多元线性回归模型的案例讲解

1、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Pi, 猪肉价格P2与牛肉价格P3的相关数据。年份Y/千克X/元P1/（元 /千克）P2/（元 /千克）P3/（元 /千克）年份Y/千P1/（元 /千克）P2/（元 /千克）P3/（元 /千克）克X/元19802.783974.225.077.8319924.189113.977.9111.4019812.994133.815.207.9219934.049315.219.5412.4119822.984394.035.407.9219944.0710214.899.4212.7619833.084593.955.53

2、7.9219954.0111655.8312.3514.2919843.124923.735.477.7419964.2713495.7912.9914.3619853.335283.816.378.0219974.4114495.6711.7613.9219863.565603.936.988.0419984.6715756.3713.0916.5519873.646243.786.598.3919995.0617596.1612.9820.3319883.676663.846.458.5520005.0119945.8912.8021.9619893.847174.017.009.3720

3、015.1722586.6414.1022.1619904.047683.867.3210.6120025.2924787.0416.8223.2619914.038433.986.7810.48（1）求出该地区关于家庭鸡肉消费需求的如下模型：lnY = 0。+ 叫 I n X + % I n R + 氏 I nF+P4l nP+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：输出结果如下:VariableCoefficientStd. Errort-StatisticProb.C0.7315200.296947-24634670 0241LOG(X)0,

4、3452570.0825654.1816490.0006LOG(P1)-0.5021220.109891-4 5692940.0002LOG(P2)0.146866.0990061.4834200 1553LOG(P3)0.0871850,0998520.8731370.3941R-squared0.982474Mean dependent var1 361301Adjusted R-sqjared0 978579S.D, dependent var0.107659S.E. of regression0.027465Akaike info criterion-4.162123Sum squar

5、ed resid0.013578Schwarz 匚riterion-3 91527BLog likelihood52.06441F-st artistic252.2633Durbin-Walson stat1 824820Prob(F-statistic)0.000000所以，回归方程为：lnY= -0.7315 + 0.3463In X -0.50211nR +0.1469In F2 +0.0872In P(-2.463)件 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

6、。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：VariableCoefficie ntStd. Errort-StatisticProb.C-1.1257970.088420-12.732370.0000LOG(X)0.4515470.02455418.389660.0000LOG(P1)-0.3727350.063104-5.9066680.0000R-squared0.980287Mean depe ndent var1.361301Adjuste

7、d R-squared0.978316S.D. dependent var0.187659S.E. of regressi on0.027634Akaike info criteri on-4.218445Sum squared resid0.015273Schwarz criteri on-4.070337Log likelihood51.51212F-statistic497.2843Durb in-Watson stat1.877706Prob(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

8、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.

9、000041451.2901973.82027.00000204732.9009228.250222.000055149.3005917.010327.0000212180.2302866.65080.0000062291.1601758.770120.0000222539.7602545.63096.0000071345.170939.100058.00000233046.9504787.900222.00008656.7700694.940031.00000242192.6303255.290163.00009370.1800363.480016.00000255364.8308129.6

10、80244.0000101590.3602511.99066.00000264834.6805260.200145.000011616.7100973.730058.00000277549.5807518.790138.000012617.9400516.010028.0000028867.9100984.520046.00000134429.1903785.91061.00000294611.39018626.94218.0000145749.0208688.030254.000030170.3000610.910019.00000151781.3702798.90083.000003132

11、5.53001523.19045.00000161243.0701808.44033.00000设定模型为：丫二AK : L :eJ(1) 利用上述资料，进行回归分析；(2) 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：In 丫 =1 n A ： ln KTn L J1)进行回归分析:Equat ion Est i*at ionSpecification OptionsEquation specifiC4tion dependent wimble foilowsd by list of regressors and PDL terms OR an axp

12、lici t e i an likeSg®) v L帕 Qc Lo电 CL)IEsti mati on setti.n.gs Method: LSSnple 1 31取消确定得到如下回归结果:VariableCoefficie ntStd. Errort-StatisticProbC1.1539940.7276111.5860040 1240LOG(©0.6092360 1763783.4541490 0018LOG(L)0.3607960 2015911.7897410 0843R-squared0.809925Me a n dependent var7.493997Ad

13、justed R-squared0 796348S.D dep endent var0.942960S.E. of regression0.42553SAkaike info criterion1 220639Sum squared resid5.070303Schwarz criterion1.359612Log likelihood-15.92300F-statistic59.65501Durbin-Watson stat0.793209Prob(F-statiistic)0.000000于是，样本回归方程为：In7=1.154 0.6091n K 0.3611 nL(1.59)(3.45

