一元線性回歸模型案例

1、第二章 一元線性回歸模型案例一、中國(guó)居民人均消費(fèi)模型從總體上考察中國(guó)居民收入與消費(fèi)支出的關(guān)系。表2.1給出了 1990年不變價(jià)格測(cè)算的中國(guó)人均國(guó)內(nèi)生產(chǎn)總值（GDPP與以居民消費(fèi)價(jià)格指數(shù)（1990年為100）所見(jiàn)的人均居民 消費(fèi)支出（CONSP兩組數(shù)據(jù)。表2.1 中國(guó)居民人均消費(fèi)支出與人均 GDP（單位：元/人）年份CONSPGDPP年份CONSPGDPP1978395.8000675.10001990797.10001602.3001979437.0000716.90001991861.40001727.2001980464.1000763.70001992966.60001949.80019

2、81501.9000792.400019931048.6002187.9001982533.5000851.100019941108.7002436.1001983572.8000931.400019951213.1002663.7001984635.60001059.20019961322.8002889.1001985716.00001185.20019971380.9003111.9001986746.50001269.60019981460.6003323.1001987788.30001393.60019991564.4003529.3001988836.40001527.00020

3、001690.8003789.7001989779.70001565.9001）建立模型，并分析結(jié)果。輸出結(jié)果為:Dependent Variable: CONSP Method: Least Squares Date: 07/02/08 Time. 20:13Sample: 1978 2000Included observations: 23VariableCoefficientStd. Error t-StatisticProb0201.1189114.8840213.512410.0000GDPP0.3861800.00722263 474710.0000R-squared0.99271

4、0Mean dependent var905.33D4Adjusted squared0.992363S.D. dependent var380.6334S.E. of regression33.26450Akaike info criterion9.929800Sum squared resid23237.06Schwarz criterion10.02854Log likelihood-112.1927F-statistic2859 644Durbin-Wat son stat0.550636Prob(F-statistic)0.000000對(duì)應(yīng)的模型表達(dá)式為：CONSP = 201.10

5、7 0.3862GDPP（13.51）（53.47）R2 =0.9927, F =2859.23,DW =0.55從回歸估計(jì)的結(jié)果可以看出，擬合度較好，截距項(xiàng)和斜率項(xiàng)系數(shù)均通過(guò)了t檢驗(yàn)。中國(guó)人均消費(fèi)增加 10000元，GDF增力口 3862元。二、線性回歸模型估計(jì)表2.2給出黑龍江省伊春林區(qū)1999年16個(gè)林業(yè)局的年木材采伐量和相應(yīng)伐木剩余物數(shù)據(jù)。利用該數(shù)據(jù)（1）畫散點(diǎn)圖；（2）進(jìn)行OLS回歸；（3）預(yù)測(cè)。表2.2 年剩余物yt和年木材采伐量xt數(shù)據(jù)林業(yè)局名年木材剩余物yt （萬(wàn)m）年木材采伐量xt （萬(wàn)m）烏伊嶺26.1361.4東風(fēng)23.4948.3新冃21.9751.8紅星11.5335

6、.9五營(yíng)7.1817.8上甘嶺6.8017.0友好18.4355.0翠巒11.6932.7烏馬河6.8017.0美溪9.6927.3大豐7.9921.5南岔12.1535.5帶嶺6.8017.0朗鄉(xiāng)17.2050.0桃山9.5030.0雙豐5.5213.8合計(jì)202.87532.00（1 ）畫散點(diǎn)圖©ETiewsFile Edi t Obj act ¥i ew Free Quick Dptionz Wiikdotf K*LpO lorkfHe: CASE1 - (d 甘倫円|PBt|objett Ptint| Sd¥e|Doti Range: 1 16 “ 16

7、obsSample: 1 16 “ 16 obs先輸入橫軸變量名，再輸入縱軸變量名Sample.Generate Series.Show GraphEmpty Group (E dit Seri es)Seri_電百 StatisticsGtour StatisticsEs timate Equat i on_ - sEstimate VAR.-.Ln百Line graph E ar graph Scatter lineTie得散點(diǎn)圖(2) OLS估計(jì)肅 EViewssiurex yFile Edit Obj ect ¥i ew Proc Quick OEtioris Window

