复习：多元线性回归模型案例

自改革开放以来,虽然中国经济平均增长速度为9.5 % ,但二元经济结构给经济发展带来的问题仍然很突出。农村人口占了中国总人口的70 %多,农业产业结构不合理,经济不发达,以及农民收入增长缓慢等问题势必成为我国经济持续稳定增长的障碍。正确有效地解决好“三农”问题是中国经济走出困境,实现长期稳定增长的关键。其中,农民收入增长是核心,也是解决“三农”问题的关键。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,寻找其根源,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建议。

农民收入水平的度量，通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。目前农业收入仍是中西部地区农民收入的主要来源。二是农业剩余劳动力转移水平。中国的农业目前仍以农户分散经营为主，农业比较效益低，尽快地把农业剩余劳动力转移出去是有效改善农民收入状况的重要因素。三是城市化、工业化水平。中国多数地区城市化、工业化水平落后于世界平均水平，这种状况极大地影响了农民收入的增长。四是农业产业结构状况。农林牧渔业对农民收入增长贡献率是不同的。随着我国“入世”后农产品市场的开放和人民生活水平的提高、农产品需求市场的改变，农业结构状况直接影响着农民收入的增长。五是农业投入水平。农民收入与财政农业支出、农村集体投入、农户个人投入以及信贷投入都有显著的正相关关系。农业投入是农民收入增长的重要保证。但考虑到农业投入主体的多元性，既有国家、集体和农户的投入，又有银行、企业和外资的投入，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： x2-财

x3 -第二、政用于农业的支出的比重，三产业从业人数占全社会从业人数的比重，x4 -非农村人口比重，x5 -乡村从业人员占农村人口的比重，x6 -农业总产值

占农林牧总产值的比重，x7 -农作物播种面积，x8—农村用电量。 年份 y 78年可比价 x2 比重 x3 % x4 % x5 比重 x6 比重 x7 千公顷 x8 亿千瓦时 1986 1987 1988 19 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 133.60 137.63 147.86 196.76 220.53 223.25 233.19 265.67 335.16 411.29 460.68 477.96 474.02 466.80 466.16 469.80 468.95 476.24 499.39 521.20 13.43 12.20 7.66 9.42 9.98 10.26 10.05 9.49 9.20 8.43 8.82 8.30 10.69 8.23 7.75 7.71 7.17 7.12 9.67 7.22 29.50 31.30 37.60 39.90 39.90 40.30 41.50 43.60 45.70 47.80 49.50 50.10 50.20 49.90 50.00 50.00 50.00 50.90 53.10 55.20 17.92 19.39 23.71 26.21 26.41 26.94 27.46 27.99 28.51 29.04 30.48 31.91 33.35 34.78 36.22 37.66 39.09 40.53 41.76 42.99 36.01 38.62 45.90 49.23 49.93 50.92 51.53 51.86 52.12 52.41 53.23 54.93 55.84 57.16 59.33 60.62 62.02 63.72 65. 67.59 79.99 75.63 69.25 62.75 .66 63.09 61.51 60.07 58.22 58.43 60.57 58.23 58.03 57.53 55.68 55.24 54.51 50.08 50.05 49.72 150104.07 146379.53 143625.87 146553.93 148362.27 149585.80 149007.10 147740.70 148240.60 149879.30 152380.60 153969.20 155705.70 156372.81 156299.85 155707.86 154635.51 152414.96 153552.55 155487.73 253.10 320.80 508.90 790.50 844.50 963.20 1106.90 1244.90 1473.90 1655.70 1812.70 1980.10 2042.20 2173.45 2421.30 2610.78 2993.40 3432.92 3933.03 4375.70 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：

Yt12X23X34X45X56X67X78X8ut

利用Eviews软件进行最小二乘估计，估计结果如下表所示：

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X1

Coefficient -1102.373 -6.635393

Std. Error 375.8283 3.781349

t-Statistic -2.933184 -1.754769

Prob. 0.0136 0.1071

X3 X4 X5 X6 X7 X8 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

18.22942 2.430039 -16.23737 -2.155208 0.009962 0.0633 2.066617 8.370337 5.4109 2.770834 0.002328 0.021276 8.8209 0.290316 -2.754847 -0.777819 4.278810 2.979348 0.0000 0.7770 0.0187 0.4531 0.0013 0.0125 345.5232 139.7117 8.026857 8.424516 374.6600 0.000000

