多元線性回歸模型

#/8'.多元線性回歸模型一．概述當今農村農民人均純收入與多個因素存在著緊密的聯系，例如人均工資收入，人均農林牧漁產值人均生產費用支出，人均轉移性和財產性收入等。本次將以安徽1995－2009年農村居民純收入與人均工資收入，人均生產費用支出，人均轉移性和財產性收入等因素的數據，通過建立計量經濟模型來分析上述變量之間的關系，強調農村居民生活的重要性，從而促進全國經濟的發(fā)展。二模型構建過程1?變量的定義被解釋變量：農民人均純收入y解釋變量：人均工資收入xi,人均農林牧漁產值x2人均生產費用支出X3人均轉移性和財產性收入X4。建立計量經濟模型：解釋農民人均純收入與人均工資收入，人均生產費用支出，人均轉移性和財產性收入的關系2?模型的數學形式設定農民人均純收入與五個解釋變量相關關系模型，樣本回歸模型為：Y=0+0X"+fX2i+$X3i+0X廣ii012343?數據的收集該模型的構建過程中共有四個變量,分別是中國從1995－2009年人均工資收入，人均農林牧漁產值人均生產費用支出，人均轉移性和財產性收入，因此為時間序列數據，最后一個即2009年的數據作為預測對比數據，收集的數據如下所示：

ot)sY盤X3X4□bsYX1X2X3糾19951302.950234.21001977.920466.400093.6100019961607.720346.360022&3.660466.8300107.210019971808750405.49002461.090480.150077.9700019981S&3.050457.17002463.460440.1500104.610019991900.290470.69002458030409.1000120080020001^34.670547.83002416.660449.1100127.410020012020000610.G5002482.260486.0800141.790020022118.000707.68002570.050526.5100166.1700紳2117.000518.32002563.820517.5800140.870020042449.300884.62003212.360698.9000165.340020062641.0001010060323&.3908&7.4100187.7100200G2969.1001184.1103449.G301035.980226.6700200735S6.30014700503974.3801119.540327.820020081737.8404656.6901395.220437.360020094504.3001882.4204M6.5401334.260477.45004.用OLS法估計模型回歸結果，散點圖分別如下:Sample13952-0(19Includedofeser/aiions1&廿甲ri誦切唧CoefficientErr^rt-StstigticPrgb-c33632621S6.260CQ130567p8603xn0.6590540.13146936317660.0046X20.6902470.t244924741256O.OQOS工3-0.274S820.203S81-13471160.207716239W0騒慟3&7413300233(^squared0997116Meandependsntvar246&323AdjustedR-s<]uared099.5&63SOdependentvar9476277SEofregression6021151AKaikembcriterion112S461Sumsquwdresid3525426SchwarzGriterion1153082LagFike-lihood-79.7110^F-&Cauntie8B4.430SDurbinVJatsonsLatPnq!>(Rst3ij￡tiq)0OODOOOY.=33.632+0.659X+0.59X-0.274X+0.152Xi1234d.f.=10,R2=0.997116,Se=(186.261)(0.1815(0.1245)(0.2037)(0.5699)t=(0.1805)(3.632)(4.741)(-1.347)(2.674)

□Group:UNTFLEDWorkfile;UNTTTLE^UrrtrtledView]ProcObjectPrintNameFreezeSampleSheet!5tats\SpecOX1OX1?X2mX3X4三、模型的檢驗及結果的解釋、評價2?擬合優(yōu)度檢驗及統計檢驗R2=0.997，可以看到模型的擬合優(yōu)度非常高，說明農民人均純收入與上述四個解釋變量之間總體線性關系顯著。模型總體性檢驗(F檢驗)：給定顯著水平a=0.05，查自由度為(4,10)的F分布表，得F(4,10)=3.48,可見該模型的F值遠大于臨界值，因此該回歸方程很明顯是顯著的。但由于X3系數不顯著且符號為負，與經濟意義不符，因此我們認為解釋變量之間存在多重共線性。變量的顯著性檢驗(t檢驗)：給定顯著水平a=0.05，查自由度為10的t分布表，得t10=1.812，大于該臨界值的的顯著變量為X1,x2,x4；x3解釋a/2變量未通過檢驗，說明x3與被解釋變量之間不存在顯著的線性相關關系。3?多重共線性的檢驗⑴相關系數檢驗法

