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的相關數據。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）求出該地區(qū)關于家庭雞肉消費需求的如下模型：lnY = 0。+ 叫 I n X + % I n R + 氏 I nF+P4l nP+u（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，回歸方程為：lnY= -0.7315 + 0.3463In X -0.50211nR +0.1469In F2 +0.0872In P(-2.463)件 182)(-4.569)(1.483)(0.873)由上述回歸結果可以知道，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

6、。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，就應該去掉該解釋變量。去掉豬肉價格P2與牛肉價格P3重新進行回歸分析，結果如下：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通過比較可以看出，AIC值和SC值都變小了，所以應該去掉豬肉價格 P2與牛 肉價格P3這兩個解釋變量。所

8、以該地區(qū)豬肉與牛肉價格確實對家庭的雞肉消費 不產生顯著影響。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)總產值 Y，資產合計K及職工人數L。工業(yè)總產 資產合計職工人數工業(yè)總產 資產合計 職工人數序號值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.

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設定模型為：丫二AK : L :eJ(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)規(guī)模報酬不變狀態(tài)嗎? 將模型進行雙對數變換如下：In 丫 =1 n A ： ln KTn L J1)進行回歸分析: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電 CL)IEsti mati on setti.n.gs Method: LSSnple 1 31取消確定得到如下回歸結果: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，樣本回歸方程為：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從回歸結果可以看出，模型的擬合度較好，在顯著性水平0.1的條件下，各項系數均通過了 t檢驗。從F檢驗可以看出，方程對 Y的解釋程度較少。R二0.7963表明，工業(yè)總產值對數值的79.6%的變化可以由資產合計對數與職工的對數值的變化來解釋，但仍有 20.4%的變化是由其他因素的變化影響的。從上述回歸結果看，?=0.97 ” 1，即資產與勞動的產出彈性之和近似為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檢驗對約束關系進行檢驗。過程如下:®E¥ie»s - Equat

15、ion: UHTITLED Torkfile: UMTITLED:Until leiEstimatTI尸口l File Edit Oljact Vifree Quick 0tions 址indo咐 Kelp 輸翊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 電 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.結果如下:Vald Test由對應概率可以知道，不能拒絕原假設，即資產與勞動的產出彈性之和為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突變點檢驗佃95-2012年中國家用汽車擁有量（y，萬輛）與城鎮(zhèn)居民家庭人均可支配收入Xt ）數據（人，元），數據見表3年份y （萬輛）X （元）年份yt （萬輛）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中國家用汽車擁有量（yt）與城鎮(zhèn)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下圖是關于y和xt的散點圖:1Q00800-600-400-200-D0 1000300050007000從上圖可以看出，2006年是一個突變點，當城鎮(zhèn)居民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 佃96年是不是一個突變點。Ho:兩個字樣本（1995 2005年，2006 2012年）相對應的模型回歸參數相等 Hi :備擇假設是兩個子樣本對應的回歸參數不等。在 佃95 2012年樣本范圍內做回歸。在回歸結果中作如下步驟：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輸入突變點:得到如下驗證結果：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，拒絕原假設，即兩個樣本（佃95- 2005年，20062012年）的回歸參數不相等。所以，2006年是突變點。2穩(wěn)定性檢驗以表3為例，在用1995 2009年數據建立的

23、模型基礎上，檢驗當把 2010 2012 年數據加入樣本后，模型的回歸參數時候出現(xiàn)顯著性變化。因為已經知道2006年為結構突變點，所以設定虛擬變量：0 1995-2005 D11 2006-2012對1995 2012年的數據進行回歸分析：做鄒氏穩(wěn)定性檢驗: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輸入要檢驗的樣本點:得到如下檢驗結果: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由上述結果可以知道，F(xiàn)值對應的概率為0.73,所以接受原假設，模型加入2010、 2011和2012年的樣本值后，回歸參數沒有發(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：DEBT； =tGDR ：2DEFt ：3REPAY ut其中GDPt表示國內生產總值（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年數據見表4。表4國

27、債發(fā)行總量 DEBTt、GDPt、財政赤字額 DEFt、年還本付息額（REPAY）數據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對以上數據進行回歸分析:得到如下輸出結果：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對應的回歸表達式為: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現(xiàn)在用似然比（LR）統(tǒng)計量檢驗約束GDPt對應的回歸系數等于零是否成立。 過程如下：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蟲UmBnxh,® E Vi ews - Equation: UKTITLED Torkf±1bz CASE6B:zCaseGbV Fil*

34、 Edit Obj«et View Fr*e £ui ek Otiens Window Help劉祿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輸入要檢驗的變量名：0>i+1 ed-Redundant Va

35、ri且ble »0n.& or more test seri esCJicel得到如下輸出結果：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兒Omitted Variables.Test - Likdibood Ra