0.995823 Mean dependent var 0.993165 S.D. dependent var 11.55028 Akaike info criterion 1467.498 Schwarz criterion -68.25514 F-statistic 1.993270 Prob(F-statistic)

表1 最小二乘估计结果

回归分析报告为：

ˆ -1102.373-6.6354X+18.2294X+2.4300X-16.2374X-2.1552X+0.0100X+0.0634XYi23456783.78132.066618.370345.412.77080.002330.02128t -2.9331.7558.820900.203162.7550.7784.278812.9793R20.995823R20.993165Df19DW1.99327F374.66二、计量经济学检验

(一)、多重共线性的检验及修正

SE375.83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x4 x6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

X2 X3 X4 X5 X6 X7 X8

X2

X3

X4

X5

X6

X7

X8

1.000000 -0.717662 -0.695257 -0.731326 0.737028 -0.332435 -0.594699 -0.717662 1.000000 0.922286 0.935992 -0.945701 0.742251 0.883804 -0.695257 0.922286 1.000000 0.986050 -0.937751 0.753928 0.974675 -0.731326 0.935992 0.986050 1.000000 -0.974750 0.687439 0.940436 0.737028 -0.945701 -0.937751 -0.974750 1.000000 -0.603539 -0.887428 -0.332435 0.742251 0.753928 0.687439 -0.603539 1.000000 0.742781 -0.594699 0.883804 0.974675 0.940436 -0.887428 0.742781 1.000000

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

A、 一元回归

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X2 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient 820.3133 -51.37836 Std. Error 151.8712 16.123 t-Statistic 5.401374 -3.173614 Prob. 0.0000 0.0056 345.5232 139.7117 12.40822 12.50763 10.07183 0.005554 0.372041 Mean dependent var 0.335102 S.D. dependent var 113.9227 Akaike info criterion 220632.4 Schwarz criterion -115.8781 F-statistic 0.4400 Prob(F-statistic) 表3 y对x2的回归结果

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -525.81 19.46031 Std. Error .11333 1.416043 t-Statistic -8.202492 13.74274 Prob. 0.0000 0.0000 345.5232 139.7117 10.37950 10.472 188.8628 0.000000

0.917421 Mean dependent var 0.912563 S.D. dependent var 41.31236 Akaike info criterion 29014.09 Schwarz criterion -96.60526 F-statistic 0.598139 Prob(F-statistic)

表4 y对x3的回归结果

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X4

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood

Coefficient -223.1905 18.65086

Std. Error 69.92322 2.242240

t-Statistic -3.191937 8.317956

Prob. 0.0053 0.0000 345.5232 139.7117 11.25018 11.34959 69.18839

0.802758 Mean dependent var 0.791155 S.D. dependent var 63.84760 Akaike info criterion 69300.77 Schwarz criterion -104.8767 F-statistic

Durbin-Watson stat 0.282182 Prob(F-statistic) 0.000000 表5 y对x4的回归结果

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X5 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -494.1440 15.77978 Std. Error 118.1449 2.198711 t-Statistic -4.182526 7.176832 Prob. 0.0006 0.0000 345.5232 139.7117 11.47978 11.57919 51.50691 0.000002

0.751850 Mean dependent var 0.737253 S.D. dependent var 71.61463 Akaike info criterion 87187.14 Schwarz criterion -107.0579 F-statistic 0.3159 Prob(F-statistic)

表6 y对x5的回归结果

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X6

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient 1288.009 -15.52398

Std. Error 143.8088 2.351180

t-Statistic 8.956395 -6.602635

Prob. 0.0000 0.0000 345.5232 139.7117 11.60250 11.70192 43.59479 0.000004

0.719448 Mean dependent var 0.702945 S.D. dependent var 76.14674 Akaike info criterion 98571.54 Schwarz criterion -108.2238 F-statistic 0.3953 Prob(F-statistic)

表7 y对x6的回归结果

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X7

R-squared

Adjusted R-squared S.E. of regression Sum squared resid

Coefficient -4417.766 0.031528

Std. Error 681.1678 0.004507

t-Statistic -6.485577 6.994943

Prob. 0.0000 0.0000 345.5232 139.7117 11.51813 11.61754

0.742148 Mean dependent var 0.726980 S.D. dependent var 73.00119 Akaike info criterion 90595.96 Schwarz criterion

Log likelihood Durbin-Watson stat

-107.4222 F-statistic 0.572651 Prob(F-statistic)

