第三章多元線性回歸模型案例

1、第三章多元線性回歸模型案例一、鄒式檢驗（突變點檢驗、穩(wěn)定性檢驗）1 .突變點檢驗1985 2002年中國家用汽車擁有量 （yt ,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3.1。表3.1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（xt ）數(shù)據(jù)年份yt （萬輛）xt （元）年份yt （萬輛）Xt （元）198528.49739.11994205.423496.2198634.71899.61995249.964283198742.291002.21996289.674838.9198860.421181.41997358.365160.3198973.121375.

2、71998423.655425.1199081.621510.21999533.885854199196.041700.62000625.3362801992118.22026.62001770.786859.61993155.772577.42002968.987702.8卜圖是關于yt和x的散點圖:1000800600-A 400200-口-0 1000300050007000X從上圖可以看出，1996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1996年是不是一個突變點。H。：兩個子樣本（198519

3、95年，19962002年）相對應的模型回歸參數(shù)相等H1：備擇假設是兩個子樣本對應的回歸參數(shù)不等。在19852002年樣本范圍內做回歸。在回歸結果中作如下步驟:口 Equation; UNTITLEDTorkflie;.|print|隔nncprcEge ,E#n)a® 小吃時StdtsResh±得到如下驗證結果:Em色f電w0n七電ti oilsEstinati oil Output 助tuil, Fitted.Risidiiil ARMA Stmcinre .笠印dients and DerivativesCovaiiaitw NitriKCoef fi ci ent

4、TemtUdual Test工卜itStd Error t-StatislicProb?38.3504<292236 口0.0100StAhility Ttitx匚k&wT ：st. . .LabelCtio* F &r e c： as t T t.,-Xmiey RESET Test.購白 Estimit e-s (0L£AdjustecT H-squaredS.E. of regression0 8痢 89.615)oikljr), 一Sum squared ires id Log likelihood Durbin-Watson stat128209 5-

5、105.38030.245355Scwari chierianF statisticProbfF-statislic)12.03008148.4841 0.000000輸入突變點:Chow Breakpoint Test; 1996F-statistic2720.717 卜出 F(2J4)Q00D001Log likelihood ratio107.375BProb. Chi-Square(2)0,000000由相伴概率可以知道，拒絕原假設，即兩個樣本（19851995年，19962002年）的回歸參數(shù)不相等。所以，1996年是突變點。2 .穩(wěn)定性檢驗以表3.1為例，在用1985 1999年數(shù)

6、據(jù)建立的模型基礎上，檢驗當把20002002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1996年為結構突變點，所以設定虛擬變量：0,1985 1995 D11,1996 2002對19852002年的數(shù)據(jù)進行回歸分析:做鄒模型穩(wěn)定性檢驗：EVievs Equation: OTITLED Torkfile: CASE6A: :Casefia Eil* Edi t Objt:t fi" Frtc guickWindow Htlp必邈IProdKb|愉U 回lntM7neReet£ |E曳列日3| Fciiceg" 5tac5 帆5MDepende

7、nt Variable: Y Method Leat Squares Dats: DT/OW0 Time: 09:19 Sample: 1935 ZU2 Included absErvstionE： 1Seseixt a.t i ons.t st i mat ion OutputActual, Fitttd, Ksidual.WMA Slructurcx.4nt= utd”Covari：ance MatrixVariableCoefflcierCoefficient Tests，c-JG.O749Eesidu：al T>sts0001X0.063E52Stabillty TestsCho

8、v BrealoLiit Test.D1-852,622-rVinw Fioiraica'E t Tatf .X*D1D17501LabelRun say EZSET Tes.R-squaiedQ. 999750Mean dependent var如Ricirsiv*ijDLS only) 一一Adjusted R-squaned0,999697S.D. dependent var278.4439S E. of regression4.8473Q9Akaiike info criterion6日即60Sum scuared resid329.0174Schwarz criterion6

9、.305921Log iikdiliood1,69254F-stalistic13639 76Durbin-Wat son Mat1 7GE734Proh(F-stati£tir) innnno輸入要檢驗的樣本點:得到如下檢驗結果：Chaw Forecast Test: Forecast from 2000 to 2002F*t浦5M口,43*32尸）必 FQ11）口，在知Lag likeliJiood ratio2,011W5Prob, Chi-Sqjare（3J0.570105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2000、2001和2002年的

