回歸模型分析報(bào)告背景意義：教育是立國(guó)之本，強(qiáng)國(guó)之基。隨著改革開(kāi)放的進(jìn)行、經(jīng)濟(jì)的快速發(fā)展和人們生活水平的逐步提高，“教育”越來(lái)越受到人們的重視。一方面，人均國(guó)內(nèi)生產(chǎn)總值的增加與教育經(jīng)費(fèi)收入的增加有著某種聯(lián)系，而人口的增長(zhǎng)也必定會(huì)對(duì)教育經(jīng)費(fèi)收入產(chǎn)生影響。 本報(bào)告將從這兩個(gè)方面進(jìn)行分析。我國(guó)1991年~2013年的教育經(jīng)費(fèi)收入、人均國(guó)內(nèi)生產(chǎn)總值指數(shù)、年末城鎮(zhèn)人口數(shù)的統(tǒng)計(jì)資料如下表所示。試建立教育經(jīng)費(fèi)收入 Y關(guān)于人均國(guó)內(nèi)生產(chǎn)總值指數(shù)Xi和年末城鎮(zhèn)人口數(shù)X2的回歸模型，并進(jìn)行回歸分析。年份Y（億元）X1(1978年=100)X2（萬(wàn)人）1991731.50282256.67312031992867.04905289.723217519931059.93744326.323317319941488.78126364.913416919951877.95011400.63517419962262.33935435.763730419972531.73257471.133944919982949.05918503.254160819993349.04164536.944374820003849.08058577.644590620014637.66262621.094806420025480.02776672.995021220036208.2653735.845237620047242.59892805.25428320058418.83905891.315621220069815.30865998.7958288200712148.06631134.6760633200814500.737421237.4862403200916502.70651345.0764512201019561.847071480.8766978201123869.293561613.6169079201228655.305191730.1871182201330364.718151853.9773111資料來(lái)源：中經(jīng)網(wǎng)統(tǒng)計(jì)數(shù)據(jù)庫(kù)

2,000-nn1000020,00030,00040.00080,00070,00060,0005090040900309001U口002,000-nn1000020,00030,00040.00080,00070,00060,0005090040900309001U口0020,00030,000 40月00根據(jù)經(jīng)濟(jì)理論和對(duì)實(shí)際情況的分析可以知道，教育經(jīng)費(fèi)收入 Y依賴于人均國(guó)內(nèi)生產(chǎn)總值指數(shù)Xi和年末城鎮(zhèn)人口數(shù)X2的變化，因此我們?cè)O(shè)定回歸模型為?*=??+??????+??????+???應(yīng)用EViews的最小二乘法程序，輸出結(jié)果如下表DependentVaria&ie:YMethod:L&astSquaresDate:1321/15Time:09:30Sample:19912013Includedobservations:23VariableCoefficientStd.Errort-StatisticPfgC5059.8551B66J77 2.6S1774O.OU338.749031.811701 15.86&S60.0000X2-0.3081760.065569 -63&16330.0000R-squarad0999147Meandepenoer?t^ar9059646AdjustedR-squarea09S8062S.Ddependentvar0050.661S.E.ofregressionQ&sgoosAkaikeinfoentenon15.75217Sumsquaredresid19558484Schwarzcriterior16.90028Losjiiteiihood-139.6500Hanran-auinncriter15.78942F-statistic9114038Curbin-Watsonstat0,549316ProbtF-statistic)0.000000??=5058.835+28.7491??1?，0.3982??2??(2.68) (15.9) (-6.1)R2=0.99 ??=0.99 F=911.4異方差的檢驗(yàn)1.Goldfeld-Quandt檢驗(yàn)Xi和X2的樣本觀測(cè)值均已按照升序排列，去掉中間X1和X2各5個(gè)觀測(cè)值,用第一個(gè)子樣本回歸：DependentVariable:YMethod:LeastSquaresDate-12/21/15Time:09:33Sample:19911999includedobservations:：9VariableCoefficientStd.Errort-StatishcPron.C-3510658573.0425 -5.Z1S1160.0020Z159095401.58Z534 3.7341090.0097X29.0839200035055 2.39328500538R-squared0.593516Meandependentvar1901.S33AdjustedR-squared09Q1255S.D.dependent\ar937.965SS.E.ofregression87.21013Akaikeinfocriterion12.03572Sumsquaredresid45633.64Schwarzcriterion12,10146Loglikelihood-5116074Harnan-Quinncriter.1189^85Fatalistic4597017Durbin-Watsonstat1554407Prob(F-statistic)o.oo0000??=-3510.668+5.9096?