8、Help彈出方程設(shè)定對(duì)話框Equation EstiaationEqu&tion spcificatianDependent vari able followed by list of regressors and FDL terms QE z expli ci t 電quatioii lik«settingsMethod-Least Squarts(NLS uid A£!1AS ample 1 16得到輸出結(jié)果如圖:® EVieTs - Equation: UHTITLED lorkfile: CASEHCaselL_l Fil« Edit O

9、Lj#ct Vi ew Prcc Quick ORtionw 世 iivdovr HlpView Proc | Object Print Name FreezeEstimate Forecast 15tats Reside |Dependent Variable: YMethod: Least SquaresDate: O6Z28AJ8 Time: 18:20Sample: 1 16Included observations: 16VariableCoefficientStd. Error (-StatisticProbC-0.7629281 220966-0.6248560.5421X0.4

10、042800.03337712.112G60.0000R-squared0.912S90Mean dependent var12.67937Adjusted R-squared1906668S D dependent var6GS546GS.E. of regnession2 036319Akaike info criterion4.376G33Sum squared resid58.05231Schwarz criterion4 473207Log likelihood33.01306F-statistic14671B6Durbin-Watsan stat1 401946Prob(F-sta

11、tistic)0.000000由輸出結(jié)果可以看出，對(duì)應(yīng)的回歸表達(dá)式為：?t - -0.7629 0.4043xt(-0.625) (12.11)R2 =0.9129, F =146.7166, DW =1.48(3) x=20條件下模型的樣本外預(yù)測(cè)方法 首先修改工作文件范圍File Edi t Obj ect View Proe Qmiuk OjtiorLS Wiitd&w Help loTkfile： CASE1 - (d：深杵*航畝c鮎el vf 1)也竺J至也盍。曲亡代Print話的日恥岀117+卜shoTFRJsgru Del&te &nrSampleR 朗 S

12、t Sample.Display Filter: *cr>Structure/Rtsi ze Current Fige. Append to Current Page. Contract Current f age.R« shape Curr ent FCopy/Extract from Current Pa莒亡 Sirt Current Page.將工作文件范圍從1 16改為1 17¥orkfile structureIXIWorkfile structm-* tyttDated " regular frequencyStrtEndFrequencyCan

13、cel確定后將工作文件的范圍改為包括17個(gè)觀測(cè)值，然后修改樣本范圍File Edit Obj ect View Froc Quick Ogti oue Window Help Torkfile: CASE1 - W八課件dat a2casel. rf 1) JDisplay Filter衛(wèi)住(object Print5“eD已taik4卜5hQ3j冠IRYstar已geifSampleSet Sample.Stfuctur電/Resize Cnirrent Page. Ajpend to Cwrent Fag電.Contract Current Page. . Reshape Current

14、PageCcpy/Extract from Current Page Sort Current Fage.將樣本范圍從 116改為1 17SampleSample rafige pairs (or samp 1 e object to copy)1 17IF condition (optional)Cancel打開x的數(shù)據(jù)文件，利用 Edit+/-給x的第17個(gè)觀測(cè)值賦值為20圍CS resid0 y® E Vi era - E(iuatian= TJMTITLED lorkfile: CASE1 = :Casel將Forecast sample 選擇區(qū)把預(yù)測(cè)范圍從1 17改為171

15、7,即只預(yù)測(cè)x=20時(shí)的y的值。由上圖可以知道，當(dāng)x=20時(shí)，y的預(yù)測(cè)值是7.32，yf的分布標(biāo)準(zhǔn)差是 2.145。、表2.3列出了中國(guó)19782000年的參政收入 Y和國(guó)內(nèi)生產(chǎn)總值 GDP勺統(tǒng)計(jì)資料。做出散點(diǎn)圖，建立財(cái)政收入隨國(guó)內(nèi)生產(chǎn)總值變化的一元線性回歸方程。表2.3 中國(guó)歷年財(cái)政收入與 GD數(shù)據(jù)年份財(cái)政收入YGDP年份財(cái)政收入YGDP19781132.2603624.10019902937.10018547.9019791146.3804038.20019913149.48021617.8019801159.9304517.80019923483.37026638.1019811175.

