恩格尔系数模型

1、恩格尔系数模型恩格尔系数是根据恩格尔定律得出的比例数，是表示生 活水平高低的一个指标。其计算公式为：恩格尔系数=食物支出金额/总消费支出金额*100%一、模型设定影响因素对于系数的分子项食物支出，其影响因素主要有收 入、食品价格；对于分母项一一总消费支出，其主要影响因 素有收入、家庭财富。收入采用人均可支配收入X1食品价格采用消费物价指数X2对于家庭财富，由于其中包括家庭储蓄、住房、耐用消 费品等许多因素，而由于数据无法查找等原因，因此家庭财 富用人均住房面积X3代替。建立模型(1 )初步建立的是简单线性回归模型Y邛 1+B2X1+B3X2+B4X3(2 )数据来源于中国统计年鉴，由于城镇居民

2、人均住房 面积数据缺失，所以采用农村居民数据。年份恩格尔系人均每年可支配物价指数(X2 )人均住房面积数(Y)(%)收入(XI)90年为基期(X3)(平方米)199058.8686.3117.8199157.6708.61.03418.5199257.67841.100218.9199358.1921.61.261920.7199458.912211.56620.2199558.61577.71.833821.0199656.31926.11.98621.7199755.12090.12.041622.5199853.42162.02.025323.3199952.62210.31.99692

3、4.2200049.12253.42.004924.8200147.72366.42.019025.7200246.22475.62.002826.5200345.62622.22.026827.2200447.22936.42.105927.9200545.63254.92.143829.7200643.03587.02.175330.7200743.14140.42.279131.6200843.74760.62.412632.4200941.05153.22.396033.6201041.15919.02.475534.1201140.46977.32.609036.2(3)参数估计将整

4、理出来的数据用EVIEWS进行回归,结果如图Dependent Variable: Y Method: Least Gquares ate: 12/27/13 Time: 19：&S Sample: d Q3D 20*1 -11 ncludedl ob-servations: AAVari a bl 右CoefficientStd. Errort-StatisticProb.CQ1.SSQ232 gaoTss2-2-.55JL3D .X10.0021209o.ooaTdv3.196790.0050X21.092191 50S043C. 72:&64b .477 4X3-d 9 4 375O._

5、24CiT 突-e.0006240.0000R-s q u a re cl Adjusied R-squaredS-.t. of regression Gum -squared r&s i d Log likeiiriood F-statistirPro b(F-siatistio；;O.Q5Q 67 Q 0.9 329 59 14&9 9 33 3-S.3 6 5S2 -37.33393 -142.8067 o.oooaooLI a n ci e p a n cl nt ws r S.D. dependent var AKaiKe info criterion Schwarai crite

6、rion Hannan-ciuinn ciiter. u rh.j n-Wats n statED.0 2.-1S2 6.73-d 22. 3.7S7U30 3-.9&6 0 0-1 3-.8 0 4-3-6 0.939 434二、模型检验经济意义检验X3人均住房面积系数为负，表明随着经济发展，人民生 活条件改善，总消费性支出增加，恩格尔系数下降，符合经 济意义。但X1系数为负，表明随着收入增加，恩格尔系数 上升，不符合经济意义。可能模型虽在多重共线性等原因。统计推断检验（1）从回归结果看，R-squared =0.9597，拟合优度很高。（2）回归参数的显著性检验（t检验）由三个参数t检验值

7、来看，三个参数的 t值分别为 t1=3.194t2=0.726 t3=-8.09。在5%显著性水平下自由度为 n-k-1=22-3-1 = 18 的 t 分布临界值为 t0.025（18）=2.101, x1 x3通过了显著性检验（3）模型总体显著性检验（F检验）F检验值比较大等于142.8067相应的P值为0.0000，说明回归方程显著，即各自变量联合起来确实对因变量Y有显 著影响；给定a=0.05，x1 x3都通过了显著性检验，拒绝原 假设，接受备择假设，即人均可支配收入、住房面积都对恩 格尔系数有显著性影响。计量经济学检验（1）多重共线性检验如图所示，各个变量之间的相关系数较高，存在较为

8、严重的 多重共线性。 Group: UNTITLED Workfile: UNTITLED:Entitled、|由升刊|伊0匚|口宅&:1:| | Print | Name | Freeze | I Sample Sheet Stats SpecCorrelationX1X2X3X11.0030000.8679680.965809X20.8679631.0000000.3S7537X30.9658090.8S75371.000000修正：逐步回归法进行修正，结果如图：Dependent Variable: Y Method: Least Squares Date: 12/27/13 Time:

9、 21:09 Sample: 1990 2011 Included observations: 22Dependent Variable: Y Method: Least Squares Date: 12/27/13 Time: 21:11 Sample: 1990 2011 Included observations: 22VariableCoefficientStdl. Error t-Sta.tisticProb.C59.676541.26920347.015930.0000X1-0.0034940.000393-8.8977290.0000R-squaredl0.798325Mean

10、dependent va.r50.03182Adjusted R-squared0.788242S.D. dependent var6.731262S.E. of regression3.097540Aka ike info criterion5.185601Sum squared resid191.8951Schwaiz criterion5.284787Log likelihood-55.04162Hannan-Quinn criter.5.208967F-statistic79.16959Durbin-Watson stat0.333378Pro b(F-sta.ti stic)0.00