10、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：DEBTt 01GDPt2DEFt3REPAYt 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）。19802001年數(shù)據(jù)見表3.2。表3.2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)年份DEBTGDPDEFREPAY年份DEBTGDPDEFREPAY198043.0145.17868.928.581991461.4216.178237.14246.81981121.7448

11、.624-37.3862.891992669.68266.381258.83438.57198283.8652.94717.6555.521993739.22346.344293.35336.22198379.4159.34542.5742.4719941175.25467.594574.52499.36198477.3471.7158.1628.919951549.76584.781581.52882.96198589.8589.644-0.5739.5619961967.28678.846529.561355.031986138.25102.02282.950.1719972476.827

12、44.626582.421918.371987223.55119.62562.8379.8319983310.93783.452922.232352.921988270.78149.283133.9776.7619993715.03820.67461743.591910.531989407.97169.092158.8872.3720004180.1894.4222491.271579.821990375.45185.479146.49190.0720014604959.3332516.542007.73對以上數(shù)據(jù)進行回歸分析:得到如下輸出結果:Dependent Variable: DEBT

13、Method: Least SquaresDate: 07/0 W9 Time: 10:22Sample: 19S0 2001Included obsenrations： 22VariableCoefficientStd. Error (-StatisticProb.C4.31400B21.5G7250.1991030.6444GDP0.3J52D20 154J7D2,23475B0.0381DEF0.9954030.D3161331.8699O.DOOOREPAY0.8797600.04950817.770220.0000R-9quared0.99B955Mean dependent var

14、1216.395Adjusted P-squared0 998781S D dependent var1485.993S.E of regreesiori5L887D5Akaike info criteri on10,89898Sum squared resid43460.78Schwarz criterion11.09735Log likelihood-115.8888F-ststistic5735.346Durbiri-Watson stat211633JProb(F-st artistic)0.000000對應的回歸表達式為:DEBTt4.310.35GDPt1.00DEFt0.88RE

15、PAYt(0.2)(2.2)(31.5)(17.8)R20.999, DW2.1,F5735.3現(xiàn)在用似然比（LR）統(tǒng)計量檢驗約束 GDPt對應的回3系數(shù) i等于零是否成立。過程如下:® ETiGTs - Equation; UHTITLEI? Torkfile; G&SE6Q; ：Gasc61)II. Fi le Ed.11 0L ject Vi e> Pi-oc Qm ck Opiti ons JYliuIci* HelpYi縣疝后匚PfE gmeg&sze歸就For&c曰st 陶己 曰R君aids； JRe £r e s ent it i

16、 oilsEstimatian. OutputActmal, Fitted R*si dualARNA S true lure.Gr adi ent5 arid D er i vat i ve s匚。ywiMatrixQefEi ci ent TestsResidaal TtstsStability TestsLabelFhtri Frrnr t.Statitir PrnhConfidence Ellipse.上aid - Coefficient RestrictiLons.0mi tted ¥ari a,bles - Likelihood Ra.ti.B日dundant Vai&#

17、39;iable5 - Likelihood io.3 LJKJLJRJR-&quaredAdjusted R-scfuared S.F. of regression Sum squared resid Lug likelihood Durbin-VVatson stat0.9SS955 口月9日7B1 51.8S706 46460765.36862 11£834Mean defjendent var S.D dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-si ati slic

18、)1216.3951495.99310,9969611.097365735.346o.aooooo輸入要檢驗的變量名:得到如下輸出結果:Redundant Variables: GDPF-statisticLog likelihood ratio49941羽 5.387082Prob. F(1,16)口 口支35口 202 日 6Prob. Chi-Squarerest Equation:Dependent Variable: DEBTMethod: Least Squares Date: O7/D1/OB Tinne： 10:35 Sample： 1930 2001Included obse

19、rvations： 22VariableCoefficientStd. Error t-StatisticProb.C40,5021715.837C52.5574320.0193DEF1.0405280.0267183B.943G40.0000REPAY0.9777640.02527238.6S90E0.0000R-squared0 996065Mean dependeni var1216.395Adjusted R-squared0.996524S.D. dependent var1465.993S. E of regression57.0S3S3Akaike info criterion1

20、1,05291Sum squared resid6190S 32Schwarz criterion11 20172Log likalihood-118.5023F-statistic7106.592Durbin-Walson stat1 814741Prob(F*st atistic)0.000000輸出結果上部是關于約束 GDP系數(shù)為零的F檢驗和LR檢驗。由于兩種檢驗的相應概率均小于0.05,即拒接原假設，GDP系數(shù)1不為零，模型中應該保留解釋變量GDP。輸出結果下部是去掉了 GDP變量的約束模型估計結果。Dependent Variable; DEBTMeihod L«a>

