(jié)果不論擬合度，殘差，還是各參數(shù)的t統(tǒng)計量的值都有了顯著的改善。(4.13)(6.61)(28.69)2、表4.2列出了2000年中國部分省市城鎮(zhèn)居民每個家庭平均全年可支配收入（乂）與消費性支出（Y）的統(tǒng)計數(shù)據(jù)。表4.22000年中國部分省市城鎮(zhèn)居民每個家庭平均全年可支配收入（X）與消費性支出（Y）單位：元地區(qū)可支配收入X消費性支出Y地區(qū)可支配收入X消費性支出Y北京10349.698493.49浙江9279.167020.22天津8140.56121.04山東6489.975022河北5661.164348.47河南4766.263830.71山西4724.113941.87湖北5524.544644.5內(nèi)蒙古5129.053927.75湖南6218.735218.79遼寧5357.794356.06廣東9761.578016.91吉林48104020.87陜西5124.244276.67黑龍江4912.883824.44甘肅4916.254126.47

上海11718.018868.19青海5169.964185.73江蘇6800.235323.18新疆5644.864422.93（1）試用OLS法建立居民人均消費支出與可支配收入的線性模型（2）檢驗模型是否存在異方差性（3）如果存在異方差性，試采用適當?shù)姆椒ü烙嬆Ｐ蛯?shù)。采用OLS法建立線性模型，結(jié)果如下:DependentVariable:YMethod:LeastSquaresDate:07/03/08Time:17:31Sample:120Includedobservations:20VariableCoefficientStd.Errort-StatisticProb.C272.3635159.67731.7057130.1053X0.7551250.02331632.386900.0000R-squared0.903129Meandependentvar5199.515AdjustedR-squared0.902192S.D.dependentvar1625.275S.E.ofregression216.8900Akaikeinfocriterion13.69130Sumsquaredresid846743.0Schwarzcriterion13.79087Loglikelihood-134.9130F-statistic1048.912Durbin-Watsonstat1.301684Prob(F-statistic)0.000000進行異方差性的檢驗，本例采用White檢驗，過程如下:得到如下部分輸出結(jié)果：WhiteHeteroskedasticityTestF-statistic14.63595Prob.FP.17)0.000201Obs*R-squared12.65213Prob.Chi-Square(2)D.CID1花9

從伴隨概率值可以看出，在5%的顯著性水平下，原模型存在異方差性采用加權(quán)最小二乘法進行估計，過程如下:@EVie>sIZI回函Filt：Edi+UtjectViewFroc^luickOj?tioreWiridowHelpgenrw=1/ABS(resid)EquationEsti>ationSpecificationOptions確定取消得到如下結(jié)果:

DependentVariable:YMethod:LeastSquaresDate:07/03/08Time:17:39Sample:120Includedobservations:20Weightingseries:WVariableCoefficientStd.Errort-StatisticProb.C415.6603116.97913.5532080.0023X0.7290260.02242932.503490.0000WeightedStatisticsR-squared0.983240Meandependentvar4471.606AdjustedR-squared0.982317S.D.dependentvar7313.160S.E.ofregression77.04831Akaikeinfocriterion11.62130Sumsquaredresid106056.0Schwarzcriterion11.72096Loglikelihood-114.2130F-statistic1056.477Durbin-Watsonstat1.545424Prob(F-statistic)0.000000Unweightec1StatisticsR-squared0.981664Meandependentvar5199.515AdjustedR-squared0.980645S.D.dependentvar1625.275S.E.ofregression226.1101Sumsquaredresid920263.9Durbin-Watsonstat1.306360此時的回歸表達式為：y=415.66+0.729x(3.55)(32.5)擬合度和殘差都有所改善。二、序列相關(guān)性1、經(jīng)濟理論指出，商品進口主要由進口國的經(jīng)濟發(fā)展水平，以及商品進口價格指數(shù)與國內(nèi)價格指數(shù)對比因素決定。由于無法取得中國商品進口價格指數(shù)，我們主要研究中國商品進口M與國內(nèi)生產(chǎn)總值GDP的關(guān)系，數(shù)據(jù)見表4.3o表4.31978-2001年中國商品進口與國內(nèi)生產(chǎn)總值年份國內(nèi)生產(chǎn)總值/億元商品進口/億美元年份國內(nèi)生產(chǎn)總值/億元商品進口/億美元19783624.1108.9199018547.9533.519794038.2156.7199121617.8637.919804517.8200.2199226638.1805.919814862.4220.2199334634.41039.619825294.7192.9199446759.41156.119835934.5213.9199558478.11320.819847171274.1199667884.61388.319858964.4422.5199774462.61423.7198610202.2429.1199878345.21402.4198711962.5432.1199982067.461657

