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、中國居民人均消費模型從總體上考察中國居民收入與消費支出的關系。表2.1給出了 1990年不變價格測算的中國人均國內生產總值（GDPP與以居民消費價格指數(shù)（1990年為100）所見的人均居民 消費支出（CONSP兩組數(shù)據(jù)。表2.1 中國居民人均消費支出與人均 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，并分析結果。輸出結果為: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對應的模型表達式為：CONSP = 201.10

5、7 0.3862GDPP（13.51）（53.47）R2 =0.9927, F =2859.23,DW =0.55從回歸估計的結果可以看出，擬合度較好，截距項和斜率項系數(shù)均通過了t檢驗。中國人均消費增加 10000元，GDF增力口 3862元。二、線性回歸模型估計表2.2給出黑龍江省伊春林區(qū)1999年16個林業(yè)局的年木材采伐量和相應伐木剩余物數(shù)據(jù)。利用該數(shù)據(jù)（1）畫散點圖；（2）進行OLS回歸；（3）預測。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m）年木材采伐量xt （萬m）烏伊嶺26.1361.4東風23.4948.3新冃21.9751.8紅星11.5335

6、.9五營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202.87532.00（1 ）畫散點圖©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得散點圖(2) OLS估計肅 EViewssiurex yFile Edit Obj ect ¥i ew Proc Quick OEtioris Window

8、Help彈出方程設定對話框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由輸出結果可以看出，對應的回歸表達式為：?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測方法 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個觀測值，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個觀測值賦值為20圍CS resid0 y® E Vi era - E(iuatian= TJMTITLED lorkfile: CASE1 = :Casel將Forecast sample 選擇區(qū)把預測范圍從1 17改為171

15、7,即只預測x=20時的y的值。由上圖可以知道，當x=20時，y的預測值是7.32，yf的分布標準差是 2.145。、表2.3列出了中國19782000年的參政收入 Y和國內生產總值 GDP勺統(tǒng)計資料。做出散點圖，建立財政收入隨國內生產總值變化的一元線性回歸方程。表2.3 中國歷年財政收入與 GD數(shù)據(jù)年份財政收入YGDP年份財政收入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）做散點圖:®E¥iewsFile Edi t &bj act 矍imw FreeGraphUimitle訓 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得到散點圖如下：14000 12000100008000-A6000400020000 000GDP2）進行回歸分析：輸出結果如下: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對應的表達式是:Y

20、=556.6 0.12GDP（2.52） （22.72）R2 = 0.96, F = 516.3從上面的結果可以看出，模型的你擬合度較高，各個系數(shù)均通過了t檢驗。財政收入增加 10000元，GDP曽加1200元。四、表2.4給出了某國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，并解釋結果。表2.4某國歷年CPI與標準普爾指數(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）作散點圖:得散點圖如下: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: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對應的回歸表達式為：S& P 二-1137.83 11.08CPI(-6.39) (9.02)回歸結果顯示，CPI指數(shù)與S&P指數(shù)正相關，斜率表示當 CPI指數(shù)變化1個點，會使 S&P指數(shù)變化11.08個點；截距表示當 CPI指數(shù)為0是，S&P指數(shù)為-1137.83，此數(shù)據(jù)沒有 明顯的經濟意義。五、表2.5給出了美國30所知名學校的 MBA學生1994年基本年薪（ASF），GPA分數(shù) （從1 4 共四個等級），GMA分數(shù)，以及每年學費（X）的數(shù)據(jù)。（1）用雙變量回歸模型分析 GPA分數(shù)是否對 ASP有影響？（

25、2）用合適的回歸模型分析 GMA分數(shù)是否與 ASP有關？（3）每年的學費與 ASP有關嗎？如果兩變量之間正相關，是否意味著進到最高費用的商 業(yè)學校是有利的？(4)高學費的商業(yè)學校意味著高質量的MBA成績嗎？為什么表2.5 美國30所知名學校的MBA學生情況學校ASP/美兀GPA分數(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)是個截面數(shù)據(jù)，建立數(shù)據(jù)文件過程如下: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為自變量進行回歸分析。結果如下: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從回歸結果可以看出，GPA分數(shù)的系數(shù)是顯著的，對 ASP有正的影響。（2）以ASP為因變量，GMA為自變量做回歸分析，結果如下：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從回歸結果可以看出，GMAT分數(shù)與ASP是顯著正相關的。(3) 以ASP為因變量，X為自變量進行回歸分析，結果如下: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從回歸結果可以看出，每年的學費與ASP顯著正相關。學費高，ASP就高；但學費僅解釋了 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為自變量進行回歸分析，結果如下：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從回歸結果可以看出，盡管高學費的商業(yè)學校與高質量的MBA成績略有正相關性，但學費對GPA分數(shù)的影響是不顯著的，所以學費并不是影響GPA分數(shù)的主要原因。六、表2.6給出了 1988年9個工業(yè)國的名義利率（Y）與通貨膨脹率（X）的數(shù)據(jù)。（1）以 利率為縱軸，以通過膨脹率為橫軸作圖；（2）用OLS法進行回歸分析；（3）如果實際利率不變，則名義利率與通貨膨脹率的關系如何。表2.6 1988年九個工業(yè)國的名義利率與通貨膨脹率國家Y/%X/%國家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7.53.1英國10.36.8德國41.6美國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上述回歸結果表明，如果實際利率不變，名義利率與通貨膨脹率呈正向關系；斜率1.2503表明通貨膨脹率上升 1個點，名義利率上升 1.25個點。七、根據(jù)表中提供的數(shù)據(jù)，試建立我國最終消費支出與國內生產總值（單位：億元）之間的回歸模型，并進行參數(shù)以及總體的顯著性檢驗。當:=0.05，x2002 = 102398億元時， 對y2003進行預

43、測。表2.7 1978-2001年中國最終消費支出與國內生產總值統(tǒng)計資料年份最終消費（y）國內生產總值（x）年份最終消費（y）國內生產總值（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資料來源：國家統(tǒng)計局中國統(tǒng)計年鑒2001.北京：中國統(tǒng)計出版社，2002(1 )做散點圖如下：從x與y的散點圖可以看出，最終消費支出與國內生產總值之間存在線性關系。因此 可設定最終消費支出 yt與國內生產總值Xt的關系為yt =b

45、obiXtut(2)根據(jù)模型設定進行線性回歸，結果如下:UKTIT：Proc Object Print Name Free李| 學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從結果中可以知道最終消費支出yt與國內生產總值Xt的關系為:乂二199.8150.59597