21、t SquaresDate: D7/D1/DB Tine 10:43Sample- 198J 2001Inrluded nhaivstinns' 22YdiiableCoefftierRera s ent ati onsE st in 4.1 i cxi Output.JF it ted, & si.AEMA Structure.Gradi 4>.tE uid Dar i vitivos bCcvarime* Matrix.上cunt Iit£1rC4.3140DGOP0.34520DEF0.995REPAY0.07975Residual Tests卜Stabi

22、lity TaeIs卜L4bcl HPitntEllipse.Wild - Coefficient Restrictions.:Oni ttcid. V4ri = Ll底電工Rits -oa.K«dundanl Karitbles - LikeJ-ihmbdL Katid.三、Wald檢驗（以表3.2為例進行 Wald檢驗，對輸出結果進行檢驗。） 檢驗過程如下：E Vie vs - Eq.uaiicnn:UHTITLEDVuzrk'fUe:CASS6B i sCasefibM Fil* Tdit Ot>jsct Vi«wFrac Quick0>ti cn

23、s VidcwHtlpimate F-stR-squarsd0398955Mean d&pendent ®1216.396Adjusted P'squared0 999781S O dependent wir1485 990S.E. of fscrossion51,03705Akaiks info critorior1O,89B93Guin squared rsaid4F4G07Suchwiri criterion11.097TL)g like ihood-1 15.36F-siatislic5755.345Durbin-Watson stat2.113034P ro

24、b (F-statistic)U.UOOOOCJ輸入約束表達式:Vald Test得到如下結果:Wald 7 e5t:Equation: UntitledTest StatisticValue出ProbabilityF-stat istic0,0063923 18)口3348Ctii-square0.0068921口 933ENull Hypothesis Sunrrary;Normalized Restriction (= 0)ValueStd. Err.3*C0-C(3)0.0402030.4&4255Restrictions are linear in coefficients.

25、從輸出結果上部可以看出，相應概率非常大，遠遠大于0.05,表明原假設成立，即約束條件3*c(2) c(3)成立，2是1的3倍。輸出結果的下部給出了約束條件3*c(2) c(3) 0的樣本值和樣本標準差，分別為0.04 和 0.48。四、表3.3中列出了中國2000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(shù)L。表3.3 中國2000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(shù)L序 號工業(yè)總產值Y/億元資產合計K/ 億元職工人數(shù)L/ 萬人序 號工業(yè)總產值Y/億元資產合計K/ 億元職，人數(shù)L/ 萬

26、人13722.7003078.220113.000017812.70001118.81043.0000021442.5201684.43067.00000181899.7002052.16061.0000031752.3702742.77084.00000193692.8506113.110240.000041451.2901973.82027.00000204732.9009228.250222.000055149.3005917.010327.0000212180.2302866.65080.0000062291.1601758.770120.0000222539.7602545.6309

27、6.0000071345.170939.100058.00000233046.9504787.900222.00008656.7700694.940031.00000242192.6303255.290163.00009370.1800363.480016.00000255364.8308129.680244.0000101590.3602511.99066.00000264834.6805260.200145.000011616.7100973.730058.00000277549.5807518.790138.000012617.9400516.010028.0000028867.9100

28、984.520046.00000134429.1903785.91061.00000294611.39018626.94218.0000145749.0208688.030254.000030170.3000610.910019.00000151781.3702798.90083.0000031325.53001523.19045.00000161243.0701808.44033.00000設定模型為：Y AK L e(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(shù)變換如下：lnY ln A ln K In L1)進行回歸分

29、析:得到如下回歸結果:Dependent Variable： LOG(Y)Method： Least SquaresDate: a7/Q3ffl Ume: 1419Sample: 1 31Included observations: 31VariableCoefficientStd Error t-StatisticProbc1.1539940.7276111.5860040.1240LOG(K)0.6092360 1763783.454149wieLOG一0.360795口.20伯911.7S97410.0343R'$quared0,309925Mean dependent 向r7.