%?#0.0839??2??SSE=45633.64用第二個(gè)子樣本回歸:DependentVariable:YMethod:LeastSqua「esDate:12721/15Time:□9:34Sample：20052C13Includedobservation口gVariableCoemdentStd.Errcr 卜StatisticProc.C17653651100642 162302201557X1107,536147.71512 2.2547550.0550X2-4.74879727069S2 -17&427701299R-squared0967065Meandependentvar1S20409AdjustedR-squared09627539Ddependent^ar79H7917S.Eofregression1049.029AJcakkftInfocritsncn17.01034Sumsquaredresid6602890.Schwarzcriterion17.0760SLoglikelihood-73.54652Hannar-Quinnalter1636347F-statislic2289229Durbin-WatBonstat1.923931Prob(F-staflStic)0.000002緋178636.6+107.5861??i?74.7488??2??SSE2=6602898Ho=ut具有同方差，H1=ut具有遞增型異方差構(gòu)造F統(tǒng)計(jì)量。F=22=黑翳=114.7>F。.。5(9,9)=3.18*+5UDDW2.所以拒絕原假設(shè)，計(jì)量模型的隨機(jī)誤差項(xiàng)存在異方差2.White檢驗(yàn)因?yàn)槟Ｐ椭泻袃蓚€(gè)解釋變量，輔助回歸式一般形式如下

???=??+?????1+?2???2+?????2+?????2+????????2+???輔助回歸式估計(jì)結(jié)果如下HeteroskedasticitfTest;WhiteF-statistic2942706ProbF(517)0.0430Obs*R-&quared10,67089FronChbSquare(5)00S83ScaledexplainedSS6726284Prob.Chi-Square(5)0541STestEquafion:DependentVariable:RESIDEMethod:LeastSquaresDale:12/21/15Time:09:38Sample:19912C13Includedobservations22VariableCoefficientStd.Errort-StatisticProt).C-1245277630348332?0410328O.S067X140478.2372466.12-0.55S5S10.5S37X1*2-19919572&.91602-0.6S42Q40.6217X1*X21.36334022373120.60936505503X210674322249.9100.474454口6412X2A2-002023500384B7■0.525770oeosaR-squarea口483951Meandepena&nty/ar850368.9AdjustetlR-s(iuared0306290S.D.dependentvar1122691.S.Eolregression935081.3Akaikeinfocriterion3055411Sumsquared「esid1.49E+13SchwarzcriteriQn3685033Leglikelihood-345.3723Hannan-Ouinncriten3052961F-statistic2942706Ournin-Watsonstat2.E44510Prot)[F-&tatisticj0043027??2=-12452776-40478.23????1+1067.432????718.9196????2-0.0202??)?2+1.3633???????2因?yàn)門(mén)R2=10.67>??i(5)=9.236該回歸模型中存在異方差3.克服異方差以1/Xi做加權(quán)最小二乘估計(jì)，Dependent''Jariabile:YMethcd:Lea￡tSquaresDate:12/21/15Time:1O;Q8Sample:19912013Includedobservations:23Weighilngsenes:1/K1Weighttype:Inversevariance(averagescaling)VariableCoefficientStd.Errort-StatistiGFrob,C3678.2011412.405274581300125X127.024571.B820421B.06653a.ooooX2-0,3461540.054423-6260297o.ooaoWeightedStatisticsR-scjLia「eci0.987992Meandependentwar65B5.27SAdjustedR-squaredo.g艙7glS.D.dependsnV-ar4981.347SEofregression工-9即5Akaikeinfocriterion16.20956Sumsquaredresid-11367925Schwarzcriterion1635767Loglikelihood-183.4099Hannan-Quinnenter1624691F-statistic8227495DurbinAATdbofistat0472699Frob{F-statistic)0.000000Weighedm€andep.5093.284UnweightedStatisticsR-3Q問(wèn)的3.968433Meandependentwar9059.646AdjustedR，squa「ed0.987331S.D.dependent;ar9050.681S.E.ofregression1019.697Suesquaredresid20754674DurfilivWats&nstat0521574估計(jì)的結(jié)果還原變量，得領(lǐng)〒3878.201+27.0246??1?70.3462?%?再用上表對(duì)應(yīng)的殘差做 White檢驗(yàn)

