多元线性回归模型案例

PAGE/NUMPAGES我国农民收入影响因素的回归分析本文力图应用适当的多元线性回归模型,对有关农民收入的历史数据和现状进行分析,探讨影响农民收入的主要因素,并在此基础上对如何增加农民收入提出相应的政策建议。 农民收入水平的度量常采用人均纯收入指标。影响农民收入增长的因素是多方面的，既有结构性矛盾因素，又有体制性障碍因素。但可以归纳为以下几个方面：一是农产品收购价格水平。二是农业剩余劳动力转移水平。三是城市化、工业化水平。四是农业产业结构状况。五是农业投入水平。考虑到复杂性和可行性，所以对农业投入与农民收入，本文暂不作讨论。因此，以全国为例，把农民收入与各影响因素关系进行线性回归分析，并建立数学模型。一、计量经济模型分析(一)、数据搜集根据以上分析，我们在影响农民收入因素中引入7个解释变量。即：-财政用于农业的支出的比重，-第二、三产业从业人数占全社会从业人数的比重，-非农村人口比重，-乡村从业人员占农村人口的比重，-农业总产值占农林牧总产值的比重，-农作物播种面积，—农村用电量。　yx2x3x4x5x6x7x8年份78年可比价比重%%比重比重千公顷亿千瓦时1986133.6013.4329.5017.9236.0179.99150104.07253.101987137.6312.2031.3019.3938.6275.63146379.53320.801988147.867.6637.6023.7145.9069.25143625.87508.901989196.769.4239.9026.2149.2362.75146553.93790.501990220.539.9839.9026.4149.9364.66148362.27844.501991223.2510.2640.3026.9450.9263.09149585.80963.201992233.1910.0541.5027.4651.5361.51149007.101106.901993265.679.4943.6027.9951.8660.07147740.701244.901994335.169.2045.7028.5152.1258.22148240.601473.901995411.298.4347.8029.0452.4158.43149879.301655.701996460.688.8249.5030.4853.2360.57152380.601812.701997477.968.3050.1031.9154.9358.23153969.201980.101998474.0210.6950.2033.3555.8458.03155705.702042.201999466.808.2349.9034.7857.1657.53156372.812173.452000466.167.7550.0036.2259.3355.68156299.852421.302001469.807.7150.0037.6660.6255.24155707.862610.782002468.957.1750.0039.0962.0254.51154635.512993.402003476.247.1250.9040.5363.7250.08152414.963432.922004499.399.6753.1041.7665.6450.05153552.553933.032005521.207.2255.2042.9967.5949.72155487.734375.70资料来源《中国统计年鉴2006》。(二)、计量经济学模型建立我们设定模型为下面所示的形式：利用Eviews软件进行最小二乘估计，估计结果如下表所示：DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-1102.373375.8283-2.9331840.0136X1-6.6353933.781349-1.7547690.1071X318.229422.0666178.8208990.0000X42.4300398.3703370.2903160.7770X5-16.237375.894109-2.7548470.0187X6-2.1552082.770834-0.7778190.4531X70.0099620.0023284.2788100.0013X80.0633890.0212762.9793480.0125R-squared0.995823Meandependentvar345.5232AdjustedR-squared0.993165S.D.dependentvar139.7117S.E.ofregression11.55028Akaikeinfocriterion8.026857Sumsquaredresid1467.498Schwarzcriterion8.424516Loglikelihood-68.25514F-statistic374.6600Durbin-Watsonstat1.993270Prob(F-statistic)0.000000表1最小二乘估计结果回归分析 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为：二、计量经济学检验(一)、多重共线性的检验及修正①、检验多重共线性(a)、直观法从“表1最小二乘估计结果”中可以看出，虽然模型的整体拟合的很好，但是x4x6的t统计量并不显著，所以可能存在多重共线性。(b)、相关系数矩阵X2X3X4X5X6X7X8X21.000000-0.717662-0.695257-0.7313260.737028-0.332435-0.594699X3-0.7176621.0000000.9222860.935992-0.9457010.7422510.883804X4-0.6952570.9222861.0000000.986050-0.9377510.7539280.974675X5-0.7313260.9359920.9860501.000000-0.9747500.6874390.940436X60.737028-0.945701-0.937751-0.9747501.000000-0.603539-0.887428X7-0.3324350.7422510.7539280.687439-0.6035391.0000000.742781X8-0.5946990.8838040.9746750.940436-0.8874280.7427811.000000表2相关系数矩阵从“表2相关系数矩阵”中可以看出，个个解释变量之间的相关程度较高，所以应该存在多重共线性。