16、7904862.40019934348.95034634.4019821212.3305294.70019945218.10046759.4019831366.9505934.50019956242.20058478.1019841642.8607171.00019967407.99067884.6019852004.8208964.40019978651.14074462.6019862122.01010202.2019989875.95078345.2019872199.35011962.50199911444.0882067.5019882357.24014928.30200013395

17、.2389403.6019892664.90016909.201）做散點(diǎn)圖:®E¥iewsFile Edi t &bj act 矍imw FreeGraphUimitle訓(xùn) k Maw Pgie /*i訥臚|-皿心3鈕1:：(197廳 20:Sample 1573 2OC Sc：0 gdp3 rssidl0 yLi ne gr aphBur gj- 4.ph.ilter *ScatterXY line£ieEmpty Group (Ed.it. Sari is-s)Series ListLi st of series, groups and/or seri

18、esonsgip yQKel得到散點(diǎn)圖如下：14000 12000100008000-A6000400020000 000GDP2）進(jìn)行回歸分析：輸出結(jié)果如下:Dependent Variable: YMethod： Least SquaresDate: 07/02)8 Time: 20:48Sample: 1978 2000Included observations: 23VariableCoefficientStd. Error卜StatisticProb.C556.6477220.89432.5199730.0199GDP0J196070 0052732Z7229B0.0000R-squ

19、ared0.96091 aMean dependsrit var4188.627Adjusted R-squared0 959057S.D. dependent var3613700S.IE. of regression731.2086Akaike info criterion16.11022Sum squared resid11227988Schwarz criterion16.20896Log likelih口od-ia3.2675F-statistic516.3338Durbin-Watson stat0 347372Prob(F-statistic)0.000000對(duì)應(yīng)的表達(dá)式是:Y

20、=556.6 0.12GDP（2.52） （22.72）R2 = 0.96, F = 516.3從上面的結(jié)果可以看出，模型的你擬合度較高，各個(gè)系數(shù)均通過(guò)了t檢驗(yàn)。財(cái)政收入增加 10000元，GDP曽加1200元。四、表2.4給出了某國(guó)1990 1996年間的CPI指數(shù)與S&P500指數(shù)。（1）以CPI指數(shù)為橫 軸，S&P500指數(shù)為縱軸作圖；（2）做回歸模型，并解釋結(jié)果。表2.4某國(guó)歷年CPI與標(biāo)準(zhǔn)普爾指數(shù)年份CPI指數(shù)S&P500旨數(shù)年份CPI指數(shù)S&P500旨數(shù)1990130.7000334.59001994148.2000460.33001991136.2

21、0003764000541.64001992140.3000415.74001996159.6000670.83001993144.5000451.41001）作散點(diǎn)圖:得散點(diǎn)圖如下:680 q640600-45-1J1CPI- - - nu nu nu O 2 8 4 0 5 4 4 4sttr山 SLXDependent variable followed by list of releasors and. FDL t erms. OR an expli ci t equat i on likeEstinnati on settingsMethod： LS2）做回

22、歸估計(jì):Equalion Est i丈：ionSp&ci fi c&ti on OpticTisLeast Squares (WLS and ARMA)gampie 1990 199S確定取稍得到如下結(jié)果：Dependent Variable: SERD1Method: Least SquaresDate: 07AJ3/08 Time: 11:11Sample: 1990 1996Included observations： 7Va riableCoefficientStd Error1-StatisticProb.C*1137.826177.9488-6.394122O.DO

23、UCPI11.083611 2285559 0216620.0003R-squared0.942123Mean dependent var464 38B6Adjusted F?-squared0.930548S D. dependent var112.3728S.E. of regression29,61448Akaike info criterion9.849360Sum squared resid4305006Schwarz criterion9.833906Log likelihood-32 47276F-etatistic81 39039Durbin-Watson stat1.1870