198814928.3552.7200089442.22250.9198916909.2591.4200195933.32436.1用OLS法建立中國商品進口方程，回歸結(jié)果如下:DependentVariable:MMethod:LeastSquaresDate:03/28/09Time:13:29Sample:19782001Includedobservations:24VariableCoefficientStd.Errort-StatisticProb.C162.905746.078483.3183760.0031GDP0.0203940.00101420.116810.0000R-squared0.948440Meandependentvar826.9542AdjustedR-squared0.946096S.D.dependentvar667.4365S.E.ofregression154.9600Akaikeinfocriterion13.00387Sumsquaredresid528277.4Schwarzcriterion13.10204Loglikelihood-154.0464F-statistic404.6860Durbin-Watsonstat0.627922Prob(F-statistic)0.000000對應的表達式為:M=152.91+0.02GDPtt(3.32)(20.11)R2=0.948,R=0.946,DW=0.627進行序列相關(guān)性檢驗，作殘差項￡與時間t以及￡與%的關(guān)系圖，如下:QEquation:UNTITLED￥orkfile:UNTITLED::U...^yE3/i自匚][°bj已匚t][Prh~it][Nvm〉][Fr?已已e已][E&tim占t已](Furi匚占占tn][R已引匚k]Re^reeentationsEstimationOutputActual,Fitted,Residual*Actual,Fitted,ResidualTableAEHAStructure...Actual?Fitted?ResidualGraphGradientsandDerivatives卜ReeidualGraphCoyarianceMatrixStandardizedKesidualGraphCoefficientTests*ResidualTests*Stabi1ityTests*ILOLU.I_l1UlL-OLdLliLIL-r1ul434.511735.645657□.□□□□00.00098818.123610.0000Label8Meandependentvar642.9952得到如下關(guān)系圖:

二'qcolue-400-300-200-1000100200300400RESID二'qcolue從上圖可以看出，隨即干擾項呈現(xiàn)正序列相關(guān)性。DW檢驗結(jié)果表明，在5%的顯著性水平下，n=24,k=2,查表得d=\.27.d廣1.45由于DW=0.627"，故存在正自相關(guān)。Y下面進行拉格朗日乘數(shù)檢驗。含1階滯后殘差項的輔助回歸過程如下:

輸入滯后階數(shù):得到如下結(jié)果:F-statistic15.87518Probability0.000674Obs*R-squared10.33227Probability0.001307TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:03/2S/D9Time:13:43Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C-10.2200535.68349-0.2864080.7774GDP0.000619□.00079S0.7757270.4466RESID(-1)0.752830□.1889463.9843670.0007R-squared0.430511Meandependentvar-1.47E-13AdjustedR-squared0.376274S.D.dependentvar151.5539S.E.ofregression119.6917Akaikeinfocriterion12.52418Sumsquaredresid300S4S.0Schwarzcriterion12.67144Loglikelihood-147.2902F-statistic7.937590Durbin-Watsonstat1.164220Prob(F-statistic)0.00270S輔助回歸表達式為：e=—10.22+0.0008GDP+0.753ett—1(-0.286)(0.776)(3.98)LM=10.33，從伴隨概率值可以看出，在顯著性為5%的水平下，模型存在1階序列相關(guān)性。作2階滯后殘差項的輔助回歸結(jié)果如下：F-statistic19.52905Probability□.□□0020Obs*R-squared15.S7241Probability0.00035STestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:03/28/09Time:13:52Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C6.59475D28.5E186□.230894□.S197GDP-□.□□□3440.000683-0.5040090.6198RESID(-1)1.09359B0.1755246.2304590.0000RESID(-2)-0.7S57760.212816-3.6922780.0014R-squared0.661350Meandependentvar-1.47E-13AdjustedR-squared0.610553S.D.dependentvar151.5539S.E.ofregression94.57826Akaikeinfocriterion12.0S774Sumsquaredresid17S9QD.9Schwarzcriterion12.2S4Q9Loglikelihood-141.0529F-statistic13.01937Durbin-Watsonstat1.S73142Prob(F-statistic)0.000061輔助回歸表達式為：e=6.59-0.0003GDP+1.0946e-0.786ett-1t-2(0.231)(-0.504)(6.231)(-3.692)LM值的伴隨概率說明模型仍然存在序列相關(guān)性，et-2的參數(shù)顯著，說明存在2階序列相關(guān)性。作3階滯后殘差項的輔助回歸結(jié)果如下：

