多元线性回归模型的案例讲解

1.表1列出了某地区家庭人均鸡肉年消费量Y与家庭月平均收入X,鸡肉价格Py猪肉价格P2与牛肉价格P3的相关数据。年份Y/千X/PJ(元/P2/(7U/P3/(7U/年份Y/千PJ(元/P2/(7U/P3/(7U/克元千克）千克）千克）克X/元千克）千克）千克）19802.783974.225.077.8319924.189113.977.9111.4019812.994133.815.207.9219934.049315.219.5412.4119822.984394.035.407.9219944.0710214.899.4212.7619833.084593.955.537.9219954.0111655.8312.3514.2919843.124923.735.477.7419964.2713495.7912.9914.3619853.335283.816.378.0219974.4114495.6711.7613.9219863.565603.936.988.0419984.6715756.3713.0916.5519873.646243.786.598.3919995.0617596.1612.9820.3319883.676663.846.458.5520005.0119945.8912.8021.9619893.847174.017.009.3720015.1722586.6414.1022.1619904.047683.867.3210.6120025.2924787.0416.8223.2619914.038433.986.7810.48（1）求出该地区关于家庭鸡肉消费需求的如下模型:(2)lnY=P+PInX+PInP+PInP+PInP+u(2)请分析，鸡肉的家庭消费需求是否受猪肉及牛肉价格的影响。先做回归分析，过程如下:EquationEst1七EquationEst1七1on输出结果如下:VariableCoefficientStd.Errort-StatisticProb.C-07315200.296947■2.4634670.0241LOG(X)0.3452570.0825654.1816490.0006L0G(P1)-0.5021220.109891-4.5692940.0002L0G(P2)0.146S680.0990061.4834200.1553L0G(P3)口国侬0.099B520.8731370.3941R-squared0.9B2474Meandependentvar1.361301AdjustedR-squared0.978579S.D.dependentvar0.187659S.E.ofregression0.027465Akaikeinfocriterion-4.162123Sumsquaredresid0.013578Schwarzcriterion-3.915276Laglikelihood52.B6441F-statistic252.2633Durbin-Watsonstat1.B24S20Prob(F-statistic)0.000000所以，回归方程为：InY=—0.7315+0.3463山X—0.50211nP+0.14691nP+0.08721nP1 2 3(-2.463) (4.182) (-4.569) (1.483) (0.873)由上述回归结果可以知道，鸡肉消费需求受家庭收入水平和鸡肉价格的影响，而牛肉价格和猪肉价格对鸡肉消费需求的影响并不显著。验证猪肉价格和鸡肉价格是否有影响，可以通过赤池准则(AIC)和施瓦茨准则(SC)。若AIC值或SC值增加了，就应该去掉该解释变量。去掉猪肉价格P2与牛肉价格P3重新进行回归分析，结果如下：VariableCoefficientStd.Errort-StatisticProb.C-1.1257970.088420-12.732370.0000LOG(X)0.4515470.02455418.389660.0000LOG(P1)-0.3727350.063104-5.9066680.0000R-squared0.980287Meandependentvar1.361301AdjustedR-squared0.978316S.D.dependentvar0.187659S.E.ofregression0.027634Akaikeinfocriterion-4.218445Sumsquaredresid0.015273Schwarzcriterion-4.070337Loglikelihood51.51212F-statistic497.2843Durbin-Watsonstat1.877706Prob(F-statistic)0.000000通过比较可以看出，AIC值和SC值都变小了，所以应该去掉猪肉价格P2与牛肉价格P3这两个解释变量。所以该地区猪肉与牛肉价格确实对家庭的鸡肉消费不产生显著影响。2.表2列出了中国2012年按行业分的全部制造业国有企业及规模以上制造业非国有企业的工业总产值Y，资产合计K及职工人数L。工业总产资产合计职工人数 工业总产资产合计职工人数