②、多重共线性的修正——逐步迭代法一元回归DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C820.3133151.87125.4013740.0000X2-51.3783616.18923-3.1736140.0056R-squared0.372041Meandependentvar345.5232AdjustedR-squared0.335102S.D.dependentvar139.7117S.E.ofregression113.9227Akaikeinfocriterion12.40822Sumsquaredresid220632.4Schwarzcriterion12.50763Loglikelihood-115.8781F-statistic10.07183Durbin-Watsonstat0.644400Prob(F-statistic)0.005554表3y对x2的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-525.889164.11333-8.2024920.0000X319.460311.41604313.742740.0000R-squared0.917421Meandependentvar345.5232AdjustedR-squared0.912563S.D.dependentvar139.7117S.E.ofregression41.31236Akaikeinfocriterion10.37950Sumsquaredresid29014.09Schwarzcriterion10.47892Loglikelihood-96.60526F-statistic188.8628Durbin-Watsonstat0.598139Prob(F-statistic)0.000000表4y对x3的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-223.190569.92322-3.1919370.0053X418.650862.2422408.3179560.0000R-squared0.802758Meandependentvar345.5232AdjustedR-squared0.791155S.D.dependentvar139.7117S.E.ofregression63.84760Akaikeinfocriterion11.25018Sumsquaredresid69300.77Schwarzcriterion11.34959Loglikelihood-104.8767F-statistic69.18839Durbin-Watsonstat0.282182Prob(F-statistic)0.000000表5y对x4的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-494.1440118.1449-4.1825260.0006X515.779782.1987117.1768320.0000R-squared0.751850Meandependentvar345.5232AdjustedR-squared0.737253S.D.dependentvar139.7117S.E.ofregression71.61463Akaikeinfocriterion11.47978Sumsquaredresid87187.14Schwarzcriterion11.57919Loglikelihood-107.0579F-statistic51.50691Durbin-Watsonstat0.318959Prob(F-statistic)0.000002表6y对x5的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C1288.009143.80888.9563950.0000X6-15.523982.351180-6.6026350.0000R-squared0.719448Meandependentvar345.5232AdjustedR-squared0.702945S.D.dependentvar139.7117S.E.ofregression76.14674Akaikeinfocriterion11.60250Sumsquaredresid98571.54Schwarzcriterion11.70192Loglikelihood-108.2238F-statistic43.59479Durbin-Watsonstat0.395893Prob(F-statistic)0.000004表7y对x6的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-4417.766681.1678-6.4855770.0000X70.0315280.0045076.9949430.0000R-squared0.742148Meandependentvar345.5232AdjustedR-squared0.726980S.D.dependentvar139.7117S.E.ofregression73.00119Akaikeinfocriterion11.51813Sumsquaredresid90595.96Schwarzcriterion11.61754Loglikelihood-107.4222F-statistic48.92923Durbin-Watsonstat0.572651Prob(F-statistic)0.000002表8y对x7的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C140.162528.966164.8388350.0002X80.1198270.0145438.2395030.0000R-squared0.799739Meandependentvar345.5232AdjustedR-squared0.787959S.D.dependentvar139.7117S.E.ofregression64.33424Akaikeinfocriterion11.26536Sumsquaredresid70361.21Schwarzcriterion11.36478Loglikelihood-105.0209F-statistic67.88941Durbin-Watsonstat0.203711Prob(F-statistic)0.000000表9y对x8的回归结果综合比较表3~9的回归结果，发现加入x3的回归结果最好。以x3为基础顺次加入其他解释变量，进行二元回归，具体的回归结果如下表10~15所示：DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-754.4481149.1701-5.0576370.0001X321.788651.93268911.273750.0000X213.450708.0127451.6786630.1126R-squared0.929787Meandependentvar345.5232AdjustedR-squared0.921010S.D.dependentvar139.7117S.E.ofregression39.26619Akaikeinfocriterion10.32254Sumsquaredresid24669.34Schwarzcriterion10.47167Loglikelihood-95.06417F-statistic105.9385Durbin-Watsonstat0.595954Prob(F-statistic)0.000000表10加入x2的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-508.678175.73220-6.7168020.0000X317.882003.7521214.7658370.0002X41.7533513.8443050.4560900.6545R-squared0.918481Meandependentvar345.5232AdjustedR-squared0.908291