24、41Prob(F-statistic)0.000279對(duì)應(yīng)的回歸表達(dá)式為：S& P 二-1137.83 11.08CPI(-6.39) (9.02)回歸結(jié)果顯示，CPI指數(shù)與S&P指數(shù)正相關(guān)，斜率表示當(dāng) CPI指數(shù)變化1個(gè)點(diǎn)，會(huì)使 S&P指數(shù)變化11.08個(gè)點(diǎn)；截距表示當(dāng) CPI指數(shù)為0是，S&P指數(shù)為-1137.83，此數(shù)據(jù)沒(méi)有 明顯的經(jīng)濟(jì)意義。五、表2.5給出了美國(guó)30所知名學(xué)校的 MBA學(xué)生1994年基本年薪（ASF），GPA分?jǐn)?shù) （從1 4 共四個(gè)等級(jí)），GMA分?jǐn)?shù)，以及每年學(xué)費(fèi)（X）的數(shù)據(jù)。（1）用雙變量回歸模型分析 GPA分?jǐn)?shù)是否對(duì) ASP有影響？（

25、2）用合適的回歸模型分析 GMA分?jǐn)?shù)是否與 ASP有關(guān)？（3）每年的學(xué)費(fèi)與 ASP有關(guān)嗎？如果兩變量之間正相關(guān)，是否意味著進(jìn)到最高費(fèi)用的商 業(yè)學(xué)校是有利的？(4)高學(xué)費(fèi)的商業(yè)學(xué)校意味著高質(zhì)量的MBA成績(jī)嗎？為什么表2.5 美國(guó)30所知名學(xué)校的MBA學(xué)生情況學(xué)校ASP/美兀GPA分?jǐn)?shù)GMA分 數(shù)X/美兀Harvard102630.03.400000650.000023894.00Sta nford100800.03.300000665.000021189.00Columbia n100480.03.300000640.000021400.00Dartmouth95410.003.40000066

26、0.000021225.00Wharto n89930.003.400000650.000021050.00Northwestern84640.003.300000640.000020634.00Chicago83210.003.300000650.000021656.00MIT80500.003.500000650.000021690.00Virgi nia74280.003.200000643.000017839.00UCLA74010.003.500000640.000014496.00Berkeley71970.003.200000647.000014361.00Cornell7197

27、0.003.200000630.000020400.00NUY70660.003.200000630.000020276.00Duke70490.003.300000623.000021910.00Carn egieMello n59890.003.200000635.000020600.00NorthCaroli na69880.003.200000621.000010132.00Michiga n67820.003.200000630.000020960.00Texas61890.003.300000625.00008580.000In dia na58520.003.200000615.

28、000014036.00Purdue54720.003.200000581.00009556.000Case Wester n57200.003.100000591.000017600.00Georgetow n69830.003.200000619.000019584.00Michiga nState41820.003.200000590.000016057.00Penn State49120.003.200000580.000011400.00Souther nMethodist60910.003.100000600.000018034.00Tula ne44080.003.1000006

29、00.000019550.00Illi nois47130.003.200000616.000012628.00Lowa41620.003.200000590.00009361.000Minn esota48250.003.200000600.000012618.00Wash ington44140.003.300000617.000011436.00上述數(shù)據(jù)是個(gè)截面數(shù)據(jù)，建立數(shù)據(jù)文件過(guò)程如下:Workfilft structTire type：Unstructured / Vndate VObservationIrr egiar 息辻 andNjnes options!WF：Page：OKC

30、arLC«lPan*l workfiles. may be made from Unstructured workflifts by later specifying date and/or然后輸入數(shù)據(jù)即可。(1) 以ASP為因變量，GPA為自變量進(jìn)行回歸分析。結(jié)果如下:Dependent Variable: SERO1Method: Least SquaresDate 07X)3/00 Time: 13:D2Sample: 1 30Included observations: 30VariableCoefficientStd Errort-StatisticProbC-273722

31、.595759 31-3.1917900.0035SER02105117 626347.093 9897230 0004R-squared0.362447Mean dependeni var6S2B0.00Adjusted squared0.339677S D dependent var18187.78S E. of regression14779 44Akaike info criterion22.10420Sum squared resid6.12E-HJ9Schwarz criteinon22 19762Log likelihood-329.5630F-statistic15.91789