30、493997Adjusted squared0.796348S.D. dependent var0.942960S.E. of regression0.425538Akaike info criterion1.220909Sum squared resid5.070303Schwarz criterion1.359512Log likelihood-15.92300F-statistic59.65501Durbin-Walson stat0793209Prob(F-statistic)o.oraoo于是，樣本回歸方程為:1.154 0.6091n K 0.3611n L(1.59) (3.45

31、)(1.79)R20.8099, R0.7963, F 59.66從回歸結果可以看出，模型的擬合度較好，t檢驗。從F檢驗可以看出，方程對 Y在顯著T平0.1的條件下，各項系數(shù)均通過了 的解釋程度較少。R 0.7963表明，工業(yè)總產值對數(shù)值的79.6%的變化可以由資產合計對數(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：®EVICTS - Equation: UWTITLED Tor

32、tfile: UWTITLED:zUntitlefile Edit Qbj ect Vi enr Proc 艮口ick 口衛(wèi)ti que TTindow Help勺口。口切£田PriTiUwaM Fragza E知 m典陀肌乂自 st閨 RgsidsRejreE ent at i 互 on. OutputAc tmal! FL t tedj Jie si dual奧ARM Structure.Covar i ance Nlatr ixStfl FrfnrPmhCoefEicieiLt Tests R« si dual Tts.ts Stability TstsConfi d

33、enee Ellipse. .WaJd "ci ent Restri cti &n.sa u .Onitte-d Vwikb1 1ssi R LatelihQod Rmti* ,Red皿d皿t Variables - Likelihood Ration .Wald Test:E(|uation: UnrtitledTest StatisticValue出ProbabilityF-statiticChi'Square0.101118口.0118(1,23) 1I 0.7529 I | 0 .754 |Null Hypothesis Summary:Normalized

34、Restriction (= 0)ValueStd Err-1 + C0 + C3)-0,0299600.094242Restrictions are linear in coetficients.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 業(yè)在2000年呈現(xiàn)規(guī)模報酬不變的狀態(tài)。結果如下:1,表明中國制造五、已知數(shù)據(jù)如表:YX1X211103298351541285-61、先根據(jù)表中數(shù)據(jù)估計以下回歸模型的方程Yi01 X1iu1iY 02X2i u2iYi01Xi 2X21 Ui（1） 回答下列問題：11嗎？為什么？22嗎？為什么?對上述3個方程進行回歸分析，結果分別如下

35、:Dependent Variable. Y Method: Least SquaresDate： 07/03/08 Time: 14:56Sample: 1 5Included cbservations： 5VariableCoefficientSt(j. Error I-StatieticProb.C-8.8000003.942925-2.2310450 1116X16.6000001 188B375.551B440.0115R-squared0.911297Mean dependent var11 00000AdJjsted R&quared0.001729S.D. ijeper

36、ident var10,93151S. E of regression3.759433Akaike info criterion5775538Sum squared resid42.40000Schtwarz criterion5619063Log likelihood-12.43097F'Statislic30,82075Durbin-Watson stat1 529245Prob(F-st Mistic)0.011526即：Y 8.86.6X1Dependent Variable: Y Method: Least Squares Date： 07/D3/1S Time 14:59S

37、ample: 1 5Included obseivationsVariableCoefficientStd, Error t-StartisticProb.C17.340750.49H41 3S 018200.0000X2-1.6506100.069060 -24.161 BO0.0002R-squared0 934367Mean dependent var11 oooooAdjusted R*sc|U3red0.933163S.D. dependent war1093161S. E. of regression0.902K1Akaike info criterion2.921991Sum 判

38、umd resid1W794Schwarz criienon2766766Log likelihood-6.304977F-statistic5837525urbin-WM&ori etat1.946272Prob(F-statistic)0.000155即： Y 17.34 1.66X2從上述回歸結果可知:1 ?, 2?2。二元回歸與分別對 Xi與X2所作的一元回歸, 其對應的參數(shù)估計不相等， 主要原因在于Xi與X2有很強的相關性。其相關分析結果如下:X1X2096汨681 .oooaooX11.000000-,967668可見，兩者的相關系數(shù)為0.9679。Dependent Va

39、riable: YMethod: Least SquaresDate: 07/0308 Time: 16:01Sample: 1 5Included observations 5VariableCoefficientStd. Error t-Stati$ticProb.C21 922224.3552595.0335070.0373X1-1.1777781.113026 -1 0581760.4009-13444440.2B9315 -7.2199490.0165R-squared0.996722Mean dependent var11,00000Adjusted squared0.993445

40、S D dependent var10.93161S.E. cif regression0.005061Akaike info criteri or26773Q9Sum squared resid1.5666E7Schwarz cntsno n2 643052Log likelihood4.193473F-statislic304.1064Durbiri-Watson 雷2912357Prob(F'Statiistic)0 003273即： Y 21.92 1.18X1 1.94X2六、表3.4中列出了某地區(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P1,豬肉價格P2與牛肉價