48.92923 0.000002

表8 y对x7的回归结果

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X8 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient 140.1625 0.119827 Std. Error 28.96616 0.014543 t-Statistic 4.838835 8.239503 Prob. 0.0002 0.0000 345.5232 139.7117 11.26536 11.378 67.841 0.000000 0.799739 Mean dependent var 0.787959 S.D. dependent var .33424 Akaike info criterion 70361.21 Schwarz criterion -105.0209 F-statistic 0.203711 Prob(F-statistic) 表9 y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：

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X2

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -754.4481 21.78865 13.45070

Std. Error 149.1701 1.9326 8.012745

t-Statistic -5.057637 11.27375 1.678663

Prob. 0.0001 0.0000 0.1126 345.5232 139.7117 10.32254 10.47167 105.9385 0.000000

0.929787 Mean dependent var 0.921010 S.D. dependent var 39.26619 Akaike info criterion 24669.34 Schwarz criterion -95.017 F-statistic 0.595954 Prob(F-statistic)

表10 加入x2的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C

Coefficient -508.6781

Std. Error 75.73220

t-Statistic -6.716802

Prob. 0.0000

X3 X4 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 17.88200 1.753351 3.752121 3.844305 4.765837 0.456090 0.0002 0.6545 345.5232 139.7117 10.47185 10.62097 90.13613 0.000000 0.918481 Mean dependent var 0.908291 S.D. dependent var 42.30965 Akaike info criterion 281.71 Schwarz criterion -96.48254 F-statistic 0.596359 Prob(F-statistic) 表11 加入x4的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X5 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -498.1550 23.97516 -4.320566 Std. Error 67.21844 3.967183 3.553466 t-Statistic -7.410986 6.043370 -1.215874 Prob. 0.0000 0.0000 0.2417 345.5232 139.7117 10.39639 10.54551 97.82772 0.000000 0.924405 Mean dependent var 0.914956 S.D. dependent var 40.74312 Akaike info criterion 26560.02 Schwarz criterion -95.76570 F-statistic 0.607882 Prob(F-statistic) 表12 加入x5的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X6

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -1600.965 29.93768 9.980135

Std. Error 346.9265 3.534753 3.184176

t-Statistic -4.614709 8.469528 3.134291

Prob. 0.0003 0.0000 0.00 345.5232 139.7117 10.00606 10.15518 148.3576 0.000000

0.948835 Mean dependent var 0.942440 S.D. dependent var 33.51927 Akaike info criterion 17976.66 Schwarz criterion -92.05754 F-statistic 1.125188 Prob(F-statistic)

表13 加入x6的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004

Included observations: 19 Variable C X3 X7 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2153.028 14.40497 0.012268 Std. Error 327.1248 1.358355 0.002447 t-Statistic -6.581673 10.60472 5.014015 Prob. 0.0000 0.0000 0.0001 345.5232 139.7117 9.5403 9.85 241.0961 0.000000 0.967884 Mean dependent var 0.963869 S.D. dependent var 26.558 Akaike info criterion 11283.94 Schwarz criterion -87.63345 F-statistic 0.690413 Prob(F-statistic) Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X8 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

表14 加入x7的回归结果 Coefficient -400.5635 15.54271 0.029233 Std. Error 103.0301 2.916358 0.019233 t-Statistic -3.887832 5.329493 1.519929 Prob. 0.0013 0.0001 0.1480 345.5232 139.7117 10.34990 10.49902 102.83 0.000000

0.927840 Mean dependent var 0.918820 S.D. dependent var 39.80687 Akaike info criterion 25353.40 Schwarz criterion -95.32401 F-statistic 0.559772 Prob(F-statistic)

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：

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X7 X2 R-squared

Adjusted R-squared S.E. of regression

Coefficient -2133.921 14.96023 0.011843 2.195243 Std. Error 340.6965 2.0945 0.002786 6.170403 t-Statistic -6.263406 7.142134 4.250908 0.355770 Prob. 0.0000 0.0000 0.0007 0.7270 345.5232 139.7117 9.637224

