一元线性回归模型案例

1、第二章一元线性回归模型案例一、中国居民人均消费模型从总体上考察中国居民收入与消费支出的关系。表2.1给出了1990年不变价格测算的中国人均国内生产总值（GDPP）与以居民消费价格指数（1990年为100）所见的人均居民消费支出（CONSP）两组数据。表2.1中国居民人均消费支出与人均GDP（单位：元/人）年份CONSPGDPP年份CONSPGDPP1978395.8000675.10001990797.10001602.3001979437.0000716.90001991861.40001727.2001980464.1000763.70001992966.60001949.80019815

2、01.9000792.400019931048.6002187.9001982533.5000851.100019941108.7002436.1001983572.8000931.400019951213.1002663.7001984635.60001059.20019961322.8002889.1001985716.00001185.20019971380.9003111.9001986746.50001269.60019981460.6003323.1001987788.30001393.60019991564.4003529.3001988836.40001527.00020001

3、690.8003789.7001989779.70001565.9001）建立模型，并分析结果。输出结果为:DependentVariable:CONSPMethod:LeastSquaresDate:07/02/08Time.20:13Sample:19782000Includedobservations:23VariableCoefficientStd.Errort-StatisticProbC201.1189114.8840213.512410.0000GDPP0.3861800.00722263474710.0000R-squared0.992710Meandependentvar90

4、5.3304Adjustedsquared0.992363S.D.dependentvar380.6334S.E.ofregression33,26450Akaikeinfocriterion9.929SOOSumsquaredresid23237.06Schwarzcriterion10,02854Loglikelihood-112.1927F-statistic2859644Durbin-Watsonstart0.550636Prob(F-statistic)0.000000对应的模型表达式为：CONSP=201.1070.3862GDPP_2（13.51）（53,47）R=0.9927,

5、F=2859.23,DW=0.55从回归估计的结果可以看出，拟合度较好，截距项和斜率项系数均通过了t检验。中国人均消费增加10000元，GDP增加3862元。二、线性回归模型估计表2.2给出黑龙江省伊春林区1999年16个林业局的年木材采伐量和相应伐木剩余物数据。利用该数据（1）画散点图；（2）进行OLS回归；（3）预测。表2.2年剩余物yt和年木材采伐量xt数据林业局名年木材剩余物yt（万m）年木材米伐重xt（万m）乌伊岭26.1361.4东风23.4948.3iw21.9751.8红星11.5335.9五营7.1817.8上甘岭6.8017.0友好18.4355.0翠面11.6932.7乌

6、马河6.8017.0美溪9.6927.3大丰7.9921.5南岔12.1535.5带岭6.8017.0朗乡17.2050.0桃山9.5030.0双丰5.5213.8合计202.87532.00（1）画散点图FileEditObject¥iewFreeQuick。卫tionWindowK*LpO>orkfHe:CASE1-(dViewProcObjectPrint1DehcRange:116-16obsSample:1161BobsCresidSample.GenerateSeries.Show.,GraphEmptyGroup(EditSeries)Seri_esStatist

7、icsGtoiirStatisticsEstimateEquation.srT=HLinegraphEargraphScatter注lineTieEstimateVAR.先输入横轴变量名，再输入纵轴变量名SeriesListIDm处如仪）&口7DOKCancel得散23.2420>怕.12(2) OLS估计牌EViewsFileEditObject¥iewProcQuickClRtionsWindowHelp汨SIreM¥弹出方程设定对话框EquationEstiaationEqu&tionsp«cificatianDependentvari

8、ablefollowedbylistofregressorsandFDLtermw,QE皿explicitequationlik«tionsettingsMethod-LeastSquarts_(NLSuidA£!1ASample116确定得到输出结果如图:®EVieTs-Equation:UHTITLEDlorkfile:CASEHCaselL_lFil«Edit0Lj#ct¥iewFreeQuickOption'WindowHelpView|PrDc|objeut|Priit|NwtieFreezeE优加趾巳|Foreua5H53t

