多元線性回歸模型的案例講解

1、1.表1列出了某地區(qū)家庭人均雞肉年消費量Y與家庭月平均收入X,雞肉價格Pi,豬肉價格P2與牛肉價格P3的相關(guān)數(shù)據(jù)。年份Y/千X/P1/（元/P2/（元/P3/（元/年份Y/千P1/（元/P2/（元/P3/（元/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.

2、0111655.8312.3514.2919843.124923.735.477.7419964.2713495.7912.9914.3619853.335283.816.378.0219974.4114495.6711.7613.9219863.565603.936.988.0419984.6715756.3713.0916.5519873.646243.786.598.3919995.0617596.1612.9820.3319883.676663.846.458.5520005.0119945.8912.8021.9619893.847174.017.009.3720015.172258

3、6.6414.1022.1619904.047683.867.3210.6120025.2924787.0416.8223.2619914.038433.986.7810.48（1）求出該地區(qū)關(guān)于家庭雞肉消費需求的如下模型:lnY011nx21nB3InP241np3u（2）請分析，雞肉的家庭消費需求是否受豬肉及牛肉價格的影響先做回歸分析，過程如下：輸出結(jié)果如下:VariableCoeffioient8td.Errort-StatisticProb.C-0.7315200.2969472.4634670.0241LOG兇034K57D.082565JOG(P1)-0.

4、5021220.109691-45G92940.0002LOG(P2)0.1468E800990061.48342D0.1553LOG(P3)0.08710600998528731370.3941R-Siquared0,902474Meandepandertvar1,361301AdjustedRsquared0.978575S.D.dependentvar0.1B7G59SEofregressian0.0274&5Akaikeinfocriterion-4.162123Sumsquaredresid0.013576Schwarzcriterion-3.915276Loglikelih

5、ood52,96441F-statistic252.2633Durlbin-Watsonstat1.324620Prob(F-statistic)0.000000所以，回歸方程為：lnY0.73150.3463lnX0.5021lnP0.1469lnB0.0872lnP3(-2.463)(4.182)(-4.569)(1.483)(0.873)由上述回歸結(jié)果可以知道，雞肉消費需求受家庭收入水平和雞肉價格的影響，而牛肉價格和豬肉價格對雞肉消費需求的影響并不顯著。驗證豬肉價格和雞肉價格是否有影響，可以通過赤池準則(AIC)和施瓦茨準則(SC)。若AIC值或SC值增加了，就應(yīng)該去掉該解釋變量。去掉豬

6、肉價格P2與牛肉價格P3重新進行回歸分析，結(jié)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.ofregression0.027634Akaikein

7、focriterion-4.218445Sumsquaredresid0.015273Schwarzcriterion-4.070337Loglikelihood51.51212F-statistic497.2843Durbin-Watsonstat1.877706Prob(F-statistic)0.000000通過比較可以看出，AIC值和SC值都變小了，所以應(yīng)該去掉豬肉價格P2與牛肉價格P3這兩個解釋變量。所以該地區(qū)豬肉與牛肉價格確實對家庭的雞肉消費不產(chǎn)生顯著影響。2.表2列出了中國2012年按行業(yè)分的全部制造業(yè)國有企業(yè)及規(guī)模以上制造業(yè)非國有企業(yè)的工業(yè)總產(chǎn)值Y,資產(chǎn)合計K及職工人數(shù)L。工業(yè)

8、總產(chǎn)資產(chǎn)合計職工人數(shù)工業(yè)總產(chǎn)資產(chǎn)合計職工人數(shù)序號值Y/億元K/億元L/萬人序號值Y/億元K/億元L/萬人13722.7003078.220113.000017812.70001118.81043.0000021442.5201684.43067.00000181899.7002052.16061.0000031752.3702742.77084.00000193692.8506113.110240.000041451.2901973.82027.00000204732.9009228.250222.000055149.3005917.010327.0000212180.2302866.6508

9、0.0000062291.1601758.770120.0000222539.7602545.63096.0000071345.170939.100058.00000233046.9504787.900222.00008656.7700694.940031.00000242192.6303255.290163.00009370.1800363.480016.00000255364.8308129.680244.0000101590.3602511.99066.00000264834.6805260.200145.000011616.7100973.730058.00000277549.5807