S.D.dependentvar139.7117S.E.ofregression42.30965Akaikeinfocriterion10.47185Sumsquaredresid28641.71Schwarzcriterion10.62097Loglikelihood-96.48254F-statistic90.13613Durbin-Watsonstat0.596359Prob(F-statistic)0.000000表11加入x4的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-498.155067.21844-7.4109860.0000X323.975163.9671836.0433700.0000X5-4.3205663.553466-1.2158740.2417R-squared0.924405Meandependentvar345.5232AdjustedR-squared0.914956S.D.dependentvar139.7117S.E.ofregression40.74312Akaikeinfocriterion10.39639Sumsquaredresid26560.02Schwarzcriterion10.54551Loglikelihood-95.76570F-statistic97.82772Durbin-Watsonstat0.607882Prob(F-statistic)0.000000表12加入x5的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-1600.965346.9265-4.6147090.0003X329.937683.5347538.4695280.0000X69.9801353.1841763.1342910.0064R-squared0.948835Meandependentvar345.5232AdjustedR-squared0.942440S.D.dependentvar139.7117S.E.ofregression33.51927Akaikeinfocriterion10.00606Sumsquaredresid17976.66Schwarzcriterion10.15518Loglikelihood-92.05754F-statistic148.3576Durbin-Watsonstat1.125188Prob(F-statistic)0.000000表13加入x6的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2153.028327.1248-6.5816730.0000X314.404971.35835510.604720.0000X70.0122680.0024475.0140150.0001R-squared0.967884Meandependentvar345.5232AdjustedR-squared0.963869S.D.dependentvar139.7117S.E.ofregression26.55648Akaikeinfocriterion9.540364Sumsquaredresid11283.94Schwarzcriterion9.689485Loglikelihood-87.63345F-statistic241.0961Durbin-Watsonstat0.690413Prob(F-statistic)0.000000表14加入x7的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-400.5635103.0301-3.8878320.0013X315.542712.9163585.3294930.0001X80.0292330.0192331.5199290.1480R-squared0.927840Meandependentvar345.5232AdjustedR-squared0.918820S.D.dependentvar139.7117S.E.ofregression39.80687Akaikeinfocriterion10.34990Sumsquaredresid25353.40Schwarzcriterion10.49902Loglikelihood-95.32401F-statistic102.8643Durbin-Watsonstat0.559772Prob(F-statistic)0.000000表15加入x8的回归结果综合表10~15所示，加入x7的模型的R最大,以x3、x7为基础顺次加入其他解释变量，进行三元回归，具体回归结果如下表16~20所示：DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2133.921340.6965-6.2634060.0000X314.960232.0946457.1421340.0000X70.0118430.0027864.2509080.0007X22.1952436.1704030.3557700.7270R-squared0.968153Meandependentvar345.5232AdjustedR-squared0.961783S.D.dependentvar139.7117S.E.ofregression27.31242Akaikeinfocriterion9.637224Sumsquaredresid11189.52Schwarzcriterion9.836053Loglikelihood-87.55363F-statistic151.9988Durbin-Watsonstat0.712258Prob(F-statistic)0.000000表16加入x2的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2226.420353.4425-6.2992430.0000X315.667292.4431136.4128390.0000X70.0127030.0025894.9063730.0002X4-1.6013622.553294-0.6271750.5400R-squared0.968705Meandependentvar345.5232AdjustedR-squared0.962445S.D.dependentvar139.7117S.E.ofregression27.07472Akaikeinfocriterion9.619741Sumsquaredresid10995.60Schwarzcriterion9.818571Loglikelihood-87.38754F-statistic154.7677Durbin-Watsonstat0.704178Prob(F-statistic)0.000000表17加入x4的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2110.381306.2690-6.8906130.0000X318.601562.6173817.1069370.0000X70.0121390.0022855.3116650.0001X5-3.9648782.163262-1.8328230.0868R-squared0.973760Meandependentvar345.5232AdjustedR