HeteroskedasticityTest^statisticObs^'squaredScaledexplainedSS206942237023303985003Prob.F(5.17)Prob.Chi'Square(5)Prob.Chi-Square(5)0.119701215Q.5516TestEquation.DependertVariable：啊GT_RESID上2MethodLeastSquaresDate:12/21/15Time:10:12Sample:19912013Indudedobservations:23CollineartestregressorsdroppedfrotspeaficationVariableCoefficientStdErrorl-StafisticFrobC1113169.21479597 0,0516240.9595WGT*23574g8215S10501 0.2290100.92102.66269217.85124 0.20517903299X1fiX2*WGTft2-01374551.263073 -010970269139X2A2*WGTA20D031780.021258 01494310,S830X2*WGTn2-190.95511171.S36 -016294008725R-squared0.378362Meandependentvar494257.6adjustedR-squared0.195527S.D.dependentsar556180.0S.E.ofregression+98S511Akaikeinf。criteriun2929746Sum陷呵“resid423E+12Schwarzcriterion29,59368Loglikelihood-330.9208Hannan-Quinncrikr.23,37196F-曲aflstiu2D69422Durbin-Watsonstat2507731Prob(F-statistic)0119712由上表可知TR2=8.7<??i(5)=9.236,說(shuō)明以及克服了異方差性自相關(guān)的檢驗(yàn)1.DW檢驗(yàn)

DependentVariable:YMethcd-LeastSquaresDatK12/21/15Time-10:08Samiple:19912013Indudedobsefvations:23Weightingseries:1W11Weigtiltype:Inversevariance(averagescaling)VariableCqefficientStlErrort-StatistiGProb.c3876.2011412405 2.7458130.0125X127.024571.&82C42 16,C&6530.0000X2-0.3451540.0S4423 -6.3C0397a.ooooWeightedStatisticsR'Squared0987992Meandependent^ar5535.276MustedR-squared0.985791S.D.dependentvar4901.347S.E.ofregression753.5206Akaikeinfccriterion16,20956Sumsquaredresid11367925Schwarzcriterion1635767Loglikelihood-1B34Q99Hannan-Quinnenter.116.24691F-statistic8227495Durbin-Watsonstat0472639Prob(F^tatistic)0.000000'.Vsightedmeandep5093.234UnweightedSt曲stiesR-squared0.983483Meandependentvar90瑜645AdjustedR^squared0.987331S.D.dependentvar9050.6B1S.E.ofregression1019.697Sumsquaredresid20764874Durbln-Wateon」at0521574因?yàn)橐阎狣W=0.47,若給定產(chǎn)0.05,查表得DW檢驗(yàn)的臨界值dL=1.17,du=1.54。因?yàn)镈W=0.47<1.17,根據(jù)判別規(guī)則，認(rèn)為誤差項(xiàng) ut存在嚴(yán)重的正自相關(guān)。2.LM檢驗(yàn)

3.F-statisticObsfcR-squared3.4597216364184Prob<FC217)ProbChi-Square(2)3.F-statisticObsfcR-squared3.4597216364184Prob<FC217)ProbChi-Square(2)0.054900415TestEquation.Depend&ntVariatile:RESIDruiethodLeastSquaresDate:12/21/15Time:12:31Sample:19922013Inducedobsen/attons:22Presamplemissingvaluelaggedresidualssettozero.VariableCoefficientStd.Errort-StatisticProbC-706.27061164736 聞￡06378口552mX1-0765*X1(-1)4602248331B005 -1.2054070.2446X2-0765"X2(-1)0.1320500151557 0.8713560.3557RESID(-1)0.9578070364S09 5.62910S0.0176RESID(-2)-1.2615700576139 -2J896970.0428R-sguaredQ.20H281n&and€pen{i&ntvar^2.48E-13AdjustedR-squared0122053SlD.dependentvar6528291SEcfregression6115932Akaikeinfocrlierlon15.B6706Sumsquaredresid6360866Scnwarzcrltenon10.11502Loglikelihood-1G9,5376Hannan-Quinnenter.1M2547F-statistic172985QDwrtjin-Watsanstat1709870臼。