10、518.790138.000012617.9400516.010028.0000028867.9100984.520046.00000134429.1903785.91061.00000294611.39018626.94218.0000145749.0208688.030254.000030170.3000610.910019.00000151781.3702798.90083.0000031325.53001523.19045.00000161243.0701808.44033.00000設(shè)定模型為：YAKLe(1) 利用上述資料，進行回歸分析；(2) 回答：中國2000年的制造業(yè)總體呈現(xiàn)

11、規(guī)模報酬不變狀態(tài)嗎?將模型進行雙對數(shù)變換如下：lnYlnAlnKInL1)進行回歸分析：得到如下回歸結(jié)果:VariableCoefficientStdErrort-StatisticProbcLOG(K)LCiG(L)11539940,6092360.36079607276111.586004017637S3464U902016911.7697410.12400.00180.0643R-squared,809925Meandependentvar7493997AdjustedP-squared0795348S.Ddependentvar0.942950BEofregression口425538

12、Akaikeinfocriterion1.220S39Sumsquaredresid5.070303Schwarzcriterion1.359612Laglikelihiood15,92300F-statistic59,65501Durbin-Watsonstat0793209Prcih-statistic)0.000000于是，樣本回歸方程為：lnY?1.1540.609lnK0.361lnL(1.59)(3.45)(1.79)_2R0.8099,R0.7963,F59.66從回歸結(jié)果可以看出，模型的擬合度較好，在顯著性水平0.1的條件下，各項系數(shù)均通過了t檢驗。從F檢驗可以看出，方程對Y的

13、解釋程度較少。R0.7963表明，工業(yè)總產(chǎn)值對數(shù)值的79.6%的變化可以由資產(chǎn)合計對數(shù)與職工的對數(shù)值的變化來解釋，但仍有20.4%的變化是由其他因素的變化影響的。從上述回歸結(jié)果看，？0.971,即資產(chǎn)與勞動的產(chǎn)出彈性之和近似為1,表明中國制造業(yè)在2000年基本呈現(xiàn)規(guī)模報酬不變的狀態(tài)。下面進行Wald檢驗對約束關(guān)系進行檢驗。過程如下:®EYiers-Eiuation:WiriTLEDlorkfile:UHTITLED:Untitlei口FileIditObjectYieiEtoc如OptionslindtmHelp匚t回時岫0周|Fr無卻eb)mateForec6st|StatB】Re

14、sidsEstimition.OutputFitttd,Rasidual>Structure.SIHRrrnrKSiatic+irPrnhC_on£ideneeEllipse.ydCoefficientRestrictians.,OiftitteiiVariaLbles-LilEelihoodRatio.a.Rgdund皿tVariableeLikelihoodRatio.Qradi«ntsuidDerivatives.CoviriancftMatrixCoe££icientTeStsR«sidualTests£tAbility

15、TestsLib*LI-Pi-ii-ii-iir-ir-htIIa.taiHbGLkCoe££icirestric11oils,separatedbycommas匚。也包二11E翼amplesc口H|y|*式結(jié)果如下:WaldTest:Equation:UntitledTestStatisticValuedfProbabilityF-statistic0.101118(1.20)I075231Chi-square01011181|CL而同NullHypothesisSummary:NormalizedRestriction(=0)ValueStdErr.-1-+0(2)-+

16、C®<30299600.094242Restrictionsarelinearincoefficients.由對應(yīng)概率可以知道，不能拒絕原假設(shè)，即資產(chǎn)與勞動的產(chǎn)出彈性之和為1,表明中國制造業(yè)在2000年呈現(xiàn)規(guī)模報酬不變的狀態(tài)。一、鄒式檢驗（突變點檢驗、穩(wěn)定性檢驗）1 .突變點檢驗1995-2012年中國家用汽車擁有量（黃，萬輛）與城鎮(zhèn)居民家庭人均可支配收入（xt,元），數(shù)據(jù)見表3。表3中國家用汽車擁有量（黃）與城鎮(zhèn)居民家庭人均可支配收入（Xt）數(shù)據(jù)年份yt（萬輛）Xt（元）年份yt（萬輛）Xt（元）199528.49739.12004205.423496.2199634.718

17、99.62005249.964283199742.291002.22006289.674838.9199860.421181.42007358.365160.3199973.121375.72008423.655425.1200081.621510.22009533.885854200196.041700.62010625.3362802002118.22026.62011770.786859.62003155.772577.42012968.987702.8下圖是關(guān)于yt和xt的散點圖:800bOD-400-7D0HX2nn-ioonsoon從上圖可以看出，2006年是一個突變點，當城鎮(zhèn)居民