F-statistic12.37576Probability0.000102Obs*R-squared15.875E1Probability□.□□1203TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:03/2B/09Time:13:57Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C6.69163829.319490.2282320.S219GDP-□.□□□3490.000703-0.4967170.6251RESID(-1)1.1078380.2439554.5411510.0002RESID(-2)-0.8192950.444735-1.B422100.0311RESID(-3)0.0322970.3733510.0865070.9320R-squared0.661484Meandependentvar-1.47E-13AdjustedR-squared0.590217S.D.dependentvar151.5539S.E.ofregression97.01614Akaikeinfocriterion12.17Q6SSumsquaredresid178830.5Schwarzcriterion12.41611Loglikelihood-141.0482F-statistic9.2S1S22Durbin-Watsonstat1.888605Prob(F-statistic)□.□□□247輔助回歸表達式為：e=6.692-0.0003GDP+1.108e-0.8196e+0.032ett—1t—2t—3(0.228)(-0.497)(4.541)(-1.842)(0.087)LM值的伴隨概率說明模型仍然存在序列相關(guān)性，但e,—3的參數(shù)不顯著，說明不存在3階序列相關(guān)性。運用廣義差分法進行自相關(guān)的處理，采用科克倫一奧科特迭代法。2階廣義差分的估計過程為：回歸結(jié)果如下：DependentVariable:MMethod:LeastSquaresDate:03/28/09Time:14:07Sample(adjusted):19802001Includedobservations:22afteradjustmentsConvergenceachievedafter5iterationsVariableCoefficientStd.Errort-StatisticProb.C169.321044.390073.8143890.0013GDP0.0197920.00107318.452500.0000AR(1)1.1081770.1812466.1142270.0000AR(2)-0.8011940.221892-3.6107360.0020R-squared0.982325Meandependentvar890.0591AdjustedR-squared0.979379S.D.dependentvar661.6499S.E.ofregression95.01304Akaikeinfocriterion12.10887Sumsquaredresid162494.6Schwarzcriterion12.30724Loglikelihood-129.1976F-statistic333.4596Durbin-Watsonstat1.853364Prob(F-statistic)□.000000InvertedARRoots.55+.70i.55-.兀li表明模型已經(jīng)不存在序列相關(guān)性。2、中國1980-2000年投資總額X與工業(yè)總產(chǎn)值Y的統(tǒng)計資料如表4.4所示。表4.4中國1980-2000年投資總額X與工業(yè)總產(chǎn)值Y單位：億元年份全社會固定資產(chǎn)投資X工業(yè)增加值Y年份全社會固定資產(chǎn)投資X工業(yè)增加值Y1980910.90001996.50019915594.5008087.1001981961.00002048.40019928080.10010284.5019821230.4002162.300199313072.3014143.8019831430.1002375.600199417042.1019359.60

19841832.9002789.000199520019.3024718.3019852543.2003448.700199622913.5029082.6019863120.6003967.000199724941.1032412.1019873791.7004585.800199828406.2033387.9019884753.8005777.200199929854.7135087.2119894410.4006484.000200032917.7339570.3019904517.0006858.000（1）當設(shè)定模型為lnY=p+PlnX+u時，是否存在序列相關(guān)性t01tt（2）若按一階自相關(guān)假設(shè)，使用廣義最小二乘法估計原模型。（3）采用差分形式X;=X"Xf1，Y*=Y-Y1作為新數(shù)據(jù)，估計模型Y*=a+aX*+v，該模型是否存在序列相關(guān)?t01對方程進行回歸分析，結(jié)果如下：DependentVariable:LOG(Y)Method:LeastSquaresDate:07/03/08Time:19:30Sample:19802000Includedobservations:21VariableCoefficientStd.Errort-StatisticProb.C1.4521090.1909257.6056450.0000LOG(X)0.8704190.02172740.061870.0000R-squared0.988300Meandependentvar9.031179AdjustedR-squared0.987684S.D.dependentvar1.062296S.E.ofregression0.117089Akaikeinfocriterion-1.347752Sumsquaredresid0.264059Schwarzcriterion-1.248274Loglikelihood16.15140F-statistic1604.953Durbin-Watsonstat0.451709Prob(F-statistic)0.000000由上面的結(jié)果可以看出，DW=0?45，小于顯著性水平為5%下，樣本容量為21的DW分布的下限臨界值d=1.22。因此，可判定模型存在一階序列相關(guān)。i（2）運用廣義最小二乘法估計模型，過程如下：