9、MRe与汨51DependentVariable:YMethod:LeastSquaresDate:O6/28AJ8Time:18:20Sample:116Includedobservations:16VariableCoefficientStd.Errort-StatisticProbC-0.7629281220966-0.6248560.5421X0.4042800.03337712.112G60.0000R-squared0.912S90Meandependentvar12.67937AdjustedR-squared0.906668SDdependentvar6665466S.E.of

10、regnession2036319Akaikeinfocriterion4.376633Sumsquaredresid58.05231S匚hwdracriterion4473207Loglikelihood33.01306F-statistic14671B6Durbin-Watsonstat1401946Prob(F-statistic)0.000000由输出结果可以看出，对应的回归表达式为：%-0.76290.4043xt(-0.625)(12.11)R2=0.9129,F=146.7166,DW=1.48(3) x=20条件下模型的样本外预测方法首先修改工作文件范围ViewsFileEdi

11、tObjectViewProeRuickOjtiorLSWiitd&wHelplorkfile：CASE1-(1二课件宜大vf1)DisplayFilter:*SampleMe,im应耳ObjectPrintsaved温Ik+Usho内因记RJstore口目曰te归bm：RanStSample.=Structure/ResireCurrentFige.CwAppendtoCurrentPage.XContractCurrentfigs.yReshapeCurrentPageCopy/ExtractfromCurrentPw管士SirtCurrentP4g«.将工作文件范围从1

12、16改为117IXIWorkfilestructyrttypeD4tewpeeiEiestionDated-regularfrequencyStirtEndFrequency¥orkfilestructure确定后将工作文件的范围改为包括17个观测值，然后修改样本范围EVievsFileEditObjectViewFroc业tickOgtioueWindowHelpforkfile:CASE1-(dtKttXdata2casel.rf1)回Vtew忸其小33次日Print5weDe3ik+/-5h口碑57115七011GerirSampleSetSample.Structurft/E

13、&sizeCurrentF自售电.AppendtoCurrentPage.ContractCurrentFage.ReshapeCm-rentFsgeCcpy/EutractfromCurrentPageSortCurrentFag4r,.将样本范围从116改为117Samplerangepirs(orw铀pleobjecttocopy)1LTIFc«ndition(options!)£anc±1打开x的数据文件，利用Edit+/-给x的第17个观测值赋值为20回C必resid0Y需EVi.w-E(iuatian=WTITLEDlorkfile:CASE1

14、=:CaselQofeJFileEdit0tjectViewProcuickOtionsirtdowHelp.¥iw;(Pro匚Jobjei±叵毗岫11司曰一酬EftinnateFarecmstl./tats限sidgDependentVariable:YMethod:LeastSquaresDate:06/29/08Time:18:17Sample(adjusted):116Includedobservations:16afteradjuVariableCoefficient-0.76292B0.404280R-squaredAdjustedR-squaredS.E.of

15、regressionSumsquaredresidLoglikelihoodDurbin-Watsonstat0.9128900.906E6B2.03631958,05231-33.0130G1.43194EPath.=c：doelements:andse11ingszhuyixidocumentDE=noneWE=easel将Forecastsample选择区把预测范围从117改为1717,即只预测x=20时的y的值。由上图可以知道，当x=20时，y的预测值是7.32,yf的分布标准差是2.145。三、表2.3列出了中国19782000年的参政收入Y和国内生产总值GDP的统计资料。做出散点图

16、，建立财政收入随国内生产总值变化的一元线性回归方程。表2.3中国历年财政收入与GDP数据年份财政收入YGDP年份财政收入YGDP19781132.2603624.10019902937.10018547.9019791146.3804038.20019913149.48021617.8019801159.9304517.80019923483.37026638.1019811175.7904862.40019934348.95034634.4019821212.3305294.70019945218.10046759.4019831366.9505934.50019956242.2005847

17、8.1019841642.8607171.00019967407.99067884.6019852004.8208964.40019978651.14074462.6019862122.01010202.2019989875.95078345.2019872199.35011962.50199911444.0882067.5019882357.24014928.30200013395.2389403.6019892664.90016909.201）做散点图：得到散点图如下：14000-r12000100008000600040002000020000400006000080000100000G