18、家庭人均可支配收入突破4838.9元之后，城鎮(zhèn)居民家庭購買家用汽車的能力大大提高?，F(xiàn)在用鄒突變點檢驗法檢驗1996年是不是一個突變點。H。：兩個字樣本（19952005年，20062012年）相對應(yīng)的模型回歸參數(shù)相等Hi：備擇假設(shè)是兩個子樣本對應(yīng)的回歸參數(shù)不等。在19952012年樣本范圍內(nèi)做回歸。在回歸結(jié)果中作如下步驟：Equation;UHTITLEDVorlfHe;GA5EGA;:C軟$,.kJ回國rocIObleE巨史rsen1*tituts"zt-iraation0utputActual,FittedResidualARMASt-rtucture.itStdErroft*S

19、tatisticProbb38.37504-2.922360D.OWOStabilityTastsChowEr*alkp*intTtst.LabelChcmrForecastTa'st,.RESETTest.ItedursiveEses(OLSoialy).AdjustedK-squaredSEofregression0.89558951S1202095-105.330312.0300B148.4841.oooooSchwar:criterionF-statisticsidualTestsSumsqu3r&dr&sidLaglikelihoodDurbin-Watson

20、statCotfficientT電24635SProfc(F-statisiic)野甘dientwandDerivitjvesCoaxlazierMatrix輸入突變點:得到如下驗證結(jié)果：ChowBreakpointTest:2006NullHifpcthasisNobreaksatspecifiedbreakpointsVaryingegessQrsAllequationvariablesEquationSample19952011F-statisticLoglikelihoodratioWaldStatistic1416.40391607092832806Prat.F(2.13JProbC

21、hi-Square(2)Prob.Ghi-Square20.00000000000000由相伴概率可以知道，拒絕原假設(shè)，即兩個樣本（19952005年，20062012年）的回歸參數(shù)不相等。所以，2006年是突變點。2 .穩(wěn)定性檢驗以表3為例，在用19952009年數(shù)據(jù)建立的模型基礎(chǔ)上，檢驗當把20102012年數(shù)據(jù)加入樣本后，模型的回歸參數(shù)時候出現(xiàn)顯著性變化。因為已經(jīng)知道2006年為結(jié)構(gòu)突變點，所以設(shè)定虛擬變量：019952005D1120062012對19952012年的數(shù)據(jù)進行回歸分析:做鄒氏穩(wěn)定性檢驗:ViewProcObjectprinthameFreezeEstimateForec

22、ast5tatsResidsVariableCoefficientStdEnort-StatisticProbC-1607491945654-54571620M01X0.0636240.0013S347,024090.MMD1-855.622512.29131-69.367910.MM0175013000230173,4932900000R-squared0999750Weardependentvar284.2606AdjustedR'squared0.999697S.Ddependentvar2784439S.E.ofregression4.947008Akaikeinfocrite

23、rion6.183060Sumsquaredresid329.0174SchwarzcrrtenanG.3S5921Laglikelihood-6169254Hannan-Quinncriter6.216343F-statistic186897EDurbin-Watsonstat1.765734Prob(F-statistic)oooooocDependentVariable：YMeth。-LeastSquaresate10/25/15Time2105Sample:19952012Includedobservations.16輸入要檢驗的樣本點:得到如下檢驗結(jié)果:ChowForecastTes

24、tEquation-UNTITLEDSpecificationYCXD1DVXTestpr«dktionsforobservationsfrom2010to2012VHugdfProb.bilityF-statistic0433432（3,11）07333Jkelihoodratio2.01110430.5701由上述結(jié)果可以知道，F(xiàn)值對應(yīng)的概率為0.73,所以接受原假設(shè)，模型力入2010、2011和2012年的樣本值后，回歸參數(shù)沒有發(fā)生顯著性變化。二、似然比（LR）檢驗有中國國債發(fā)行總量（DEBTt,億元）模型如下：DEBTt01GDPt2DEFt3REPAYtut其中GDPt表示

25、國內(nèi)生產(chǎn)總值（百億元），DEFt表示年財政赤字額（億元），REPAY表示年還本付息額（億元）。19902011年數(shù)據(jù)見表4。表4國債發(fā)行總量DEBTt、GDPt、財政赤字額DEFt、年還本付息額（REPAY）數(shù)據(jù)199043.0145.17868.928.582001461.4216.178237.14246.81991121.7448.624-37.3862.892002669.68266.381258.83438.57199283.8652.94717.6555.522003739.22346.344293.35336.22199379.4159.34542.5742.4720041175