14、)(1.79)2R =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二0.7963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，?=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:®E¥ie»s - Equat

15、ion: UHTITLED Torkfile: UMTITLED:Until leiEstimatTI尸口l File Edit Oljact Vifree Quick 0tions 址indo咐 Kelp 输翊Pa object Print Name Rresent at ionsEstimatiflA OutputActual, Fitted, Residual 卜 ARJJA Structure.Gradients and Derivatives 卜Covari anc e NlatnuStd Frrrir irRtatitir PmhCoeffi ci«nt Tdual Te

16、stsStability TestsConfi dsnee Ellipse. u Wald = Co 电 ffi ci ent Restrictions.Omitted Variables " Lik&lihood RatioB B,Redundant Vari ables 一 Likelihood Rati o.Wald Test:Equation: IJntitlledTest StatisticValuedfProbabilityF-statistic0.101118（1 28）1075291Chi-square0 1011181Null Hypothesis Summ

17、ary:Normalized Restriction （二 0）ValueStd. Err-1 + C（2） + C©）*0.0299650.094242Restrictions are linear in coefficients.结果如下:Vald Test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）1突变点检验佃95-2012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Xt ）数据（人，元），数据见表3年份y （万辆）X （元）年份yt （万辆）X （

18、元）199528.49739.12004205.423496.2199634.71899.62005249.964283199742.291002.22006289.674838.9199860.421181.42007358.365160.3199973.121375.72008423.655425.1表3中国家用汽车拥有量（yt）与城镇居民家庭人均可支配收入（200081.621510.22009533.885854200196.041700.62010625.3362802002118.22026.62011770.786859.62003155.772577.42012968.9877

19、02.8下图是关于y和xt的散点图:1Q00800-600-400-200-D0 1000300050007000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 佃96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 Hi :备择假设是两个子样本对应的回归参数不等。在 佃95 2012年样本范围内做回归。在回归结果中作如下步骤：O Equation: UMTITLED Torkf lie: CASE6&:Cas.口R

20、eEresentat i amBxtinartian. OutputActual,Residual AEHA Structure.Grdi snts uid Eerivlives *Covariance MatrixCceffici ent Tex七£kResi du&l Tests卜ty Testsit Std. Error t-Statistic Prob38.87504-2.9223ED0.01009Chow Breakpoint Test.S.E. of regression Sum squared resid Log likelihood Durbin-Watson

21、 statChow Eorecast Test.Ramzey RESET I&st.Recursiva Estimates (015 only.89.515128209.5-105 3S030.245356 Prob(F-statistic)Schwarz criterion F-statistic12 03008148.48410 000000NameFreezeEstimateForecastStdtsResids输入突变点:得到如下验证结果：Chow Breakpoint Test: 2006Null Hyp口thesis No breaks at specified break

22、pointsVarying regressors: All equation variablesEquation Sample 1995 2011F-statistic1416.403Prob.F(2J3>0.0000Log likelihood ratio91 60709ProbChi-SquareQ 0000Wald Statistic2332.806Prob.Chi-Square(2)O.QOOO由相伴概率可以知道，拒绝原假设，即两个样本（佃95- 2005年，20062012年）的回归参数不相等。所以，2006年是突变点。2稳定性检验以表3为例，在用1995 2009年数据建立的

23、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：0 1995-2005 D11 2006-2012对1995 2012年的数据进行回归分析：做邹氏稳定性检验:ViewProc ObjectPrintName Freeze EstimateForecastDependent Variable YMethod- Least SquaresDate: 10/25 15 Time: 21:05Sample: 1995 2012Included observations: 18VariableCoeffi

24、cientStd Errort-StatisticProbC-16 074912.9456S45.4571620 0001X0 0636240 00135347 024090.0000D1-352 622512 29131-69 367910 0000D1*X0 1750130.00238173 493290 0000R-squared0.999750Mean dependent var284 2606Adjusted R-squared0 999697S D dependent var278 4439S E of regression4 347808Akaike info critenorr

25、6 1BS060Sum squared resid329.0174Schwarz criterian6.386921Log likelihood-51.69254Hannan-Quinn criter.6.215343F-statistic18669 76Durbin-Watson stat1.765734Prob(F-statistic)0 000D00输入要检验的样本点:得到如下检验结果:Chow Forecast TestEquation: UNTITLEDSpecification: YCXD1 D1*XTest predictions for observations from 20

26、10 to 2012ValuedfProbabilityF-statistic0 433432(3, 11)0.7333Likelihood ratio2 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（ DEBTt，亿元）模型如下：DEBT； =tGDR ：2DEFt ：3REPAY ut其中GDPt表示国内生产总值（百亿元），DEFt表示年财政赤字额（亿元），REPAY表示年还本付息额（亿元）。19902011年数据见表4。表4国