18、DP2）进行回归分析：输出结果如下:DependentVariable:YMethod：LeastSquaresDate:D7/02AJ8Time:20:40Sample:19782000Includedobservations:23VariableCoefficientStd.ErrorStatisticProb.C556.6477220.89432.5199730.0199GDP011980700052732Z7229B0.0000R-squared0.96091aMeandependsritvar4188.627AdjustedR-squared0959057S.D.dependentv

19、ar3613700S.E.ofregression731.2086Akaikeinfocriterion16,11022Sumsquaredresid11227988Schwarzcriterion16,20895Loglikelihood83.2675F-statistic516.3338Durbin-Watsonstat0347372Prob(F-statistic)0.000000对应的表达式是:Y=556.60.12GDP2（2.52）（22,72）R=0.96,F=516.3从上面的结果可以看出，模型的你拟合度较高，各个系数均通过了t检验。财政收入增加10000元，GDP增加1200

20、元。四、表2.4给出了某国19901996年间的CPI指数与S&P500指数。（1）以CPI指数为横轴，S&P500指数为纵轴作图；（2）做回归模型，并解释结果。表2.4某国历年CPI与标准普尔指数年份CPI指数S&P500指数年份CPI指数S&P500指数1990130.7000334.59001994148.2000460.33001991136.20003764000541.64001992140.3000415.74001996159.6000670.83001993144.5000451.41001）作散点图:nJFiXfi1

21、EditOhjactVi*wFitijcQaitkQtiotleWiTidowHalpS-ampla.1Gener-a.teSezrics.Vie西procc>bi白匚11|pGraph*Rwnq白：19901991EspLyGroup(EditSeries)Sample.199D1991|Serie£$ti/+iu，*q°.GtqurStistics/EstimateEauati&Ti.叵serdlE=timitftVAR.ijlqteakLin«gr>n.pRR*rgrap>tScaitt«-rKTlineFiampgt&a

22、mp;r*<UmirleclXNewFageyn130TFdth-e：docuiTiaxitsutdsattizhuvisciA-ntkDBWF=imtitladListofseries,group零11sand/orseri«sexpressionsOKCancel得散点图如下:680q640-600560-Q520480440-°°400-03BD-0320-rrrrrr160130135140145150155CPI2)做回归估计:得到如下结果：DependentVariable:SERD1Method:LeastSquaresDate:07AJ3/0

23、8Time:11:11Sample:19901996Includedobservations：7ValiableCoefficientStdError1-StatisticProb.C*1137.826177.9488-6.394122O.DOUCPI11,08361122855590216620.0003R-squared0.942123Meandependentvar46438B6AdjustedF?-squared0.930548SD.dependentvar112.3728S.E.ofregression29,61448Akaikeinfocriterion9.849360Sumsqu

24、aredresid4305.006Schwarzcriterion9.833906Loglikelihood-3247276F-statistic8139039Durbin-Watsonstat1.187041Prob(F-statistic)0.000279对应的回归表送式为：S&P=-1137.8311.08CPI(-6.39)(9.02)回归结果显示，CPI指数与S&P指数正相关，斜率表示当CPI指数变化1个点，会使S&P指数变化11.08个点；截距表示当CPI指数为0是，S&P指数为-1137.83,此数据没有明显的经济意义。五、表2.5给出了美国30所

25、知名学校的MBA学生1994年基本年薪(ASP),GPA分数(从14共四个等级)，GMAT分数，以及每年学费(X)的数据。(1) 用双变量回归模型分析GPA分数是否对ASP有影响？(2) 用合适的回归模型分析GMAT分数是否与ASP有关？(3) 每年的学费与ASP有关吗？如果两变量之间正相关，是否意味着进到最高费用的商业学校是有利的？(4)高学费的商业学校意味着高质量的MBA成绩吗？为什么表2.5美国30所知名学校的MBA学生情况学校ASP/fbrGPA分数GMAT分数X/fbrHarvard102630.03.400000650.000023894.00Stanford100800.03.3

26、00000665.000021189.00Columbian100480.03.300000640.000021400.00Dartmouth95410.003.400000660.000021225.00Wharton89930.003.400000650.000021050.00Northwestern84640.003.300000640.000020634.00Chicago83210.003.300000650.000021656.00MIT80500.003.500000650.000021690.00Virginia74280.003.200000643.000017839.00