0.968153 Mean dependent var 0.961783 S.D. dependent var 27.31242 Akaike info criterion

Sum squared resid Log likelihood Durbin-Watson stat

111.52 Schwarz criterion -87.55363 F-statistic 0.712258 Prob(F-statistic)

9.836053 151.9988 0.000000

表16 加入x2的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X7 X4 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -2226.420 15.66729 0.012703 -1.601362 Std. Error 353.4425 2.443113 0.0025 2.553294 t-Statistic -6.299243 6.412839 4.906373 -0.627175 Prob. 0.0000 0.0000 0.0002 0.5400 345.5232 139.7117 9.619741 9.818571 154.7677 0.000000

0.968705 Mean dependent var 0.962445 S.D. dependent var 27.07472 Akaike info criterion 10995.60 Schwarz criterion -87.38754 F-statistic 0.704178 Prob(F-statistic)

表17 加入x4的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X7 X5 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2110.381 18.60156 0.012139 -3.9878 Std. Error 306.2690 2.617381 0.002285 2.163262 t-Statistic -6.0613 7.106937 5.311665 -1.832823 Prob. 0.0000 0.0000 0.0001 0.0868 345.5232 139.7117 9.443544 9.2373 185.5507 0.000000 0.973760 Mean dependent var 0.968512 S.D. dependent var 24.79152 Akaike info criterion 9219.2 Schwarz criterion -85.71367 F-statistic 0.733972 Prob(F-statistic) 表18 加入x5的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C

Coefficient -2418.859

Std. Error 323.7240

t-Statistic -7.471979

Prob. 0.0000

X3 X7 X6

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

20.99887 0.009920 5.359184

3.397120 0.002495 2.571950

6.181374 3.976660 2.083705

0.0000 0.0012 0.0547 345.5232 139.7117 9.391407 9.590236 195.74 0.000000

0.975093 Mean dependent var 0.970112 S.D. dependent var 24.15359 Akaike info criterion 8750.940 Schwarz criterion -85.21837 F-statistic 1.084023 Prob(F-statistic)

表19 加入x6的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X7 X8

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -2013.355 13.01578 0.011615 0.012375

Std. Error 361.8657 2.032420 0.002558 0.013416

t-Statistic -5.563818 6.404078 4.540322 0.922401

Prob. 0.0001 0.0000 0.0004 0.3709 345.5232 139.7117 9.590455 9.7285 159.5158 0.000000

0.969608 Mean dependent var 0.963529 S.D. dependent var 26.68115 Akaike info criterion 10678.26 Schwarz criterion -87.10933 F-statistic 0.6722 Prob(F-statistic)

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x3 x6 x7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：

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X6 X7 X2 R-squared

Adjusted R-squared S.E. of regression

Coefficient -2405.108 21.26850 5.310543 0.0096 1.302605 Std. Error 339.7396 3.699787 2.665569 0.002766 5.655390 t-Statistic -7.079269 5.748573 1.992273 3.503386 0.230330 Prob. 0.0000 0.0001 0.0662 0.0035 0.8212 345.5232 139.7117 9.492888

0.975187 Mean dependent var 0.968098 S.D. dependent var 24.95411 Akaike info criterion

Sum squared resid Log likelihood Durbin-Watson stat

8717.904 Schwarz criterion -85.18244 F-statistic 1.082771 Prob(F-statistic)

9.741424 137.5567 0.000000

表21 加入x2的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X6 X7 X4 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2401.402 22.10570 9.0033 0.007086 4.417678 Std. Error 316.2980 3.420783 3.781330 0.003247 3.3488 t-Statistic -7.592215 6.462174 2.403660 2.182005 1.319147 Prob. 0.0000 0.0000 0.0307 0.0466 0.2083 345.5232 139.7117 9.379513 9.628049 154.4909 0.000000 0.977847 Mean dependent var 0.971517 S.D. dependent var 23.57887 Akaike info criterion 7783.481 Schwarz criterion -84.10537 F-statistic 1.580301 Prob(F-statistic) 表22 加入x4的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X6 X7 X5 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -2375.188 20.83493 4.629196 0.010217 -0.693692 Std. Error 430.7065 3.657414 5.252860 0.003171 4.304485 t-Statistic -5.514631 5.696629 0.881272 3.221953 -0.161156 Prob. 0.0001 0.0001 0.3930 0.0061 0.8743 345.5232 139.7117 9.494817 9.743353 137.2849 0.000000