27、债发行总量 DEBTt、GDPt、财政赤字额 DEFt、年还本付息额（REPAY）数据199043.0145.17868.928.582001461.4216.178237.14246.81991121.7448.624-37.3862.892002669.68266.381258.83438.57199283.8652.94717.6555.522003739.22346.344293.35336.22199379.4159.34542.5742.4720041175.25467.594574.52499.36199477.3471.7158.1628.920051549.76584.781

28、581.52882.96199589.8589.644-0.5739.5620061967.28678.846529.561355.031996138.25102.02282.950.1720072476.82744.626582.421918.371997223.55119.62562.8379.8320083310.93783.452922.232352.921998270.78149.283133.9776.7620093715.03820.67461743.591910.531999407.97169.092158.8872.3720104180.1894.4222491.271579

29、.822000375.45185.479146.49190.0720114604959.3332516.542007.73对以上数据进行回归分析:得到如下输出结果：Dependent Variable- DEBT Method: Least Squares Date 10/25/15 Time 21:13 Sample: 1990 2011Included observations 22VariableCoefficient Std Error t-Statistic ProbC4.31400821.667250.1991030 8444GDP0 3452020 15447Q2 2347660

30、 0384DEF099&4030 03161331 486990 0000REPA Y0 8797600 04950817 770220 0000R-squared0.998955Mean dependent var1216 395Adjusted R-squared0.998781S D dependmnt 回1485 993S E of regression51 88705Akaike info criterion10.89898Sum squared resid4346078Schwarz criterion11 09735Log likelihood-115 8388Hanna

31、n-Quinn criter.10 94571F-statistic5735.346Durbin-Watson stat2.116834Prot>(F-statistic0.000000对应的回归表达式为:DEBTt =4.31 0.35GDR 1.00DER 0.88REPAY(0.2)(2.2)(31.5)(17.8)R2 二 0.999, DW = 2.1, F 二 5735.3现在用似然比（LR）统计量检验约束GDPt对应的回归系数等于零是否成立。 过程如下：Coe£fi ci ent TestsC-onf i deuce Ellipse.Re si dxioil Tc

32、stvk世4d. - Coe££ici wikt Restr i ct.i oxkv.i ty TestskQmi tteJ Vari able? - Li k&lihood.LabelK-edumdaxit Vari ables Likelihcicd Kati o. ra-JR-J'FTTT- *1 二 二p_' c _PFL_FL_>R-gquared0.99B955Meart depeirident '/ar1216.395Adjusted R-squared0 993781只 n dependent war14B5.993S.

33、C. of regression51.88705ZKkaike info criterion10.99893Sum squared resid48460.78Schwarz crrteriori11 09735Log likelihood 115.B0BBF-statistic5735.346Durbin-Watson stat2.116834Prob(F-statistic)o aaaaoo,It Sttrll ExmirIStajJ虫UmBnxh,® E Vi ews - Equation: UKTITLED Torkf±1bz CASE6B:zCaseGbV Fil*

34、 Edit Obj«et View Fr*e £ui ek Otiens Window Help劉禄jtfJProc】|objett Print gamBFre冃頑EnimMe FoceMS可5tPtsResid$Reresemt atiEe timati on OutpiLitAc tiiiiJ.j P i 11. ： dl R* x i dlu«J.卜Structure. . tradiexits and Derivatives *Cav«3-s tc« Nl.titiie输入要检验的变量名：0>i+1 ed-Redundant Va

35、ri且ble »0n.& or more test seri esCJicel得到如下输出结果：Redundant Variables: GDPF-st atistic4 994134Prob F(1,1B).038350Log likelihood ratio53B7D32Prob. Chi-Square(l)0.0202SGRestricted Test Equation Dapendent Variable DEBTMethod: Least SquaresDate 10/25/15 Time 21 17Sample 1990 2011Included observat

36、ions: 22VariableCoefficientStd Error t-StatisticProb.C40.5021715 837052.5574320.0193DEF1 0406280 02G71S 3B 94自640 ooooREPAY0 9777S40.02527233 639060.0000Rsquarad0 99S665Mean dependant var1216.395Adjusted R-squared0.998524S.D. dependent 丿日1485.993S.E of regression57 080B8Akaike info criterion11 05294Sum squared resid61906 32Schwarz criterion11 20172Log likelihood-118.5323Hannan-Quinn criter.11.08799F亠 statistic71Q6.592Durbin-Watson stat1 814741Pro b(F-stati stic)0 000000Wald Test- Coefficient R亡stricticin儿Omitted Variables.Test - Likdibood Ra