-squared0.968512S.D.dependentvar139.7117S.E.ofregression24.79152Akaikeinfocriterion9.443544Sumsquaredresid9219.289Schwarzcriterion9.642373Loglikelihood-85.71367F-statistic185.5507Durbin-Watsonstat0.733972Prob(F-statistic)0.000000表18加入x5的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2418.859323.7240-7.4719790.0000X320.998873.3971206.1813740.0000X70.0099200.0024953.9766600.0012X65.3591842.5719502.0837050.0547R-squared0.975093Meandependentvar345.5232AdjustedR-squared0.970112S.D.dependentvar139.7117S.E.ofregression24.15359Akaikeinfocriterion9.391407Sumsquaredresid8750.940Schwarzcriterion9.590236Loglikelihood-85.21837F-statistic195.7489Durbin-Watsonstat1.084023Prob(F-statistic)0.000000表19加入x6的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2013.355361.8657-5.5638180.0001X313.015782.0324206.4040780.0000X70.0116150.0025584.5403220.0004X80.0123750.0134160.9224010.3709R-squared0.969608Meandependentvar345.5232AdjustedR-squared0.963529S.D.dependentvar139.7117S.E.ofregression26.68115Akaikeinfocriterion9.590455Sumsquaredresid10678.26Schwarzcriterion9.789285Loglikelihood-87.10933F-statistic159.5158Durbin-Watsonstat0.672264Prob(F-statistic)0.000000表20加入x8的回归结果综合上述表16~20的回归结果所示，其中加入x6的回归结果最好，以x3x6x7为基础一次加入其他解释变量，作四元回归估计，估计结果如表21~24所示：DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2405.108339.7396-7.0792690.0000X321.268503.6997875.7485730.0001X65.3105432.6655691.9922730.0662X70.0096890.0027663.5033860.0035X21.3026055.6553900.2303300.8212R-squared0.975187Meandependentvar345.5232AdjustedR-squared0.968098S.D.dependentvar139.7117S.E.ofregression24.95411Akaikeinfocriterion9.492888Sumsquaredresid8717.904Schwarzcriterion9.741424Loglikelihood-85.18244F-statistic137.5567Durbin-Watsonstat1.082771Prob(F-statistic)0.000000表21加入x2的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2401.402316.2980-7.5922150.0000X322.105703.4207836.4621740.0000X69.0890333.7813302.4036600.0307X70.0070860.0032472.1820050.0466X44.4176783.3488891.3191470.2083R-squared0.977847Meandependentvar345.5232AdjustedR-squared0.971517S.D.dependentvar139.7117S.E.ofregression23.57887Akaikeinfocriterion9.379513Sumsquaredresid7783.481Schwarzcriterion9.628049Loglikelihood-84.10537F-statistic154.4909Durbin-Watsonstat1.580301Prob(F-statistic)0.000000表22加入x4的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2375.188430.7065-5.5146310.0001X320.834933.6574145.6966290.0001X64.6291965.2528600.8812720.3930X70.0102170.0031713.2219530.0061X5-0.6936924.304485-0.1611560.8743R-squared0.975139Meandependentvar345.5232AdjustedR-squared0.968036S.D.dependentvar139.7117S.E.ofregression24.97818Akaikeinfocriterion9.494817Sumsquaredresid8734.736Schwarzcriterion9.743353Loglikelihood-85.20076F-statistic137.2849Durbin-Watsonstat1.023211Prob(F-statistic)0.000000表23加入x5的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2212.242259.5324-8.5239510.0000X322.066292.6622318.2886470.0000X69.5956532.3800884.0316380.0012X70.0061150.0022602.7059780.0171X80.0369230.0112393.2853540.0054R-squared0.985936Meandependentvar345.5232AdjustedR-squared0.981918S.D.dependentvar139.7117S.E.ofregression18.78702Akaikeinfocriterion8.925144Sumsquaredresid4941.332Schwarzcriterion9.173681Loglikelihood-79.78887F-statistic245.3639Durbin-Watsonstat2.186293Prob(F-statistic)0.000000表24加入x8的回归结果综合表21~24所示的回归结果，其中加入x8的回归结果最好，以x3x6x7x8为基础顺次加入其他的解释变量，其回归结果如表25~27所示：DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2207.020272.6061-8.0960050.0000X322.174952.9031907.6381330.0000X69.5667312.4800573.8574640.0020X70.0060280.0024512.4589490.0287X80.0368460.0116743.1561950.0076X20.5358114.4226450.1211520.9054R-squared0.985952Meandependentvar345.5232AdjustedR-squared0.980549S.D.dependentvar139.7117S.E.ofregression19.48522Akaikeinfocriterion9.029279Sumsquaredresid4935.759Schwarzcriterion9.327523Loglikelihood-79.77815F-statistic182.4791Durbin-Watsonstat2.180501Prob(F-statistic)0.000000表25加入x2的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-1373.136279.4825-4.9131370.0003X320.093301.92848610.419210.0000X60.4804012.8459720.1688000.8686X70.0084970.0016925.0214100.0002X80.0605020.0098736.1281460.0000X5-11.232922.844094-3.9495600.0017R-squared0.993607Meandependentvar345.5232AdjustedR-squared0.991148S.D.dependentvar139.7117S.E.ofregression13.14457Akaikeinfocriterion8.241984Sumsquaredresid2246.136Schwarzcriterion8.540228Loglikelihood-72.29885F-statistic404.1009Durbin-Watsonstat1.704834Prob(F-statistic)0.000000表26加入x5的回归结果DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2056.366236.8112-8.6835690.0000X320.602202.4130968.5376610.0000X65.2648342.8042921.8774200.0831X70.0088530.0023063.8394460.0020X80.0717420.0180263.9800360.0016X4-9.8612314.279624-2.3042280.0384R-squared0.990014Meandependentvar345.5232AdjustedR-squared0.986174S.D.dependentvar139.7117S.E.ofregression16.42798Akaikeinfocriterion8.687938Sumsquaredresid3508.420Schwarzcriterion8.986182Loglikelihood-76.53541F-statistic257.7752Durbin-Watsonstat1.965748Prob(F-statistic)0.000000表27加入x4的回归结果据表25~27所示，分别加入x2x4x5后R均有所增加，但是参数的T检验均不显著，所以最终的计量模型如下表所示：DependentVariable:YMethod:LeastSquaresSample:19862004Includedobservations:19VariableCoefficientStd.Errort-StatisticProb.C-2212.242259.5324-8.5239510.0000X322.066292.6622318.2886470.0000X69.5956532.3800884.0316380.0012X70.0061150.0022602.7059780.0171X80.0369230.0112393.2853540.0054R-squared0.985936Meandependentvar345.5232AdjustedR-squared0.981918S.D.dependentvar139.7117S.E.ofregression18.78702Akaikeinfocriterion8.925144Sumsquaredresid4941.332Schwarzcriterion9.173681Loglikelihood-79.78887F-statistic245.3639Durbin-Watsonstat2.186293Prob(F-statistic)0.000000表28多重共线性修正后的最终模型回归分析报告为：(二)、异方差的检验A、相关图形分析图1图2图3图4从图1~4可以看出y并不随着x的增大而变得更离散，表明模型可能不存在异方差。残差分析图图5图6图7图8从图5~8看出，e2并不随x的增大而变化，表明模型可能不存在异方差。ARCH检验ARCHTest:F-statistic0.558635Probability0.652331Obs*R-squared1.960709Probability0.580602TestEquation:DependentVariable:RESID^2Method:LeastSquaresSample(adjusted):19892004Includedobservations:16afteradjustingendpointsVariableCoefficientStd.Errort-StatisticProb.C279.7407120.18892.3275090.0382RESID^2(-1)0.0519710.2514140.2067170.8397RESID^2(-2)-0.2234090.241815-0.9238870.3737RESID^2(-3)-0.1579920.249154-0.6341150.5379R-squared0.122544Meandependentvar204.2351AdjustedR-squared-0.096820S.D.dependentvar286.6884S.E.ofregression300.2464Akaikeinfocriterion14.45940Sumsquaredresid1081774.Schwarzcriterion14.65255Loglikelihood-111.6752F-statistic0.558635Durbin-Watsonstat1.767931Prob(F-statistic)0.652331表29ARCH检验D、White检验WhiteHet