11、0000VariableCoefficientStd. Error t-StatisticProb.C73.588943.55339420.709480.0000X2-12.195311.791441-6.8075390.0000R-squared0.698535Mean dependent var50.03132Adjusted! R-squared0.683461S.D. dependent var6.731262S.E. of regression3.787120Akaike info criterion5.5S7601Sum squared residl2S6.S467Schwarz

12、criterion5.686786Log likelihood-59.46361Hannan-Quinn criter.5.610966F-statistic46.34259Durbin-Watson stat0.306764Pro b(F-stati stic)0.000001Dependent Variable: Y Method: Least Squares Date: 12/27/13 Time: 21:14 Sample: 1990 2011 Includled obsen/ations: 22VariableCoefficientStd. Error t-StatisticProb

13、.X3-1.1751520.069564-16.093100.0000C80.436201.33872743.745590.0000R-squared0.934500Mean dependent var50.03182Adjusted! R-squared0.931233S.D. dependent var6.731262S.E. of regression1.765169Akaike info criterion4.060878Sum squared resid6231642Schwa iz criterion4.160063Log likelihood-42.66966Hannan-Qui

14、nn criter.4.0S4243F-statistic285.3795Durbin-Watson stat0.577409Prob (F-statistic)0.000000依据可调系数最大原则，选择x3作为回归模型的第一个解释变量。再次进行回归，如下图。Dependent Variable: Y Method: Least Squares ate: 12/27/13 Time: 21:48 Sample: 1990 2011 Included obsen/ations: 22由于X1调整系数大于X2 ,综合参考决定，使用X1与X3作为解释变量。但由于X1和X3之间线性关系仍比较明显，随着

15、可支配收入增加，住房面积增加，家庭财富不断增加，并且X1的系数存在符号问题，所以仅用X3作为解释变量。Dependent Variable: Y Method: Least Squares Date: 12/27/13 Time: 21:58 Sample: 1990 2011 Included! obseivations: 22VariableCoefficientStd. Errort-Sta.tisticProb.C92.135613.83618524.017510.0000X10.0023360.0007053.3142440.0036X3-1.8766100.219142-8.563

16、4320.0000R-squaredl0.958500Mean dependent var50.03182Adjusted R-squared0.954131S.D. dependent var6.731262S.E. of regression1 441634Akaike info criterion3.695555Sum squared resid39.48785Schwarz criterion3.844333Log likelihood-37.65110Hannan-Quinn criter.3.730603F-statistic219.4141Durbin-Watson stat0.

17、908837Prob(F-statistic)0.000000VariableCoefficientStd. Errort-StatisticProb.C80407851.85307343.391630.0000X21.5258021.8261170.8355450.4138X3-1.2879720.152136-8.4659230.0000R-squaredl0.936829Mean dependent var50.03182Adljustedl R-squared0.930179S.D. dlependlent var6.731262S.E. of regression1.778643Ak

18、aike info criterion4115702Sum squared resid60.10782Schwa it criterion4.264480Log likelihood-42.27272Hannan-Quinn criter.4.150750F-statistic140.8851Durbin-Watson stat0.571592Prob (F-statistic)0.000000此时从结果看可决系数为：0.931233，拟和优度较好，F值285.3795，在5%显著性水平下拒绝原假设，即变量对恩 格尔系数有显著影响。再看变量T检验值通过了 5%的显著 性检验拒绝原假设，即变量对

19、恩格尔系数显著影响。由于此 时为一元回归，所以不存在多重共线性。(2)异方差检验使用White检验如图Heteroskedasticity T est:WhiteF-statistic0.3594S3Frob. FC2.19)0.70ZZObsR-squaredC.-B 02091Prob. Ghi-S-quare(2)0.6696Scaled explained SSProb. Ohi-S-quare(2)0.8247Test Equation:Dependent Variable: RESIDA2Method: Least-Squares ate: 1229/13 Time:22:19S-

20、ample: 1990Included observations: 22.VariableCoeffi cientStd. Errort-StatisticProb.G-11.1333617 947+7-D.623395 .540 4-X1.1243161.3B99910.8038660.4286XA2-0.021 5S70 026024-0.823736 .+175从结果得obs*R-squard=0.802091 ,大于5%显著性水平 下自由度为1的2分布的相应临界值3.14846，故接受原假 设，表明模型随机误差项不存在异方差。（2）自相关检验使用DW检验，根据回归结果，DW检验值为0.

21、577409， 查DW统计表得dl为1.239，由于检验值小于dl值，判断 模型存在正自相关。修正自相关，如图所示。ependeni VariabIe: Y-C.730Method: Least Squaresate: 12/29/13 Time: 23:33Sample (adjusted): 1991 2011Included observations: 2A after adjustmentsVariableCoefficientStd. Error t-StatisticProbC20.346591.35694515.362390.0000X-0 73a334*X(-1)-1.0500810.172041-6.103682o.ooaoR-squared.662252Mean dependent var12.73937Adjusied R-squared0.644476S.D. dependent v