0.975139 Mean dependent var 0.968036 S.D. dependent var 24.97818 Akaike info criterion 8734.736 Schwarz criterion -85.20076 F-statistic 1.023211 Prob(F-statistic)

表23 加入x5的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X6 X7 X8 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2212.242 22.06629 9.595653 0.006115 0.036923 Std. Error 259.5324 2.662231 2.380088 0.002260 0.011239 t-Statistic -8.523951 8.2887 4.031638 2.705978 3.285354 Prob. 0.0000 0.0000 0.0012 0.0171 0.0054 345.5232 139.7117 8.925144 9.173681 245.3639 0.000000 0.985936 Mean dependent var 0.981918 S.D. dependent var 18.78702 Akaike info criterion 4941.332 Schwarz criterion -79.78887 F-statistic 2.186293 Prob(F-statistic) 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x3 x6 x7 x8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：

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X6 X7 X8 X2 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2207.020 22.17495 9.566731 0.006028 0.036846 0.535811 Std. Error 272.6061 2.903190 2.480057 0.002451 0.011674 4.4225 t-Statistic -8.096005 7.638133 3.8574 2.4549 3.156195 0.121152 Prob. 0.0000 0.0000 0.0020 0.0287 0.0076 0.9054 345.5232 139.7117 9.029279 9.327523 182.4791 0.000000 0.985952 Mean dependent var 0.980549 S.D. dependent var 19.48522 Akaike info criterion 4935.759 Schwarz criterion -79.77815 F-statistic 2.180501 Prob(F-statistic) 表25 加入x2的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X6

Coefficient -1373.136 20.09330 0.480401

Std. Error 279.4825 1.928486 2.845972

t-Statistic -4.913137 10.41921 0.168800

Prob. 0.0003 0.0000 0.8686

X7 X8 X5

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.008497 0.060502 -11.23292

0.001692 0.009873 2.844094

5.021410 6.128146 -3.949560

0.0002 0.0000 0.0017 345.5232 139.7117 8.241984 8.540228 404.1009 0.000000

0.993607 Mean dependent var 0.991148 S.D. dependent var 13.14457 Akaike info criterion 2246.136 Schwarz criterion -72.29885 F-statistic 1.704834 Prob(F-statistic)

表26 加入x5的回归结果 Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X6 X7 X8 X4

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -2056.366 20.60220 5.2834 0.008853 0.071742 -9.861231

Std. Error 236.8112 2.413096 2.804292 0.002306 0.018026 4.279624

t-Statistic -8.683569 8.537661 1.877420 3.839446 3.980036 -2.304228

Prob. 0.0000 0.0000 0.0831 0.0020 0.0016 0.0384 345.5232 139.7117 8.687938 8.986182 257.7752 0.000000

0.990014 Mean dependent var 0.986174 S.D. dependent var 16.42798 Akaike info criterion 3508.420 Schwarz criterion -76.53541 F-statistic 1.965748 Prob(F-statistic)

表27 加入x4的回归结果 据表25~27所示，分别加入x2 x4 x5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：

Dependent Variable: Y Method: Least Squares Sample: 1986 2004 Included observations: 19 Variable C X3 X6 X7 X8 R-squared

Coefficient -2212.242 22.06629 9.595653 0.006115 0.036923 Std. Error 259.5324 2.662231 2.380088 0.002260 0.011239 t-Statistic -8.523951 8.2887 4.031638 2.705978 3.285354 Prob. 0.0000 0.0000 0.0012 0.0171 0.0054 345.5232

0.985936 Mean dependent var

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.981918 S.D. dependent var 18.78702 Akaike info criterion 4941.332 Schwarz criterion -79.78887 F-statistic 2.186293 Prob(F-statistic)

139.7117 8.925144 9.173681 245.3639 0.000000

表28 多重共线性修正后的最终模型 回归分析报告为：

ˆ -2212.242+22.0663X+9.5956X+0.00612X+0.03692XYi36782.66222.3800.002260.011239t8.5239518.288654.0322.705983.285354R20.985936R20.981918Df19DW2.186293F245.3639SE259.5324

(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。

B、 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。 C、 ARCH检验

ARCH Test: F-statistic Obs*R-squared

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Sample(adjusted): 19 2004

Included observations: 16 after adjusting endpoints Variable C RESID^2(-1) RESID^2(-2) RESID^2(-3)