得到如下輸出結(jié)果：DependentVariable:LOG(Y)Method:LeastSquaresDate:07/03/08Time:19:44Sample(adjusted):19812000Includedobservations:20afteradjustmentsConvergenceachievedafter21iterationsVariableCoefficientStd.Errort-StatisticProb.C1.1262130.4141942.7190470.0146LOG(X)0.9035380.04430320.357950.0000AR(1)0.6495610.1456794.4588360.0003R-squared0.995501Meandependentvar9.102781AdjustedR-squared0.995061S.D.dependentvar1.036599S.E.ofregression0.072853Akaikeinfocriterion-2.263273Sumsquaredresid0.090228Schwarzcriterion-2.113913Loglikelihood25.63273F-statistic1914.828Durbin-Watsonstat1.348354Prob(F-statistic)0.000000InvertedARRoots.65則回歸模型表達式為：lnY=1.126+0.904lnX+0.65AR(1)運用拉格朗日乘數(shù)檢驗模型是否還存在一階序列相關(guān)性，過程如下:

glEViews-[Equation:UKTITLEDTorkfile:UNTITLED::Untitled\]FileEditObjectViewProc^uick0e+idileWindowHelp鼬辿Prmjgbj巳匚11|pri「it||Na「「iE：||FfE8EE||Esti「「i，iiE||For口匚ast||5tats||RmnidsFlejirh5entationsEstimatioilOutputActu：dl.Fitted,Reeidual*AFJilAStructure...GradienteandDerivatives*Cuy：di-i：±tlcpMatrix-erationsCoefficientTests*itStd.Errort-StatisticProb.ReeidualTests*Correlogj-:din-Q-etatisticeStabi1ityTests*CorrelogramSquaredResidualsLabelHistugi-:din-lTurm：dlityTestSerialCorrelationLMTest...R-squared0.99551AdjustedR-squared0.99501S.E.ofregression0.0728：ARCHLMTest...WhiteHeteruskedasticityl：nncroeeterms.]WMthHeteruskedasticityI'.crussterms.'1nL-i---..nddnnd檢驗一階滯后項檢驗部分結(jié)果如下:Breusch-GodfreySerialCorrelationLMTestF-statistic2.197567Prob.F(1,16)0.157656Obs*R-squared2.415232Prob.Chi-Square(l)0.120160從伴隨概率值可以看出，此時已經(jīng)不存在一階序列相關(guān)性了。（3）采用差分形式估計模型，過程如下：

回歸結(jié)果如下：DependentVariable:D(Y)Method:LeastSquaresDate:07/03/08Time:19:55Sample(adjusted):19812000Includedobservations:20afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C291.0347350.10330.8312820.4167D(X)0.9920730.1602356.1913800.0000R-squared0.680473Meandependentvar1878.690AdjustedR-squared0.662721S.D.dependentvar1835.505S.E.ofregression1065.983Akaikeinfocriterion16.87582Sumsquaredresid20453746Schwarzcriterion16.97539Loglikelihood-166.7582F-statistic38.33318Durbin-Watsonstat1.620070Prcib(F-statistic)0.000008此時的DW值為1.62，大于5%的顯著性水平下容量為20的DW檢驗的臨界值上限氣1.41，因此差分形式的模型不存在一階序列相關(guān)性。3、某上市公司的子公司的年銷售額Y與其總公司年銷售額X的觀測數(shù)據(jù)見表4.5.表4.5某上市公司的子公司的年銷售額Y與其總公司年銷售額X序號XY序號XY1127.320.9611148.324.54213021.412146.424.33132.721.9613150.2254129.421.5214153.125.64513522.3915157.326.366137.122.7616160.726.987141.223.4817164.227.528142.823.6618165.627.789145.524.119168.728.2410145.324.0120171.728.78用最小二乘法估計Y關(guān)于X的回歸方程；用DW檢驗分析隨機干擾項的一階自相關(guān)性；直接用差分法估計回歸模型的參數(shù)。首先運用OLS法估計Y關(guān)于X的回歸方程，結(jié)果如下:DependentVariable:YMethod:LeastSquaresDate:07/03/08Time:20:09Sample:120Includedobservations:20VariableCoefficientStd.Errort-StatisticProb.C-1.4547500.214146-67932610.0000X0.1762830.001445122.01700.0000R-squared0.998792Meandependentvar24.56900AdjustedR-squared0.998725S.D.dependentvar2.410396S.E.ofregression0.086056Akaikeinfocriterion-1.972991Sumsquaredresid0.133302Schwarzcriterion-1.873418Loglikelihood21.72991F-statistic14888.14Durbin-Watsonstat□734726Prob(F-statistic)0.000000回歸方程表達式為:Y=—1.4548+0.1763Xtt(-6.79)(122.01)

在5%的顯著性水平下，容量為n=20的DW分布的臨界值為《=1.201,氣=1.411，由于DW=0.735<dL，所以該模型存在一階正自相關(guān)。直接運用差分法對原模型進行估計，過程如下：