序号值Y/亿元K/亿元L/万人序号值Y/亿元K/亿元L/万人13722.7003078.220113.000017812.70001118.81043.0000021442.5201684.43067.00000181899.7002052.16061.0000031752.3702742.77084.00000193692.8506113.110240.000041451.2901973.82027.00000204732.9009228.250222.000055149.3005917.010327.0000212180.2302866.65080.0000062291.1601758.770120.0000222539.7602545.63096.0000071345.170939.100058.00000233046.9504787.900222.00008656.7700694.940031.00000242192.6303255.290163.00009370.1800363.480016.00000255364.8308129.680244.0000101590.3602511.99066.00000264834.6805260.200145.000011616.7100973.730058.00000277549.5807518.790138.000012617.9400516.010028.0000028867.9100984.520046.00000134429.1903785.91061.00000294611.39018626.94218.0000145749.0208688.030254.000030170.3000610.910019.00000151781.3702798.90083.0000031325.53001523.19045.00000161243.0701808.44033.00000设定模型为：Y=AKaL3e^利用上述资料，进行回归分析；回答：中国2000年的制造业总体呈现规模报酬不变状态吗?将模型进行双对数变换如下：InK=lnA+ocln^+PlnL+|Li1)进行回归分析:EquationEstiBationSpeci11cationj-ptiQHL二Equationspecific：i+£onIieiiemlentv：±riablefoilowedbylistotregi-essors：xtl>1PELtermSjORulgfjlicitequati'jnL口gj%)二L「gJ：k；iLog：CL)IEetimationsettirL^sHethod：]_S.-tMqmir白工(HLS皿&WJA)S：iniple131取消确定

取消得到如下回归结果:VariableCoefficientStd.Errort-StatisticProb.cLOG的LOG(L)1.1539940.6092360.3607960.727611 1.6060040.176378 3.4641490.201691 17897410.12400.0018O.OS43R-squared0.809925Meandependentvar7.493997AdjustedR-squared0.796348S.D.dependentvar0.942960S.E.ofregression0.42553SAkaikeinfocriterion1.220839Sumsquaredresid5.070303Schwarzcriterion1.359612Laglikelihood-15.92300F-statistic59,65501Durbin-Watsonstat0.793209Prab(F-statistic)0.000000于是，样本回归方程为：八Inr=1.154+0.6091nK+0.3611nL(1.59)(3.45) (1.79)R2:0.8099,R=0.7963,F=59.66从回归结果可以看出，模型的拟合度较好，在显著性水平0.1的条件下，各项系数均通过了t检验。从F检验可以看出，方程对Y的解释程度较少。R=0.7963表明，工业总产值对数值的79.6%的变化可以由资产合计对数与职工的对数值的变化来解释，但仍有20.4%的变化是由其他因素的变化影响的。从上述回归结果看，&+8=0.97“1,即资产与劳动的产出弹性之和近似为1,表明中国制造业在2000年基本呈现规模报酬不变的状态。下面进行Wald检验对约束关系进行检验。过程如下:®EVievs-[Equation:UNTITLEDlorkfile:UNTITLED::UntilleiFileEditObjectViewFreeQuickO^tionsTfindowMeLp过后圆Pr口匚][口团日匚H[Prinl:][rJmnn日][Fr日曰w1][Enimatn 日匚aiHGtmwRREsidE]RsEi-esentationsEztinitiunOutputActualFitted.Residual.MlMAStruetm-&...gradient与wdDerivatives”ii!--!tnFrrnr =十i匚:h「 Pii!--!tnFrrnr =十i匚:h「 PrnhC_urLtideneeEllipse...jVald-Coe±ticientRestricticms...OmittedTariabl已三一Lihelihoc«dEatio...ReduiLilsmtVari：iblqe_Likelihoo■!Ratlo...Coe±ticierbtTestsReeidn：dlTestsStati.1ityLih^LtaI I Ii -r-L・L・LiLi-T

结果如下:WaldTest:Equation:UntitledTestStatistic ValuedfProbabilityF-statistic 0.10111B(1,冽I07529)Chi-square 0.10111B1|口75口5|NullHypothesisSummary:NormalizedRestriction(=□)ValueSid.Err.-1+0(2)十C(3)-0.0299680.094242Restrictionsarelinearincoefficients.由对应概率可以知道，不能拒绝原假设，即资产与劳动的产出弹性之和为1,表明中国制造业在2000年呈现规模报酬不变的状态。一、邹式检验（突变点检验、稳定性检验）.突变点检验1995-2012年中国家用汽车拥有量（y，万辆）与城镇居民家庭人均可支配收入t（X，元），数据见表3。t表3中国家用汽车拥有量（y）与城镇居民家庭人均可支配收入（1）数据t t年份y（万辆）tX(元)t年份y（万辆）tX(元)t199528.49739.12004205.423496.2199634.71899.62005249.964283199742.291002.22006289.674838.9199860.421181.42007358.365160.3199973.121375.72008423.655425.1