27、UCLA74010.003.500000640.000014496.00Berkeley71970.003.200000647.000014361.00Cornell71970.003.200000630.000020400.00NUY70660.003.200000630.000020276.00Duke70490.003.300000623.000021910.00CarnegieMellon59890.003.200000635.000020600.00NorthCarolina69880.003.200000621.000010132.00Michigan67820.003.20000

28、0630.000020960.00Texas61890.003.300000625.00008580.000Indiana58520.003.200000615.000014036.00Purdue54720.003.200000581.00009556.000CaseWestern57200.003.100000591.000017600.00Georgetown69830.003.200000619.000019584.00MichiganState41820.003.200000590.000016057.00PennState49120.003.200000580.000011400.

29、00SouthernMethodist60910.003.100000600.000018034.00Tulane44080.003.100000600.000019550.00Illinois47130.003.200000616.000012628.00Lowa41620.003.200000590.00009361.000Minnesota48250.003.200000600.000012618.00Washington44140.003.300000617.000011436.00上述数据是个截面数据，建立数据文件过程如下:然后输入数据即可。(1) 以ASP为因变量，GPA为自变量进

30、行回归分析。结果如下:DependentVariable:SER01Method：LeastSquaresDate07X)3/00Time:13:D2Sample:130Includedobservations:30VariableCoefficientStdErrort-StatisticProbC-273722.59575931-3.1917900.0035SER02105117626347.09398972300004R-squared0.362447Meandependentvar68260.00Adjustedsquared0.339677SDdependentvar18187.78

31、SE.ofregression1477944Akaikeinfocriterion22.10420Sumsquaredresid6.12E-HJ9Schwarzcriteinon2219762Loglikelihood-329.5630F'Statistic15.91789Durbin-Watsonstat1006276Prob(F-statistic)0000432从回归结果可以看出，GPA分数的系数是显著的，对ASP有正的影响。(2) 以ASP为因变量，GMAT为自变量做回归分析，结果如下：DependentVariable:SER01Method:LeastSquaresDate

32、:07ABO8Time:13:07Sample:130Includedobservations:30VariableCoefficientStd.Errort-StartisticProbC-33230G.847572.09-6.9853320.0000SER03641.6598Zb.15036842622200000R-squared0.717175Meandependentvar6S260.00AdjustedR-squared0.707074S.D.dependentvar18187.78S.E.ofregression9343701Akaikeinfocriterion21.29139

33、Sumsquaredresid271E-KBSchwarzcriterion2138480Logllikelihood-317.3709F-statistic7100122Durbin-Watsonstat1128809Prob(F-statistic)0000000从回归结果可以看出，GMAT分数与ASP是显著正相关的。（3）以ASP为因变量，X为自变量进行回归分析，结果如下:DependentVariable:SER01Method:LeastSquaresDate:073/08Time：13:09Sample:130Includedobservations:30VariableCoeff

34、icientStd.Errort-StatisticProb.C23126329780.86323644460.0252SER0426334630.55160147742520.0001R-squared0448748Meandependentvar68260.00AdjustedR-squared0.429061S.D.dependentvar10107.73S.Eofregression1374278Akaikeinfocriterion21.95876Sumsquaredresid5.29E-KJ9Schwarzcriterion2205217Loglikelihood-327.3013

35、F-statistic22.79348Durbin-Watsonstat1142178Prob(F-statistic)0000051从回归结果可以看出，每年的学费与ASP显著正相关。学费高，ASP就高；但学费仅解释了ASP变化的一部分，明显还有其他因素影响着ASP。（4）以GPA为因变量，X为自变量进行回归分析，结果如下：DependentVariable:SER02Method:LeastSquaresDate.07ADM8Time:13:14Sample:130Includedobservations:30VariableCoefficientStdErrort-StatisticPro

36、b.C3J475790.07255943.379360.0000SER04617E-D6409E-0615079520U28R-squared0.075112Meandependentvar3.253333AdjustedR-squared0042080SD.dependentvar0104166Sofregression0,1101951Akaikeinfocriterion1.664311Sumsquaredresid0.291032Schwarzcriterion-1.570897Loglikelihood26.96466F-statistic2.273920Durbin-Watsoni