回歸結(jié)果如下：DependentVariable:D(Y)Method:LeastSquaresDate:07/03/08Time:20:14Sample(adjusted):220Includedobservations:19afteradjustmentsVariableCoefficientStd.Errort-StatisticProb.C0.0405280.0226421.7099590.0913D(X)0.1507830.00724821.907560.0000R-squared0.965791Meandependentvar0.411579AdjustedR-squared0.963778S.D.dependentvar0.344146S.E.ofregression0.065498Akaikeinfocriterion-2.514302Sumsquaredresid0.072929Schwarzcriterion-2.414088Loglikelihood25.80587F-statistic479.9410Durbin-Watsonstat1.740834Prcib(F-statistic)0.000000在5%的顯著性水平下，容量為19的DW檢驗的臨界值的下限與上限分別為《=1.18,du=1.40，DW=1.75＞匕，所以模型已經(jīng)不存在一階序列相關(guān)性。估計的回歸模型表達式為：D(Y)=0.0405+0.1588D(X)(1.8)(21.9)4、下表列出了1978年至1998年我國城鄉(xiāng)居民儲蓄存款年底余額(單位：億元)和GDP指數(shù)(1978年=100)的歷史統(tǒng)計資料：建立居民儲蓄存款模型檢驗模型的自相關(guān)性

