多元线性回归模型

#/8'.多元线性回归模型一．概述当今农村农民人均纯收入与多个因素存在着紧密的联系，例如人均工资收入，人均农林牧渔产值人均生产费用支出，人均转移性和财产性收入等。本次将以安徽1995－2009年农村居民纯收入与人均工资收入，人均生产费用支出，人均转移性和财产性收入等因素的数据，通过建立计量经济模型来分析上述变量之间的关系，强调农村居民生活的重要性，从而促进全国经济的发展。二模型构建过程1•变量的定义被解释变量：农民人均纯收入y解释变量：人均工资收入xi,人均农林牧渔产值x2人均生产费用支出X3人均转移性和财产性收入X4。建立计量经济模型：解释农民人均纯收入与人均工资收入，人均生产费用支出，人均转移性和财产性收入的关系2•模型的数学形式设定农民人均纯收入与五个解释变量相关关系模型，样本回归模型为：Y=0+0X"+fX2i+$X3i+0X广ii012343•数据的收集该模型的构建过程中共有四个变量,分别是中国从1995－2009年人均工资收入，人均农林牧渔产值人均生产费用支出，人均转移性和财产性收入，因此为时间序列数据，最后一个即2009年的数据作为预测对比数据，收集的数据如下所示：

ot)sY盘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法估计模型回归结果，散点图分别如下:Sample13952-0(19Includedofeser/aiions1&廿甲ri诵切唧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、模型的检验及结果的解释、评价2•拟合优度检验及统计检验R2=0.997，可以看到模型的拟合优度非常高，说明农民人均纯收入与上述四个解释变量之间总体线性关系显著。模型总体性检验(F检验)：给定显著水平a=0.05，查自由度为(4,10)的F分布表，得F(4,10)=3.48,可见该模型的F值远大于临界值，因此该回归方程很明显是显著的。但由于X3系数不显著且符号为负，与经济意义不符，因此我们认为解释变量之间存在多重共线性。变量的显著性检验(t检验)：给定显著水平a=0.05，查自由度为10的t分布表，得t10=1.812，大于该临界值的的显著变量为X1,x2,x4；x3解释a/2变量未通过检验，说明x3与被解释变量之间不存在显著的线性相关关系。3•多重共线性的检验⑴相关系数检验法

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上图是Eviews输出所有变量的相关系数矩阵，可发现Y与所有解释变量都是正相关的关系，所以进一步确定了上面的回归存在共线性问题。另外，我们发现XI和X2的相关系数很高，两变量很可能存在共线性。⑵多个解释变量的相关性检验由上面的分析可知，X1和X2有很高的相关性，那么我们这里就用X1做被解释变量，X2和X3做解释变量，可得回归模型如下：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，回归模型的拟合优度非常高，F值也远大于临界值。如果将显著水平

扩大到=10%的话，X2系数显著，X3系数不显著。因此x1,x2存在共线性。四、模型的建立这里我们用逐步回归法得到农民人均纯收入模型。1•分别用四个解释变量对Y进行回归，回归结果分别如下: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诣诣l憶tic723S712□urbin-WatsonstatQ707G42ProtXF-statistic)0.000000Includedobse-rvatioris:15VarJable-Co■術凤ntSrdEnart-Ststisn-cProb'C^74576579756245■丁牺3冊QX2106543800312523422033O.C00&R-square<J0963021Meandepentfentvar2466323AdjustedR-s口阴陀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与x拟合优度R2最大，因此将这个方程作为基本方程,2nrizrCl可以看出，Y与x拟合优度R2最大，因此将这个方程作为基本方程,2加入其他变量。然后往里2•引入第二个变量入变量x1后,SarnpJ?1S952009includedobse^Wns-'15甘申JI甲函6理艸对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阳AJkaikein怖uitenonSctiwaF^crit^riDnF-s(ati^iicProb{Fstatistic)246ft3239476277t1.SS817118Q973伽1068O.OOOC0O3.17＜临界值3.18，其系数通不过显著性检验。IncludedubsePi'atiGns^5VariableCaeffieienlSLd.ErrorProb.C1013.43792.85^631C.5137&flOCQQ幽?509fi«d0J0S2501£轻畀了0oooaR-squjredO.SS30D6Llearadependant2466.323AdjustedR-squared05601603Ddependent啣347.6277S.EoFregressiara189.1455Akarkeinfocritsrion13.44643Sumre^id血丽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郦0Log[ikeli^ood$97O06GF-staiistic5427696Ojrbin-WstEdnstat1525416ProbfF^tatistiC)O.OOOQOCSample19952008Includetfo-bservaitions'15引入变量x后，t值-0.22444V临界值3.18，其系数通不过显著性检验。3Samp,&19952C09lncFud^deb^grv^tieri专’15CoeffiGi^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引入变量x后，t值2.715V临界值3.18，其系数通不过显著性检验。4综上所述，本次模型只引入变量x，其最终输出结果如下:2Sample:199£>2009IncfudfrddtorvaiionsISVsn^bleCoefficientStel.Enort-Sta(tisriePwbG245・F应3闕tloooooX21D69436003125234?2033a他0R-^quared09的悚1lvi-&andependeniwar2W&.323AdjustedR-squared0.988176S.D.depend电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模型的最终结果为

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