32、Durbin-Watson stat1 006276Prjb(F-stsrtislic)0 000432從回歸結(jié)果可以看出，GPA分?jǐn)?shù)的系數(shù)是顯著的，對(duì) ASP有正的影響。（2）以ASP為因變量，GMA為自變量做回歸分析，結(jié)果如下：Dependent Variable: SER01Method： Least SquaresDate; 07/03/09 Time: 13:07Sample: 1 30Included observations: 30VariableCoetficientStd. Error t-StatisticProbC-332306.847572.09-6.9053320.0

33、000SERBS641.65987b. 150368 4262220 0000R-squared0.717175Mean dependent var6S260.00Adjusted R-squared0.707074S.D. dependent var18187.78S.IE. of regression9343.701Akaike info criterion21.29139Sum squared resid271E-KBSchwarz criterion21 38480Log likelihood-317.3709F-statistic71 00122Durbin-Watson stat1

34、 120809Prob(F-statistic)0 000000從回歸結(jié)果可以看出，GMAT分?jǐn)?shù)與ASP是顯著正相關(guān)的。(3) 以ASP為因變量，X為自變量進(jìn)行回歸分析，結(jié)果如下:Dependent Variable: SER01Method: Least SquaresDate: 073/08 Time: 13:09Sample: 1 30Included observations: 30VariableCoefficientStd. Error t-StatisticProb.023126.329780.8632 3644460.0262SER042.6334630.5516014 774

35、2520.0001R-squared0.448740Mean dependent var68260.00Adjusted R-squared0 429061S.D. dependent var18187.78S.E of regression13742.78Akaike info criterion21.95876Sum squared resid5.29E4J9Schwarz criterion22 05217Log likelihood-327.3813F-statistic22.79348Durbin-Watson stat1.142178Prob(F-statistic)0 00005

36、1從回歸結(jié)果可以看出，每年的學(xué)費(fèi)與ASP顯著正相關(guān)。學(xué)費(fèi)高，ASP就高；但學(xué)費(fèi)僅解釋了 ASP變化的一部分，明顯還有其他因素影響著ASP（4）以GPA為因變量，X為自變量進(jìn)行回歸分析，結(jié)果如下：Dependent Variable： SER02Method: Least Squares Date: 07ADM8 Time: 13:14Sample: 1 30Included observations: 30VariableCoefficientStd. Error t-StartisticProb.C3.1475790.07255943 379360.0000SER046J7E-064 09E

37、-061.5079520 U28R-squared0.075112Mea n depends nt var3.253333Adjusted R-squared0 042080S D. dependent var0104166S.E. of regression0.101951Akaike info criterion1.664311Sum squared resid0.291032Schwarz criterion-1.570897Log likelihood26.96466F-statistic2.273920Durbin-Watson stat1.702755Prob(F-statisti

38、c)0.142769從回歸結(jié)果可以看出，盡管高學(xué)費(fèi)的商業(yè)學(xué)校與高質(zhì)量的MBA成績(jī)略有正相關(guān)性，但學(xué)費(fèi)對(duì)GPA分?jǐn)?shù)的影響是不顯著的，所以學(xué)費(fèi)并不是影響GPA分?jǐn)?shù)的主要原因。六、表2.6給出了 1988年9個(gè)工業(yè)國(guó)的名義利率（Y）與通貨膨脹率（X）的數(shù)據(jù)。（1）以 利率為縱軸，以通過(guò)膨脹率為橫軸作圖；（2）用OLS法進(jìn)行回歸分析；（3）如果實(shí)際利率不變，則名義利率與通貨膨脹率的關(guān)系如何。表2.6 1988年九個(gè)工業(yè)國(guó)的名義利率與通貨膨脹率國(guó)家Y/%X/%國(guó)家Y/%X/%澳大利亞11.97.7墨西哥66.351加拿大9.44瑞典2.22法國(guó)7.53.1英國(guó)10.36.8德國(guó)41.6美國(guó)7.64.4X

39、意大利11.34.8(1 )作線圖®EVievsreidseiO1Sampl a.GTenerate Seri esP,三Sh&w . . .E牝旳h卜Series)Series StatisticsGtr ouji StatisticsEstin&te Equ&ti&n. 眇RIk L*qjQuivk Oti ons Window HelpRange: 1 9Sample; 1 9Linu graph Bar graph Scatter 冀 line PieFile： Edi t Object Vi* Froc口 Vorkfile:|¥ie