EZ1□Group:UmTB.EDWorkfle:UNTTTLED\UntitledEZ1'■.■'ier.'P「(?cObjectPrint'NoneFreezeSample：Sheet.Stats5pecCorrelationMatrix￥X4X3X1￥1.0000000.9-8132909657660.9944960991140X409813291.000000094807709713000967125X30.9657&60.9480771.0000000.9726930.966013X20.99449509713000.9726931.0000000982869X10.9911400.%71250.9650130.9828691000000□□J上圖是Eviews輸出所有變量的相關系數矩陣，可發(fā)現Y與所有解釋變量都是正相關的關系，所以進一步確定了上面的回歸存在共線性問題。另外，我們發(fā)現XI和X2的相關系數很高，兩變量很可能存在共線性。⑵多個解釋變量的相關性檢驗由上面的分析可知，X1和X2有很高的相關性，那么我們這里就用X1做被解釋變量，X2和X3做解釋變量，可得回歸模型如下：Sample19952009Includedobs&rvationa-15VariableCoefficientStiErrorbStatisticProb□C-757X510173.1762■4.3727210.0009X20477174013031736616510.0033X30245400032^83007440180.4712R-squared0.967529Meandependentvar851.20&3AdjustedR-squared0962118SDdeperdentwar5123819S.E.ofregression9972701Akaikeinfocriterion1221951Sumsqusre-dresitf1193457Selw/srserilerion12.35122Loglikelihood8B.64705Ssi^tistic178.7821Durbin-Watsonstatu1.190711ProtXF-statistic)u0oooooo■口X=-757.251+0.477X+0.2454X123t=(-4.373)(3.662)(0.744)R2=0.9675,R2=0.9621，F=178.78,DW=1.19。可以看到，回歸模型的擬合優(yōu)度非常高，F值也遠大于臨界值。如果將顯著水平

擴大到=10%的話，X2系數顯著，X3系數不顯著。因此x1,x2存在共線性。四、模型的建立這里我們用逐步回歸法得到農民人均純收入模型。1?分別用四個解釋變量對Y進行回歸，回歸結果分別如下:Sample199-52009tncludedobservations-15VariableCoefficientStd..Errort-StatisticProb.C906.0043670372413.60437OJQOOIQX11,8330690068131^6904910.0000R-squaredO9S235SMeancEmpend自nii/ar2466.323AdjustedR-squared09B1G01S.D.dependentvarMT.6277S.E.ofregression130.&186Akaikeintocriterion1270601Sumsquaredresid2217959Schwarzcriterjon12.80041Luglikelihood-9329505F詣詣l憶tic723S712□urbin-WatsonstatQ707G42ProtXF-statistic)0.000000Includedobse-rvatioris:15VarJable-Co■術鳳ntSrdEnart-Ststisn-cProb'C^74576579756245■丁犧3冊QX2106543800312523422033O.C00&R-square<J0963021Meandepentfentvar2466323AdjustedR-s口陰陀d0.TO176S.Ddependentvar9d7.E2778Eofregres-sion103W3iAkaikeinfocriteiion12.23174Sumsquaredresid13SCJ333SchwarzcriteTion12.3261^Lagkikelihoad-B?73808F-sUbslic1171oilDufbin-WstsonMProb(F-Malistic)0.000000j-4q-rverbp■pSampleIncludedob&ervatEons15Cibeffiti&rtiStdErtCirt-SlatistieProb.C589laps1&4.58393.8110620.4022X32.62&64601S5S231342307G.&OOflR-sqifar&dQ.93Z7fl5Meand&p€rLdem版2466323AdjustedR-squared09S7520SDfhapandacitvarM7.?27TS.E.ofregression2S6iOffiA^aik&infocriteffcn14.(M4SjSumsquared『e霑id846033.4Schwarzentenort14.13921Loglikelihood-10333€1F-staftistie130178SDurbiri-Wstsonst3l0.777246Pr-ob(F-￡t3thlic>O.OOQOOO