37、stat1.702755Prob(F-statistic)0142768从回归结果可以看出，尽管高学费的商业学校与高质量的MBA成绩略有正相关性，但学费对GPA分数的影响是不显著的，所以学费并不是影响GPA分数的主要原因。六、表2.6给出了1988年9个工业国的名义利率（Y）与通货膨胀率（X）的数据。（1）以利率为纵轴，以通过膨胀率为横轴作图；（2）用OLS法进行回归分析；（3）如果实际利率不变，则名义利率与通货膨胀率的关系如何。表2.61988年九个工业国的名义利率与通货膨胀率国家Y/%X/%国家Y/%X/%澳大利亚11.97.7墨西哥66.351加拿大9.44瑞典2.22法国7.53.1英

38、国10.36.8德国41.6美国7.64.411.34.8意大利(1)作线图®EVievsFile：EditObjectViwFroc口¥orkfile:“|曰网|.|-口匚ObjectguickOttjeng立indo*H&lpRange:19Sample;19ra回！3120reidserO1SeriesStatisticsGroujiStatisticsEstift&teEquation.E3inateVAR.Sampla.GenerateSeriesP,PShaw电叩hEmptyGroup(EditSeries)UrHTl档NewPag日/Fath=

39、c：Xdocumentswds»ttirLgszhuyiximyd&cmntnts-non«WF-untitled-403020on£3sSeriesList10-0、IIIIIB0102030405060SERO2(2)作OLS回归，结果如下:D叩白ndentVariable:SER01Method:LeastSquaresDate:U7J03J08Time:13:37Sample:19Includedobservations:9VariableCoefficientStd.Errort-StatisticPmb.C2.63617406913033.81

40、33400.0066SERD21.2502860.0393263179270.0000R-squared0.993122Meandependentvar14,50000AdjustedR-squared0.992140S.D.dependentvar1969160SE.ofregression1.745010Akaikeinfocriterion4145445Sumsquaredresid21.33498Schwarzcriterion4189272Loglikelihood16.65490F-statistic10W778Durbin-Watsonstat1.B1937EPrab(F-sta

41、tistic)0000000上述回归结果表明，如果实际利率不变，名义利率与通货膨胀率呈正向关系；斜率1.2503表明通货膨胀率上升1个点，名义利率上升1.25个点。七、根据表中提供的数据，试建立我国最终消费支出与国内生产总值（单位：亿元）之间的回归模型，并进行参数以及总体的显著性检验。当a=0.05,x2002=102398亿元时，对丫2003进行预测。表2.71978-2001年中国最终消费支出与国内生产总值统计资料年份最终消费（y）国内生产总值（x）年份最终消费（y）国内生产总值（x）19782239.13624.1199011365.218547.919792619.44038.2199

42、113145.921617.819802976.14517.8199215952.126638.119813309.14862.4199320182.134634.419823637.95294.7199426796.046759.419834020.55934.5199533635.058478.119844694.57171.0199640003.967884.619855773.08964.4199743579.474462.619866542.010202.2199846405.978345.219877451.211962.5199949722.782067.519889360.11

43、4928.3200054616.789442.2198910556.516909.2200158952.695933.3资料来源：国家统计局.中国统计年鉴2001.北京：中国统方t出版社，2002(1)做散点图如下：从x与y的散点图可以看出，最终消费支出与国内生产总值之间存在线性关系。因此可设定最终消费支出见与国内生产总值Xt的关系为yt=b0biXtut(2)根据模型设定进行线性回归，结果如下:Viez.Proc:ObjectPrintNameFreezeEstimateForecastStats：ResidsDependentvariable:YMethod:LeastSquaresDat

44、e：10/20/09Time：20.45Sample：19782001Includedobservations;24VariableCoefficientStd.Errort-StatisticProb.C199.S150204.55510.976B270.3393X0.5959770.004501132.42450.0000R-squared0.993747Meandependentr；ar19897.37AdjustedR-squared0,998690S.D.dependentvar19006.77SE.ofregression687.9103hkaikeinfocritenon15,984S5Sumsquaredresid10410853Schwarzcriterion16,08302Loglikelihood-189.8182Ha