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient 279.7407 0.051971 -0.223409 -0.157992

Std. Error 120.18 0.251414 0.241815 0.249154

t-Statistic 2.327509 0.206717 -0.923887 -0.634115

Prob. 0.0382 0.8397 0.3737 0.5379 204.2351 286.6884 14.45940 14.65255 0.558635 0.652331

0.558635 Probability 1.960709 Probability

0.652331 0.580602

0.122544 Mean dependent var -0.096820 S.D. dependent var 300.24 Akaike info criterion 1081774. Schwarz criterion -111.6752 F-statistic 1.767931 Prob(F-statistic)

表29 ARCH检验 在显著性水平0.05的情况下，237.81473,npR21.960709,则有npR21.960709237.81473,所以接受源假设，表明模型中不存在异方差。

D、White检验

White Heteroskedasticity Test: F-statistic Obs*R-squared

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Sample: 1986 2004 Included observations: 19

Variable C X3 X3^2 X3*X6

Coefficient 83312.19 39.976 92.15690 -23.05086

Std. Error 792151.1 3785.514 56.34778 26.32794

t-Statistic 0.105172 2.361628 1.635502 -0.875529

Prob. 0.9213 0.0775 0.1773 0.4307

5.378778 Probability 18.04165 Probability

0.058152 0.2041

X3*X7 X3*X8 X6 X6^2 X6*X7 X6*X8 X7 X7^2 X7*X8 X8 X8^2 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

-0.094926 -0.965260 14984.21 -53.822 -0.045905 -0.395631 -10.35154 5.76E-05 -7.38E-06 62.90965 0.001515 0.045467 0.775504 4130.063 41.43287 0.033365 0.371854 12.19311 4.21E-05 0.000266 32.25761 0.002697 -2.087801 -1.244688 3.628083 -1.300036 -1.375837 -1.063942 -0.8466 1.369855 -0.0277 1.950226 0.561479 0.1051 0.2812 0.0222 0.2634 0.2409 0.3473 0.4437 0.2426 0.9792 0.1229 0.6044 260.0701 337.4753 13.01876 13.737 5.378778 0.058152

0.949560 Mean dependent var 0.773022 S.D. dependent var 160.7806 Akaike info criterion 103401.7 Schwarz criterion -108.6782 F-statistic 3.254288 Prob(F-statistic)

表30 White检验 在显著性水平0.05的情况下，21423.6848,npR218.04165,则有npR218.041652323.6848,所以接受源假设，表明模型中不存在异方差。

综合上述4种方法得出的结论，说明模型中不存在异方差。

(三)、自相关检验及修正 ①自相关的检验 A、DW检验

已知DW= 2.18535949259，查表得DL=0.859 ，DU=1.848，所以4-DU=2.152 B、图示法：

图 9

从图中可以看出大部分点落在1、3象限，表明存在正自相关。

图 10

从图中可以看出， 随着t的变化逐次变化，并不频繁改变符号，而是正的后面跟着几个负的，表明存在正自相关。

综上所述，说明模型存在自相关性。 ②自相关的修正——德宾两步法

将广义方程表示为：Yt112X33X64X75X82X33X64X75X8Yt1vtt1t1t1t1

ˆ作为的估计值。将上述式子作为一个多元模型进行普通最小二乘估计，将Y的t1Dependent Variable: Y Method: Least Squares Sample(adjusted): 1987 2003

Included observations: 17 after adjusting endpoints Variable C X3 X6 X7 X8 X3(1) X6(-1) X7(-1) X8(-1) Y(-1) R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Coefficient -2604.551 7.131100 8.1283 0.019522 -0.006728 17.36973 3.379990 -0.012486 0.095023 -0.208123 Std. Error 1325.611 6.627590 3.462779 0.007068 0.071817 3.915884 3.045985 0.003263 0.114697 0.495599 t-Statistic -1.9793 1.075972 2.347532 2.762059 -0.093687 4.435710 1.109654 -3.826887 0.828466 -0.419942 Prob. 0.0902 0.3176 0.0513 0.0280 0.9280 0.0030 0.3038 0.0065 0.4347 0.6871 348.9382 132.18 7.766572 8.256698 302.5781 0.000000 0.997436 Mean dependent var 0.994140 S.D. dependent var 10.17327 Akaike info criterion 724.4683 Schwarz criterion -56.01586 F-statistic 2.4208 Prob(F-statistic) 表 31 广义方程估计结果 由上表可知0.208123，下一步使用广义差分法进行修正：令Y1Yt-Yt1,X31X3t-X3t-1,X61X6t-X6t-1,X71X7t-X7t-1,X81X8t-X8t-1;1111,212,313,414,515;则模型可表示为：Y11121X3131X6141X7151X81vt