41、格P3的相關數(shù)據(jù)。表3.4某地區(qū)家庭人均雞肉年消費量丫與家庭月平均收入 X,雞肉價格P1,豬肉價格P2與牛肉價格P3年份Y/千克X/元Pi/（元/千克）P2/（元/千克）P3/（元/千克）年份Y/千克X/元Pi/（元/千克）P2/（元/千克）P3/（元 /千克）19802.783974.225.077.8319924.189113.977.9111.4019812.994133.815.207.9219934.049315.219.5412.4119822.984394.035.407.9219944.0710214.899.4212.7619833.084593.955.537.921995

42、4.0111655.8312.3514.2919843.124923.735.477.7419964.2713495.7912.9914.3619853.335283.816.378.0219974.4114495.6711.7613.9219863.565603.936.988.0419984.6715756.3713.0916.5519873.646243.786.598.3919995.0617596.1612.9820.3319883.676663.846.458.5520005.0119945.8912.8021.9619893.847174.017.009.3720015.1722

43、586.6414.1022.1619904.047683.867.3210.6120025.2924787.0416.8223.2619914.038433.986.7810.48（1） 求出該地區(qū)關于家庭雞肉消費需求的如下模型:lnY 01 ln X 2lnE 3 ln P24 ln P3 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：Dependant Vansble: LOC(Y) Method: Least SquaresDate: O7/Q3A38 Tim9 15:32Sample: 199D 2c02Included observation

44、s. 23VariableCoefficientSid. Errort-StatisticProb,C-0 7315200.296947-2.4534670.02419G的0.3452570.0025654151549O.OM6L0G(P1)43.5021220.109391-4.5092340.0002L0G(P2)0.1463680.0990061.4834200.1553L0G(P3)0.0871050.0998520.8731370.3941R-squared0.992474Mean dependent 劃1.361301Adjusted R-squaned0.978579S.D. d

45、ependent0 137K9S.E of regreesiori0.D27465Akmik口 info匕府山口門4162123Sum squared reed0.013578Schwarz cmterion-3.916276Log likelihood62 36441F-statistic252.2533Durbiri-Watson stat1.824320Prob(F-ststistic)0000000所以，回歸方程為:輸出結果如下:1nY 0.7315 0.34631n X 0.50211n P 0.14691n P20.08721n P3(-2.463)(4.182)(-4.569)(

46、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。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:VariableCoefficientStd. Errort-StatisticProb.C-1.1257970.088420-12.732370.0000LOG(X)0.4515470.02455418.389660.0000LOG(P1)-0.3727350.

47、063104-5.9066680.0000R-squared0.980287Mean dependent var1.361301Adjusted R-squared0.978316S.D.dependent var0.187659S.E. of regression0.027634Akaike info criterion-4.218445Sum squared resid0.015273Schwarz criterion-4.070337Log likelihood51.51212F-statistic497.2843Durbin-Watson stat1.877706Prob(F-stat

48、istic)0.000000通過比較可以看出，AIC值和SC值都變小了，所以應1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生顯著影響。七、某硫酸廠生產的硫酸的透明度指標一直達不到優(yōu)質要求，經(jīng)分析透明度低與硫酸中金屬雜質的含量太高有關。影響透明度的主要金屬雜質是鐵、鈣、鉛、鎂等。通過正交試驗 的方法發(fā)現(xiàn)鐵是影響硫酸透明度的最主要原因。測量了 47組樣本值，數(shù)據(jù)見表 3.5。表3.5硫酸透明度y與鐵雜質含量x數(shù)據(jù)序數(shù)XY序數(shù)XY131190256050232190266041334180276152435140286334536150296440

49、6371203065257391103169308408132742094210033744010428034762511431103579301243803685251348683787161449803889161550503999201652704076201752504110020185360421002019544443110152054544411015215648451222722565046154202358564721020245852硫酸透明度y與鐵雜質含量的散點圖如下所以應該建立非線性回歸模型。1.通過線性化的方式估計非線性模型。(1)建立倒數(shù)模型，在 Equation Specification(方程設定)框中輸入得到輸出結果為EVievs - Equation; UWTITLED lorkfile: CASE2:Case2a£ilt Edit Qbject Visw oc grick 0jtitjns Window JJelpM蹴I JPr白亡。國電小PKritJPame|FF2cn E此ima忸底。叫七自專耳叵激可回已加kDe pendeni Variable: 1/YMethod; Least SquaresDate; 0G/2