表4.6我國城鄉(xiāng)居民儲蓄存款與GDP指數(shù)統(tǒng)計資料年份存款余額YGDP指數(shù)X年份存款余額YGDP指數(shù)X1978210.610019895146.9271.31979281107.619907034.2281.71980399.511619919107307.61981523.7122.1199211545.4351.41982675.4133.1199314762.39398.81983892.5147.6199421518.8449.319841214.7170199529662.25496.519851622.6192.9199638520.84544.119862237.6210199746279.858219873073.3234.3199853407.47638.219883801.5260.7運用OLS法估計模型，并選擇統(tǒng)計檢驗結(jié)果較好的模型。經(jīng)過比較，取雙對數(shù)模型。估計的結(jié)果如下：DependentVariable:LOG(Y)Method:LeastSquaresDate:10Z27/09Time:09:19Sample:19781998Includedobservations:21VariableCoefficientStd.Errort-StatisticProb.C-8.1081590.253921-31.931870.0000LOG莒2.9651970.04581564720710.0000R-squared0.995485Meandependentvar8.236497AdjustedR-squared0.995247S.D.dependentvar1.756767S.E.ofregression0.121116Akaikeinfocriterion-1.293736Sumsquaredresid0.278715Schwarzcriterion-1.194257Loglikelihood15.58422F-statistic4188770Durbin-Watsonstat0.740145Prob(F-statistic)0.000000回歸方程表達式為：InY=—8.1082+2.9652lnXtt(-31.93)(64.72)R2=0.995,R=0.995,DW=0.740進行序列相關(guān)性檢驗，作殘差項圖:RejireeentatioileActual.,Fitted,ReeidualEetimatiunLhitputActual.Fitted,Actual.,Fitted,ReeidualActual.Fitted,Rp5idualGraphUUHLL1ClHILLiHiLiReeidu：dlTests?:90.253921-31.931870.0000StabilityTests10.04581564.720710.0000Label15Meandependentvar8.236497AdjustedR-squared0.995247S.D.dependentvar1.756767S.E.ofregression0.121116Akaikeinfocriterion-1.293736Sumsquaredresid0.270715Schwarzcriterion-1.194257Loglikelihood15.50422F-statistic4188.770Durbin-Watsonstat□740145Prob(F-statistic)0.000000b.AJJilAStructm-p...Gradientsand.Derivatives>Cnvari:±tlceMatrixReeidu：dlGraphSt:dXLil：±rdizedF：gmidnalGraphILOLU.I_IIUIL-OLdlliLIL-FIU得到:殘差項二呈現(xiàn)有規(guī)律的波動，預示著可能存在自相關(guān)性。以及e盧e1的關(guān)系圖:RESID(-1)運用相關(guān)圖和Q統(tǒng)計量進行檢驗：滯后期選擇10：Ce,e,,e得到殘差t與滯后值t-1t-2t-10的各期相關(guān)系數(shù)和偏相關(guān)系數(shù)圖:Date:10/27/09Time:09:42Sample:19781998Includedobservations:21AutocorrelationPartialCorrelationACPACQ-StatProbI11110.5160.5166.42580.011I11112-0.092-0.4886.64120.03611_11113-0.359-0.06610.1030.01811_11114-0.326-0.12213.1230.0111匚11匚15-0.240-0.20814.8680.0111匚11匚16-0.180-0.16715.9090.0141111[17-0.073-0.07716.0930.0241□111180.1240.04816.6600.0341_111■190.3360.16221.2040.0121ZJ1111100.299-0.05425.1350.005可以明顯地看出，存在著一階和二階自相關(guān)性。各階滯后的Q統(tǒng)計量的p值都小于0.05,說明在5%的顯著性水平下，拒絕原假設(shè)，殘差序列存在序列相關(guān)。運用B-G檢驗法：RejireserLtatioileEstimationOutputActual.,Fitted,Residual*AFJilAStrueti±re...GradientsandDerivatives*C口variar”：eMatrixitStd.Errort-StatisticProb.CuefficipniTestsResidualTestsStabilityTestsLabelAdjustedR-squaredS.E.ofregressionSumsquaredresidLoglikelihoodDurbin-Watsonstat]Correlogram-Q-statisties*CorrelngramSqu：di-edRe，idualsHiEti：igr：diri一Ni：irm：dlityTest0.99520.12110.270715.504：Seri：±1CorrelationLMTest...ARCHLMTest...WhiteHeteroskedasticity(nocrossterms)Whi_teHeteroskedaEticity(crosEterms)□740145Pr北(F-statistic)0.000000并選擇滯后期為2期，估計結(jié)果如下:Breusch-GodfreySerialCorrelationLMTestF-statisticObs*R-squared9.00845110.80492Prob.F(2,17)Prob.Chi-Square(2)0.0021500.004505TestEquation:DependentVariable:RESIDMethod:LeastSquaresDate:10/27/09Time:09:58Sample:19781998Includedobservations:21Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProb.C0.0017430.1910140.0091230.9928LOG(X)-□.□□□5860.034536-0.0169770.9867RESIDf-1)0.8920100.2121854.2039370.0006RES*)-0.5909380.215317-2.744499R-squared0.514520Meandependentvar-1.52E-15AdjustedR-squared0.428847S.D.dependentvar0.118050S.E.ofregression0.089216Akaikeinfocriterion-1.825877Sumsquaredresid0.135310Schwarzcriterion-1.626920Loglikelihood23.17171F-statistic6.005634Durbin-Watsonstat1.511332Prob(F-statistic)0.005546其中，LM⑵=nR2=10.80492，臨界概率p=0.004505，認為輔助回歸模型是顯著的，即存在自相關(guān)性。又因為匕一「匕一2的回歸系數(shù)均顯著地不為0，表明居民存款模型存在一、二階自相關(guān)性。自相關(guān)性的具體形式為：e=0.0017-0.0006lnx+0.8920e廣0.5909e2(0.0091)(-0.0170)(4.2039)(-2.7445)三、多重共線性1.根據(jù)表4.7數(shù)據(jù)，分析我國居民家庭電力消耗量與可支配收入及居住面積的關(guān)系，以用來預測居民家庭對電力的需求量。表4.7我國居民家庭電力消耗量與可支配收入及居住面積資料年度年人均家庭電力消耗量(千瓦小時)Y人均居住面積(平方米)X1年人均可支配收入指數(shù)(1978年=100)X2198521.212.45243.17198623.213.02254.28198726.413.49265.39198831.213.94277.61198935.314.42273.49

199042.414.87281.33199146.915.44289.71199254.615.64307.66199361.216.99321.07199472.716.65339.33199583.517.25356.58199693.117.82383.951997101.818.33399.85(1)分別作Y對X1、Y對X2的回歸方程:Y對X1的回歸結(jié)果為：DependentVariable:YMethod:LeastSquaresDate:10/27/09Time:10:31Sample:19851997Includedobservations:13VariableCoefficientStd.Errort-StatisticProb.C-161.285915.87817-10.157720.0000X113.929491.02331413.612140.0000R-squared0.943961Meandependentvar53.34615AdjustedR-squared0.938866S.D.dependentvar27.26563S.E.ofregression6.741502Akaikeinfocriterion6.795081Sumsquaredresid499.9263Schwarzcriterion6.881996Loglikelihood-42.16802F-statistic185.2903Durbin-Watsonstat1.031819Prob(F-statistic)0.000000Y對X2的回歸結(jié)果為：DependentVariable:YMethod:LeastSquaresDate:10/27/09Time:10:31Sample:19851997Includedobservations:13VariableCoefficientStd.Errort-StatisticProb.C-113.00225.580299-20.364380.0000X20.5441270.01797530.271220.0000R-squared0.980138Meandependentvar53.34615AdjustedR-squared0.987060S.D.dependentvar27.26563S.E.ofregression3.101594Akaikeinfocriterion5.242340Sumsquaredresid105.0188Schwarzcriterion5.329263Loglikelihood-32.07526F-statistic916.3460Durbin-Watsonstat1.071197Prob(F-statistic)0.000000可見，收入和住房面積對電力都有很好的解釋作用。(2)作Y對X1和X2的二元回歸方程：