200081.621510.22009533.885854200196.041700.62010625.3362802002118.22026.62011770.786859.62003155.772577.42012968.987702.8下图是关于y和x的散点图:t t1000800-6DD-4001000800-6DD-4002000-□1QQQ30005QQQ?□□□X从上图可以看出，2006年是一个突变点，当城镇居民家庭人均可支配收入突破4838.9元之后，城镇居民家庭购买家用汽车的能力大大提高。现在用邹突变点检验法检验1996年是不是一个突变点。Ho：两个字样本（1995—2005年，2006—2012年）相对应的模型回归参数相等H1：备择假设是两个子样本对应的回归参数不等。在1995—2012年样本范围内做回归。0.245355ProL(F-statistic)在回归结果中作如下步骤：0.245355ProL(F-statistic)Jlajii-eserLtatiotleTstini：±tiunUutputActual,Fittei,Re5idu；1l ,AEMAStructure...Gr：id1 e：=lTlJD&r1vat1vee♦C至Matfi器itStd.Errort-StatisticProb.CueffinientTests ，Eeziilu：=J_Teste .—9 38.87504 -2.9223S0 0.0100StaLilityTests lChowBre：=Jq：iointTest...LabelChij'iii'ForecastT也=t...MSETT口与t.一Rguur-iv.EstimatesI.IJLS口工LLy」...AdjustedR-squared 0,B966-S.E.ofregression 89.515l12日2口9石-105.380312,03000140.48410.000000SchwarzcritericinF-statisticSumsquaredresidLoglikelihoodDurtiin-Watsonstat□Equation:UNTITLEDTorkfile:HASEBa，:Cas.・・匚〕回国Objc匚[Prinl:][Narne

输入突变点:得到如下验证结果：ChowBreakpointTest:200&NullHypothesis:NobreaksatspecifiedbreakpointsVaryingregressors:AllequationvariablesEquationSample:19952011F-statistic1416.403Prob.F(2.13J0.0000Loglikelihoodratio91.G0709Prob-.Chi-Square(2)0.0000WaldStatistic2832.806Prob-.Ghi-Square(2)0.0000由相伴概率可以知道，拒绝原假设，即两个样本（1995—2005年，2006—2012年）的回归参数不相等。所以，2006年是突变点。.稳定性检验以表3为例，在用1995—2009年数据建立的模型基础上，检验当把2010—2012年数据加入样本后，模型的回归参数时候出现显著性变化。因为已经知道2006年为结构突变点，所以设定虚拟变量：101995—2005D1\|12006—2012对1995—2012年的数据进行回归分析:

EquationE£timstian.确定耳乂消做邹氏稳定性检验:ViewProcObjectPrintN^meFreezeEstimateForecastStatsResids□ependentVariable:YMethod二LeastSquares□ate:10/25/1STime:21:05Sample:19%2012Includedobservations:18VariableCoefficientStd.Errort-StatisticProb.C-16.074912.945554 -5.4571620.0001X0.0535240.001353 47.02409O.QOOOD1-852.6.22512.29131 -69.36791O.ftOOOtm0.1750130.002381 73.49329o.aoooR-squared0.999750Meandependentvar234.260GAdjustedR-squared0.999697S.D.dependentvar27S.4439S.E.ofregression4.047800Akaikeinfocriterion6.108060Sumsquaredresid3290174Schwarzcriterion6.385921Loglikelihood-51692MHannan-Quinncriter6.215343F-statistic1868976Durbin-Watsonstat1.766734ProbfF-statistic)0.000000输入要检验的样本点:得到如下检验结果:ChowForecastTestEquation:UNTITLEDSpecification:YCXD1DI^XTestpredi-ctionsforobservationsfrom2010to2012Value出ProIbabiIityF-statistic0.433432PJ1)0.7333Likelihooti「atio2.01110430.5701由上述结果可以知道，F值对应的概率为0.73,所以接受原假设，模型加入2010、2011和2012年的样本值后，回归参数没有发生显著性变化。二、似然比（LR）检验有中国国债发行总量（DEBT，亿元）模型如下：tDEBT=P+PGDP+PDEF+0REPAY+ut0 1t2t3 tt其中GDP表示国内生产总值（百亿元），DEF表示年财政赤字额（亿元），REPAY表示t t t年还本付息额（亿元）。1990—2011年数据见表4。表4国债发行总量DEBT、GDP、财政赤字额DEF、年还本付息额（REPAY）数据t t t t199043.0145.17868.928.582001461.4216.178237.14246.81991121.7448.624-37.3862.892002669.68266.381258.83438.57199283.8652.94717.6555.522003739.22346.344293.35336.22199379.4159.34542.5742.4720041175.25467.594574.52499.36199477.3471.7158.1628.920051549.76584.781581.52882.96199589.8589.644-0.5739.5620061967.28678.846529.561355.031996138.25102.02282.950.1720072476.82744.626582.421918.371997223.55119.62562.8379.8320083310.93783.452922.232352.921998270.78149.283133.9776.7620093715.03820.67461743.591910.531999407.97169.092158.8872.3720104180.1894.4222491.271579.822000375.45185.479146.49190.0720114604959.3332516.542007.73对以上数据进行回归分析:EquationEstimation EquationEstimation 次得到如下输出结果：DependentVariable:DEBTMethod:LeastSquaresDate;10/25/15Time;21;13Sample:19902011Includedobservations:22VariableCoefficientStd.Errort-StatisticProb.C4.3MOOB21一667250.1991030.8444GDP0.3462020.1644702.2347560.0384DEF0.9954030.031E1331.486990.0000REPAY0.3797600.04950817770220.0000R-squa.red099S955Meandependentvar1216.355AdjustedR-squared0.998781S.D.dependentvar1405.993S.E.ofregression51.88705Akaikeinfiocriterion10.09398Sumsquaredresid484E0.7SSchwarzcriterio-n11.09735Loglikelihood-115.8888Hannan-Quinncriter10.94571F-statistic5735.346Durbin-Watsonstat2.116834Prob(F-statistic)0000000对应的回归表达式为：DEBT=4.31+0.35GDP+1.00DEF+0.88REPAYt t t t(0.2)(2.2) (31.5) (17.8)R2=0.999,DW=2.1,F=5735.3现在用似然比（LR）统计量检验约束GDP对应的回归系数P等于零是否成立。t 1过程如下：S'ETievs-[Eauation:UKTITLEDTorkflie:CASEGB::CaseGb\]FileEditObJactViawProcQuiukQ^tiansWind-owHalpM县以介「□匚gblE匚H[p门nHIrJamc][Freeze][Estimate]/口叱匚至川5H5][艮6与出]ReuresentaticmsE5tirria+1oilUixtpij.t.Actual,Fitted..Eesidual ♦ATlfilAS,tmctur..Gra.d.1ant.eand.Darivativac，匚口文ari：mi2eM：itrix1+ !二：+「』匚rrnv t-!=：ti==+ir- Pfti1--■ConfiiienceZEllipEe_..-ild-CoohhiaisritBLqetrlct£ofle...QmitiedV：±t-iablee-LikelihoodRati'ii...CoefficientTests .Ra=idualTqe-Le +StabilityTests 卜LatiplRediiriilaxLt.VariabLes一Lik^lihood.Rati□...1—■,—,■,*+--1*,—,,•*--*■*■1—■二二1-1—11R-squared 0.990955 Meandependentvar 1216,395AdjustedR-squared D998731 S□.dependentvar 1435993S.E.ofregression 51.B8705 Akaikeinfocriterion 10.89S9BSumsquaredresid 40460.70 SchwarzciiLeiion 11.09735Loglikslihood -11G.aasaF-statietic 5735.34BDurbin-Watscnstat 2.11BB34 Prob(F-statiGtic) 0.000000输入要检验的变量名:得到如下输出结果：RedundantVariables:GDPF-statistic4.994134Prob.F(1.18)0.033350Loglikelihoodratio5.387082Prob.Chi-Square(l)口.口2口2加R.estrictedTestEquation:□■ependentVariatjle.DEBTMethod:LeastSquaresMe-10/25/15Time12+17Sample:19902011Includedobservations:22VariableCoefficientStd.Errort-StatisticPr&b.c40.5021715.83705 2.557432U..0193□EF1040S280026718 38948G400000REPAY0.9777S4□025272 3S.6.890G□,0000R.-sqjared0998665Meandependentvar1216.396AdjustedR-squar@d099S524S.D.dependentvar1485.993S.E.ofriegression57.oaoaaAkiaikeinfocriterion11.05294Sumsquaredresi-dB1906.32Schwarzcriterion11.20172L&glik且lih口ad-11BS823Hannan-Quinncriter1100799F-3tatistic710G.5^2Durbin-Wat3onstat1.014741ProbfF-statistic)0000000

输出结果上部是关于约束GDP系数为零的F检验和LR检验。由于两