nrizrCl可以看出，Y與x擬合優(yōu)度R2最大，因此將這個方程作為基本方程,2nrizrCl可以看出，Y與x擬合優(yōu)度R2最大，因此將這個方程作為基本方程,2加入其他變量。然后往里2?引入第二個變量入變量x1后,SarnpJ?1S952009includedobse^Wns-'15甘申JI甲函6理艸對st#日町卞自制i詡GPrak口cXIX2>10122950744S930.643743198734802142270124561-05053703.4771235.1SE08S0.61570.00460.0002□R'SquarsdArfjv-ste-dR-squ^redSE.ofr&gr&s￡ienSunsquaredr^sidLaglik&litioodDurbin-Wal&anwt^t0.994531099361975.6955Q68T57>1B4511280.6771fi2Meand^pendtrtlvsrSDckpentfent陽AJkaikein怖uitenonSctiwaF^crit^riDnF-s(ati^iicProb{Fstatistic)246ft3239476277t1.SS817118Q973伽1068O.OOOC0O3.17＜臨界值3.18，其系數通不過顯著性檢驗。IncludedubsePi'atiGns^5VariableCaeffieienlSLd.ErrorProb.C1013.43792.85^631C.5137&flOCQQ幽?509fi?d0J0S2501￡輕畀了0oooaR-squjredO.SS30D6Llearadependant2466.323AdjustedR-squared05601603Ddependent啣347.6277S.EoFregressiara189.1455Akarkeinfocritsrion13.44643Sumre^id血麗2SqIw串rg^rit^ri^n13,54WSLogHkatihood9894S57F-stfltitttit33E4079Dirbin-Watsnrist-st08043fl1PfotXFslslistic)o.aooma

VariablyCae-fliQi^ntSUErrorVariablyCae-fliQi^ntSUErrort-S-UtLslicPrab.C-7807225185.8S1942W77S0001?109957Q013935E7S6WS700000X3-0.0794450.3S3972^0.2244400S2G20989(66Meandependent弼2466.323AdjustedR-squared0997344S.Ddependentvar9476277BEofregiession107.0266Akarkeinfoenterion12.360BSSumsqu-aredresid137d5G3Schwsrzcritenon12酈0Log[ikeli^ood$97O06GF-staiistic5427696Ojrbin-WstEdnstat1525416ProbfF^tatistiC)O.OOOQOCSample19952008Includetfo-bservaitions'15引入變量x后，t值-0.22444V臨界值3.18，其系數通不過顯著性檢驗。3Samp,&19952C09lncFud^deb^grv^tieri?！?5CoeffiGi^mStd.Enort-SlaEisticPrab口G烷X4^295.53870.7865792Q79S771S40626-1605642a10765972^72a.78S9^52715315013430.00000.0138R-^quareJAdjustedR-SQuaredStofregressioriSumsquarednesBLwglikelitioodDwfbip-Wst-SQn0993199Q.99296fi344100635500.70464450013SGSSeMean<j&p?nderitvarS.D.depenctoFitwrinfocriterionSchwarzcriterionF-statistfcPr?b(F-st-stislit)2166323S4762773361102^72376.23^7a.ooo(no引入變量x后，t值2.715V臨界值3.18，其系數通不過顯著性檢驗。4綜上所述，本次模型只引入變量x，其最終輸出結果如下:2Sample:199￡>2009IncfudfrddtorvaiionsISVsn^bleCoefficientStel.Enort-Sta(tisriePwbG245?F應3闕tloooooX21D69436003125234?2033a他0R-^quared09的悚1lvi-&andependeniwar2W&.323AdjustedR-squared0.988176S.D.depend電mtvarS47.&277StQfrpgr?5siQn103.M34AR甲ik耳iiiifp^ntQrig>n12,23174SiafMsquarer?iid138033.3Schwiratcit^riort12.32615Loglik^liltood*0973808F-statrstic1171.031Durbin-Watsonsi.at1j4€116SPraib(F-std3Elic}a.oaoooo模型的最終結果為

Y=-745.76+1.069X2(-7.644)(34.22)R2=0.989,R2=0.988,F=1171.031,DW=1.4611i-iSarnple115IncludedQtseryatiofis:15VanableCoeflicieiitStdErrorProbc?72圖03365?244￡-t11737702857X252.42523W4餌321.29^52202195X2*2■B.0fi773&0