DependentVariable:YMethod:LeastSquaresDate:10Z27/09Time:10:36Sample:19851997Includedobservations:13VariableCoefficientStd.Errort-StatisticProb.C-125.35300.362488-14.909920.0000X12.8085951.6059941.7400200.1109X20.4400500.0613167.1897490.0000R-squared0.990916Meandependentvar53.34615AdjustedR-squared0.989100S.D.dependentvar27.26563S.E.ofregression2.846668Akaikeinfocriterion5.129350Sumsquaredresid01.03519Schwarzcriterion5.259723Loglikelihood-30.34077F-statistic545.4302Durbin-Watsonstat1.338435Prob(F-statistic)0.000000回歸方程表達式為：j=—125.3530+2.808595氣+0.44085%(-14.98992)(1.74882)(7.189749)R2=0.990916R2=0.9891DW=1.338435F=545.4382X1的回歸系數(shù)不顯著；F統(tǒng)計量的值顯示X1和X2對Y的共同影響是顯著的。多重共線性檢驗CorrelationMatrixYX1X2Y1.0000000.9715760.994051X10.9715761.0000000.963124X20.9940510.9631241.000000X1和X2的相關(guān)系數(shù)高達0.963124，兩者高度正相關(guān)。輔助回歸模型檢驗，作X1與X2的散點圖，并將X1對乂2作簡單回歸:

400-360-320-230-回歸結(jié)果為:400-360-320-230-DependentVariable:X1Method:LeastSquaresDate:10Z27/09Time:10:40Sample:19851997Includedobsen/ations:13VariableCoefficientStd.Errort-StatisticProb.C4.1126560.9629224.2710160.0013X20.0367720.00309711.872260.0000R-squared0.927608Meandependentvar15.40846AdjustedR-squared0.921027S.D.dependentvar1.901766S.E.ofregression0.534437Akaikeinfocriterion1.725433Sumsquaredresid3.141853Schwarzcriterion1.012340Loglikelihood-9.215313F-statistic140.9505Durbin-Watsonstat0.856038Prcib(F-statistic)0.000000x=4.1127+0.0368%1t2t(4.2710)(11.8723)R2=0.9276DW=0.8560F=140.9505住房面積與收入之間存在顯著的線性關(guān)系。VIF=—^=1=13.81368因此該模型存在嚴重方差膨脹因子11-R121-0-9276082，大于10，的多重共線性。因此該模型存在嚴重變換模型的形式

對原設(shè)定的模型進行適當?shù)淖兓?，可以消除或削弱原模型中解釋變量之間的相關(guān)關(guān)系。利用對數(shù)模型擬合上述數(shù)據(jù)，結(jié)果顯示如下：DependentVariable:LOG（Y）Method:LeastSquaresDate:10/27/09Time:10:57Sample:19051997Includedobservations:13VariableCoefficientStd.Errort-StatisticProb.C-10.090901.151730-8.7615100.0000LOG(X1J3.0080500.5745245.2357370.0004LOG(X2J1.0035090.4541522.2096340.0516R-squared0.908284Meandependentvar3.850234AdjustedR-squared0.905941S.D.dependentvar0.532496S.E.ofregression0.063139Akaikeinfocriterion-2.407767Sumsquaredresid0.039866Schwarzcriterion-2.357394Loglikelihood19.17049F-statistic421.7588Durbin-Watsonstat2.193484Prob(F-statistic)0.000000回歸方程表達式為：Inj=—10.0910+3.0081ln氣+1.0035ln七（6.4.4）（-8.7615）（5.2357）（2.2096）R2=0.9883R2=0.9859dW=2.1935F=421.7588在對數(shù)模型中，收入和住房面積系數(shù)在統(tǒng)計上都是顯著的，回歸模型在整體上也是顯著的。說明我們原先設(shè)計的線性回歸模型是有誤的。2、根據(jù)理論和經(jīng)驗分析，影響糧食生產(chǎn)（Y）的主要因素有農(nóng)業(yè)化肥使用量（X1）、糧食播種面積（X2）、成災面積（X3）、農(nóng)業(yè)機械總動力（X4）、農(nóng)業(yè)勞動力（X5），其中，成災面積的符號為負，其余均應為正。表4?8列出了中國糧食生產(chǎn)的相關(guān)數(shù)據(jù)，擬建立中國糧食生產(chǎn)函數(shù)。表4.8中國糧食生產(chǎn)與相關(guān)投入資料年份糧食產(chǎn)量/萬噸農(nóng)業(yè)化肥使用量/萬千克糧食播種面積/千公頃成災面積/公頃農(nóng)業(yè)機械總動力/萬千瓦農(nóng)業(yè)勞動力/萬人198338728.001659.800114047.016209.3018022.0031645.10198440731.001739.800112884.015264.0019497.0031685.00198537911.001775.800108845.022705.3020913.0030351.50198639151.001930.600110933.023656.0022950.0030467.00198740208.001999.300111268.020392.7024836.0030870.00198839408.002141.500110123.023944.7026575.0031455.70198940755.002357.100112205.024448.7028067.0032440.50199044624.002590.300113466.017819.3028708.0033330.40199143529.002806.100112314.027814.0029389.0034186.30199244264.002930.200110560.025894.7030308.0034037.00