3仔褊而鼠冏口,189832Breusch-CactfrejjSerialCorrelationLMTest:LM=6.36>??Q5（2）=5.991所以誤差項(xiàng)存在二階自相關(guān)克服自相關(guān)首先估計(jì)自相關(guān)系數(shù)??=????1-h一1一0.472=0.765對(duì)原變量做廣義差分變換。令GDYt=Yt-0.765Yt-iGDXit=Xit-0.765Xit-iGDX2t=X2t-0.765X2t-i以GDYt,GDXit,GDX2t(i992~20i3年)為樣本再次回歸Depend&ntVariable:Y-0765*Y(*1)Method.LeastSquaresDate:12/21H5Time:12:19Sample但咖磯皿19922013mciutiedobservations:22afteradjjslmentsVariableCoefficientStd.Error1-StatisticProb.CX1-0,766*X1r-1}X2^,765*X2G1)241,32202742970525 0.1S18563.&46518 75221630.156S39 -192S490249900030D0691R-squared0954230Meandependent\/ar3248.404AfljusteelR-squamd0.949412SD.dependentwar3051.463S.E.ofregression6S63290Afcaifcemfocriterion16C2672SumsquaredresidS949904.Schwarzcriterion161764^Loglivelihood-1732939Hannan-Quinnenter.160&VGFatalistic193.0535Durbin-Watsanstat1.368592ProbtF-statistic)0.009000得到GDYt=241.322+27.4297GDXit-0.3024GDX2tDW=1.4,介于dL=1.17,du=1.54之間，所以不能判別 3是否存在一階自相關(guān)，自相關(guān)性沒(méi)有消除由上一步LM統(tǒng)計(jì)量知誤差項(xiàng)存在二階自相關(guān)，采用直接擬合的估計(jì)結(jié)果是，D&pendentVariaBle:YMetfiodLeastSquaresDate:12J21/15Time:1238Sample(□djusted):19932013kidudadobseivations;21aft?radjustmentsConvergenceachisvadafter7iterationsVariableCoeflicientStdEnart-StetisticPrabc2610.3131928.3541.3536410.1947X124,908302.34358110,52830o.oocaX2-028989S0.073680-3.9345180.0012.網(wǎng)1)13898020.2990884.6374980.0003AR⑵-1.1527360.465390-2.47692400248R-squared0.096549Meandependent/ar9844.344AdjustedR-squared0.396696S.D.dependentvsr&09OJ249S.E.regressiori597.0527Akaikeinfocriterion1582614Gumsquaredresid5703551SchwaiT.criterion1807434Loglik&llilioad-1611745Hannan-Cuinnc「it白「.15,83012F-statistic1155.034□urbin-Wdsonstaf1751639Prob(F-statistic)0000000Inverted.iKRoots.69-.62I.69*82iEstimated isnonstationary舜2610.313+24.9083??i?70.2899??2??+1.3898????-i-1.1527????-2+???DW=1.75介于du=1.54和4-du=2.46,依據(jù)判別規(guī)則，誤差項(xiàng)已消除自相關(guān)

多重共線性的檢驗(yàn).Klein判別法Covariance.Analysis:OrdinaryDate:12/23y15Time:12:10Sample:19912013includedobservations:23CovarianceCorrelationXIX2X123237431.000000X2&22T7851.77E+0B99717271.000000因?yàn)閨rxix2|=0.97<R2=0.99,所以不存在多重共線性.修正Frisch法用每個(gè)解釋變量對(duì)被解釋變量做最小二乘回歸DependentVariable:YLlettiad.LeastSquaresDate：12/23TI5Time:12:16Sample:19912013Inductedobservations:23VariableCoefficientStd.Eror t-S1atistucProt>.C-&C97.3336600321 -3.966242acoocX118077700703142 25.709S90.0000R-squar&d0.969208Meandependentvar9059646AdjustedR-squared0.967742s.d.dependentvar9050.631S.E.ofregression1625.552Akaikeinfocriterion17.70S02Sumsquaredresid55490797Schwarzcriterion17,80676Laglikelihood-20t&423Hannan'Quinncriter.17732&6Ffitaifstia&60,9596Durbin-Watsonslat0.239431Prob(F-statistic)0.000000—??=-6097.333+18.0786??iR2=0.969 ??=0.968