40、H|Prcid ObjectDelete | 匕 eir bam pie |Filter: *Until険(I X 忖ewFnqe /Fath = c： Xdocuments aid se11ingsihuyiximy do-cujments聞-non«將=gtit“dSeries ListLi st of series, groups, artd/or seri es escpresEiojisSERO2(2)作OLS回歸，結(jié)果如下:D即endent Variable: SER01Method: Least SquaresOate: 07?0310 Time: 13:37Sampl

41、e: 1 9Included observations: 9VariableCoefficiientStd. Error t-StatisticPmb.C2.6361740 6913033.8133400.0066SERD21.2502860.03932631 7927J0.0000R-squared0.993122Mean dependent var14.50000Adjusted R-squared0.992140S.D. dependent var1969160S E. of regression1.745010Akaike info criterion4 145445Sum squar

42、ed resid21.33498Schwarz criterion4 189272Log likelihood16.65490F-statistic1010,778Durbin-Wats口n stat1.B1937EPrab(F-statistic)0 000000上述回歸結(jié)果表明，如果實(shí)際利率不變，名義利率與通貨膨脹率呈正向關(guān)系；斜率1.2503表明通貨膨脹率上升 1個(gè)點(diǎn)，名義利率上升 1.25個(gè)點(diǎn)。七、根據(jù)表中提供的數(shù)據(jù)，試建立我國(guó)最終消費(fèi)支出與國(guó)內(nèi)生產(chǎn)總值（單位：億元）之間的回歸模型，并進(jìn)行參數(shù)以及總體的顯著性檢驗(yàn)。當(dāng):=0.05，x2002 = 102398億元時(shí)， 對(duì)y2003進(jìn)行預(yù)

43、測(cè)。表2.7 1978-2001年中國(guó)最終消費(fèi)支出與國(guó)內(nèi)生產(chǎn)總值統(tǒng)計(jì)資料年份最終消費(fèi)（y）國(guó)內(nèi)生產(chǎn)總值（x）年份最終消費(fèi)（y）國(guó)內(nèi)生產(chǎn)總值（x）19782239.13624.1199011365.218547.919792619.44038.2199113145.921617.819802976.14517.8199215952.126638.119813309.14862.4199320182.134634.419823637.95294.7199426796.046759.419834020.55934.5199533635.058478.119844694.57171.019964000

44、3.967884.619855773.08964.4199743579.474462.619866542.010202.2199846405.978345.219877451.211962.5199949722.782067.519889360.114928.3200054616.789442.2198910556.516909.2200158952.695933.3資料來(lái)源：國(guó)家統(tǒng)計(jì)局中國(guó)統(tǒng)計(jì)年鑒2001.北京：中國(guó)統(tǒng)計(jì)出版社，2002(1 )做散點(diǎn)圖如下：從x與y的散點(diǎn)圖可以看出，最終消費(fèi)支出與國(guó)內(nèi)生產(chǎn)總值之間存在線性關(guān)系。因此 可設(shè)定最終消費(fèi)支出 yt與國(guó)內(nèi)生產(chǎn)總值Xt的關(guān)系為yt =b

45、obiXtut(2)根據(jù)模型設(shè)定進(jìn)行線性回歸，結(jié)果如下:UKTIT：Proc Object Print Name Free李| 學(xué)timaftej Forecast Stats 口estds|Dependentvariable: Yf,Method: Least SquaresDate： 10/20/09 Time： 20.45Sample: 1978 2001Included obsercatians: 24VariableCoefficientStd. Errort-StatisticProb.C199.8150204.55510.976B270.3393X0.5959770.004501

46、132.42450.0000R-squared0.998747Mean dependentr；ar19897.37Adjusted R-squared0.998690S.D. dependent var19006.77S E. of regression687.91Q3skaike infc criterion15.984S5Sum squared resid10410853Schwarz criterio n16.03302Log likelihood-189.8182Hannan-Quinn criter1601089F-statistfC1753624Durbin-Watso n stat0.333719Prob(F-statistic)0.000000從結(jié)果中可以知道最終消費(fèi)支出yt與國(guó)內(nèi)生產(chǎn)總值Xt的關(guān)系為:乂二199.8150.59597