199345649.003151.900110509.023133.0031817.0033258.20199444510.003317.900109544.031383.0033802.0032690.30199546662.003593.700110060.022267.0036118.0032334.50199650454.003827.900112548.021233.0038547.0032260.40199749417.003980.700112912.030309.0042016.0032434.90199851230.004083.700113787.025181.0045208.0032626.40199950839.004124.300113161.026731.0048996.0032911.80200046218.004146.400108463.034374.0052574.0032797.50設(shè)糧食生產(chǎn)函數(shù)為:Y=。+pX+pX+pX+pX+pX+u01122334455用OLS法估計模型，結(jié)果如下：DependentVariable:YMethod:LeastSquaresDate:07/03/08Time:20:41Sample:19832000Includedobservations:18VariableCoefficientStd.Errort-StatisticProb.C-12815.7514078.90-0.9102800.3806X16.212562□7408818.3853730.0000X20.4213800.1269253.3199190.0061X3-0.1662600.059229-2.8070650.0158X4-0.0977700.067647-1.4452990.1740X5-0.0284250.202357-0.1404710.8906R-squared0.982798Meandependentvar44127.11AdjustedR-squared0.975630S.D.dependentvar4409.100S.E.ofregression688.2984Akaikeinfocriterion16.16752Sumsquaredresid5685056.Schwarzcriterion16.46431Loglikelihood-139.5077F-statistic137.1164Durbin-Watsonstat1.810512Prob(F-statistic)0.000000對應的回歸表達式為：Y=—12815.75+6.213X+0.421X-0.166X—0.098X—0.028X(-0.91)(8.39)123(3.32)(-2.81)45(-1.45)(-0.14)從上面的結(jié)果可以看出，X4和X5前的參數(shù)估計值未能通過t檢驗，而且符號的經(jīng)濟意義也不合理，故認為解釋變量間存在多重共線性。對X1，X2,X3,X4,X5進行簡單的相關(guān)系數(shù)檢驗，過程如下:X1X2X3X4X5X11.0000000.0118230.6401750.9602780.545450X20.0118231.000000-0.454908-0.0384790.182359X30.640175-0.4549001.0000000.6095650.355735X40.960270-0.0384790.6895651.0000000.454169X50.5454500.1823590.3557350.4541691.000000得到如下結(jié)果:得到如下結(jié)果:從上面的結(jié)果可以看出，X1與X4之間存在高度相關(guān)性。接下來找出最簡單的回歸形式。分別做出V與X1,X2,X4,%間的回歸，結(jié)果如下：

DependentVariable:YMethod:LeastSquaresDate:07/04/08Time:13:47Sample:19832000則：則：Y=30867.31+4.576XVariableCoefficientStd.Errort-StatisticProb.C30867.311206.36425.587060.0000X14.5761150.39819911.492020.0000R-squared0.891941Meandependentvar44127.11AdjustedR-squared0.885187S.D.dependentvar4409.100S.E.ofregression1493.984Akaikeinfocriterion17.56072Sumsquaredresid35711799Schwarzcriterion17.65965Loglikelihood-156.0465F-statistic132.0666Durbin-Watsonstat1.855174ProbfF-statistic)0.000000i(25.59)(11.49)R2=0.8919,F=132.1,DW=1.86(2)DependentVariable:YMethod:LeastSquaresDate:07/04/08Time:13:48Sample:19832000Includedobservations:18VariableCoefficientStd.Errort-StatisticProb.C-33822.4168409.15-0.4944140.6277X20.6988800.6132731.1395900.2712R-squared0.075073Meandependentvar44127.11Adjus