DependentVariable:YMethodLeastSquaresDate:12/23/15Time12:20Sample:13912013Inducedobsen/aticins:23VariableCoefficientStd.Errort-SlatisticProb.C-21977092913204 -75439570.0000X20.5147250,055798 11,0170。0.0000R-squared5852502Meandep?ndentvar9059.646AdjustedR-squared0.&4547BS.D.dependentvar9050.601S.E.ofregression3557750Akaikeintocriterion19,27459Sumsquaredresid266E*O6Schwarzcriterion1037333Loglikelihood-219.657BHannan-Quinncriter.19,29942Fatalistic1213744Durbin-Watsonstat0122423Prob(Rstalistic)0.000000??=-21977.09+0.6147??2R2=0.852 ??=0.845取第一個(gè)方程為基本回歸方程，引入 X2,對(duì)Y做關(guān)于Xi和X2的最小二乘回歸,DependentVariable:￥Metliod:LeastSquaresDate：12/23/15Time:1224Sample:19S12013includedobservations:23VariableCoefficientStd.Errort-StatisticProb.C5058.8351885.377 2兆1774oai43K12874D0S1,311701 15.858560.0000X2-0-3W1760.065638 -5.0616380.0000R-squared09S9147M&andependentvar3059.646AdjuStsdR-sqtared0969062SDd&p&nd&ntvar90E0.681SEofr&gression968,9005-Xksikcinfocriterion16.75217Sumsquaredresid19558484Schwarzcriterion1&90028Loglikelihood-189.6500Hannan-Quinncriter.1673942F-statistic911.4038Durbin-Watsonstat054981&FirobtF-Elati￡tic)0000000??=5058.835+28.7491??i-0.3982??R2=0.989 ??=0.988可以看出，加入X2后，R2和？？均有所增加，Xi系數(shù)顯著性不受影響，所以在模型中保留X2綜上:

D&pendentVariaolB：VMetTiod.LeastSquaresDate:12J21/15Time：1238Sample(adjusted)-19932013Includedobservations:21aft?radjustmentsConvergerceachiev&dafter7iterationsR-squared0.996549R-squared0.996549AdjustedR-squared0995696S.E.ofregressionS97.0627Sumsquaredies：id5703551.Loglik&llihood-1611745F-statistic1155.034Prob(F-st3tistic>0000000Meandependentrar98M,3超S.D.dependentvar&C9C249Akaikeinfocriterion1502614Schwarzcriterion16,37434Hannan-Cuinnc「it白「.15S8012Durbin-Batsonstat1751639VariableCoefficientStdErrwt-StatisticPretb.c2610.3131928.3541.35364101947X124,9083023435sl10,52830o.oocoX2-0.2893960.073690-3.9345180.0012AR(1)1.3998020.2996884.637498O.OOC3.崛⑵-11527360.465350-2.4769240.0248Invertc口ARRoots .&9-.B2I .69^82iEstimated-Rprocessisnonstationary估計(jì)的回歸模型為緒2610.313+24.9083??1?70.2899?%?+1.38987???/-1.1527????-2+???模型總顯著性的F檢驗(yàn)Ho=31=匝=?=3??=0H1=例不全為零F=1155.034>Fo.o5(2,17)=3.59,拒絕H。，總體回歸方程存在顯著的線性關(guān)系模型單個(gè)回歸參數(shù)顯著性的 t檢驗(yàn)由上表看出，截距項(xiàng)的t檢驗(yàn)未通過(guò)，接受 H。，位=0皿？2610.313+24.9083??1?70.2899??2??+1.3898????-1-1.1527????-2+???Dependentvariable:YMethod:LeastSquaresDate：12^3/15Time：12:40Sample(adjusted):19932013Includedobservations:21afteradjustmentsConvergenceadiievedafter6iterationsVariableCoefficientStd.Error sticProb.X122.2470314B63B3 14967720.0000X24Mqs2fr20323459 ￡.3231070.0000AR(1)1.497*37Q.3410S1 43&C6470.0004AR⑵-1.1725550542590 ^216103200453R-sqjared0996187Meandependentvsr5846348AdjustedR-squared0935515SDdependentvarS090249SEofregression606.3020Akaikeinfocriterion1583051Sumsquaredresid63Q067SSchwarzcriterion16,02946La9likelihood-152.2203Hannan-Quinncriter.15,&7369Durbin-Wartsonstat1632305In^'ertedARRoots75<79i75^.781EstimatedAR口「ocessisncnstation