26、.25467.594574.52499.36199477.3471.7158.1628.920051549.76584.781581.52882.96199589.8589.644-0.5739.5620061967.28678.846529.561355.031996138.25102.02282.950.1720072476.82744.626582.421918.371997223.55119.62562.8379.8320083310.93783.452922.232352.921998270.78149.283133.9776.7620093715.03820.67461743.59

27、1910.531999407.97169.092158.8872.3720104180.1894.4222491.271579.822000375.45185.479146.49190.0720114604959.3332516.542007.73對以上數(shù)據(jù)進行回歸分析:得到如下輸出結(jié)果：DependentVariable:DEBTMethod.LeastSquaresDate：10/25/15Time：21:13Sample19902011Includedobservations.22VariableCoefficientStdError(-StatisticPrab.C431400821.

28、GS7250.1991030.8444GDP0.3452020.5544705.23475GD.03&4DEF0.99S4030.03161331.486990t)COOREPAY08797500W9503177702200000R-squared0.998955Meandependentvar1216.395AdjustedR-squared。998781S.Ddependentvar1485,993SEofregression5188705Akaikeinfocnterion1089898Sumsquaredresid48460.78Schwarzcriterion11.09735

29、Loglikelihood-115.3838Hannan-Q.Liinnenter.10.94571F-statistic5735346Durbin-Watsonstat2116834Prcb(F-Stati&tiG)0.000000對應(yīng)的回歸親達式為：DEBTt4.310.35GDPt1.00DEFt0.88REPAYt(0.2)(2.2)(31.5)(17.8)_2_R0.999,DW2.1,F5735.3現(xiàn)在用似然比(LR)統(tǒng)計量檢驗約束GDPt對應(yīng)的回歸系數(shù)1等于零是否成立。過程如下：®ETiewslEw*ion:UMTITLEDinEy.liIKileIlFurel

30、自乂13E,31*國出*Qh貨七£iawE1rMy.111?？趖iwWind*wH*lpRtwEheLLluIIuumEs11ma±i.onOiitpu.t.Actual,Fitted,KeEi«iualARFflASlx-u.cture.Gr-adianteand.DariatittceCaiifarianceNatrixCoefflciuxtTesti“iTn!11IIkT-：illCuuif1JiifiLueE1Lip".nri=TiirlrnnRqsiT口二七二*oli=Coc££ictiarit£.4e,Lr1c1

31、1oils.StabilityTests*QjnitiedVariables-Likelihcod.JLatio.Hed.TLn.d.-&ntVaria.bXe3-LiRdj.hooAR.-e.t.io.wj*mwrr.frwR-quarsd0.999955itvji1216.395Artju5tpdR-qmared990761S.Pdependsnt',-rI4的993S.E.ofrcgroaaion51.BS706Akaikeinfocriterion10,89036Sumsquaredresid46460.76chwarzcriterion111.09735Inglike

32、lihand-115SRS8F-5tsti=ticS73534KDurbin-Watsonatat2.11EB34ProbfF-Btstistic)oooooo輸入要檢驗的變量名:得到如下輸出結(jié)果：RedundantVariables:GDPF-atatistic4994134Prab.F(1,16)口一口3的50Loglikelihoodratio5.307082Prob.Ci-Square(l)口120286R.estnctedTestEquation.DependentVariable.DEBTMethod.LeastSquaresr>ate_10/25/15Timp2117Sam

33、ple:19S02011Includedobservations:22VariableCoAfficientStd.Errort-StatisticProb.c40.502V15.837052.55743200193DEF104QG7S009671S3S54864onooREPAYQ977754Q0252723B605OG0OQOQR-squared099B6&5Meandependentvar1216395AdjustedRsquared0998524SDdep町nd臼ntv3r14S5993S已ofregression57OQQ06Akaikeinfotriterion1105294SumsquHredresid6190632Schwarzcrrterion1120172l?？趌ikplihotHl-11Sna內(nèi)HAnnan-Ouinncriter11nR799F-statistic71065呢Durbin-Watsonstat1014741ProbfF-statistic)0OOQOOD輸出結(jié)果上部是關(guān)于約束GDP系數(shù)為零的F檢驗和LR檢驗。由于兩種