Dependent Variable: Y+Y(-1)*0.208123 Method: Least Squares Sample(adjusted): 1987 2004

Included observations: 18 after adjusting endpoints Variable C

X3+X3(-1)*0.208123 X6+X6(-1)*0.208123 X7+X7(-1)*0.208123 X8+X8(-1)*0.208123 R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient -2681.442 23.58766 11.09440 0.005062 0.041248 Std. Error 307.9863 2.433154 2.238039 0.002144 0.010478 t-Statistic -8.706368 9.694275 4.957199 2.361369 3.936757 Prob. 0.0000 0.0000 0.0003 0.0345 0.0017 427.42 162.1550 8.942761 9.190086 310.7127 0.000000

0.98 Mean dependent var 0.9863 S.D. dependent var 18.86626 Akaike info criterion 4627.166 Schwarz criterion -75.48485 F-statistic 2.0380 Prob(F-statistic)

表32 广义差分估计结果 此时DW= 2.038，查表得DL=0.820，DU=1.872，DU(四)、模型的最终结果为：

因为1111,212,313,424,515;所以12681.442/10.208123-2219.5108,223.5877,311.0944,40.005062,50.041248;因此最终的模型为：ˆ -2219.5180 23.5877X11.0944X 0.005062X0.041248XYi36782.43354 2.238039 0.002144 0.010478t 8.70 9.6943 4.957199 2.361369 3.936757R20.98R20.9863F310.7127Df18DW2.038se307.9863

三、经济意义检验 模型估计结果表明：

在假定其他解释变量不变的情况下，当第二、三产业从业人数占全社会从业人数的比重增长一个百分点，农民人均纯收入就会增加23.5877元；

在假定其他解释变量不变的情况下，当农业总产值占农林牧总产值的比重增长一个百分点，农民人均纯收入就会增加11.0944元；

在假定其他解释变量不变的情况下，当农作物播种面积增长一千公顷，农民人均纯收入就会增加0.002144元；

在假定其他解释变量不变的情况下，当农村用电量增长一亿千瓦时，农民人

均纯收入就会增加0.4124元；

（四） 、统计检验 A、拟合优度检验

由表32可知，R20.98，R20.9863，说明模型的拟合优度很好。

B、F检验

针对H0：23450，在给定的显著性水平0.05下，查表可得自由度为k-1=4和n-k=14的临界值为F414,3.11。由表32中得到F=310.7127，由于F=310.7127>F414,3.11，应拒绝源假设23450，说明方程显著。

C、t检验

针对H0：j0j1、、23、、45，在给定的显著性水平0.05下，查表可ˆ、ˆ、得自由度为n-k14的临界值为t/2142.145。由表32中得到与12ˆ、ˆ、ˆ对应的t统计量分别为-8.70、9.6943、4.9572、2.3614、3.9368，其 345绝对值均大于t/2142.145，这说明分别拒绝H0：j0j1、、23、、45说明各个解释变量对方程均有显著影响。(五)、回归预测 ①、点预测

使用Eview软件进行点预测：

首先在Workfile窗口点击Range，出现Change Workfile Range窗口，将End Data改为2005；

然后在Workfile 中点击Sample，将窗口中的1986 2004改为1986 2005； 使用命令Data X3 X6 X7 X8，在数据表中将2005年的数据输入； 在Equation中，点击Forecast，确定后可得到如下图所示的结果，同时在Workfile中生成一新的序列Yf。

在Workfile中双击Yf就可以看到其具体的数值。 最终得到的点预测的纸如下表33所示：

1987 1988 19 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 实际值 137.6259 147.86166 196.76153 220.53342 223.25142 233.19453 265.66734 335.16333 411.2878 460.67926 477.955 474.01886 466.80042 466.15639 469.80345 468.95245 476.2441 499.38776 预测值 112.227 183.74 192.343 224.91 228.018 241.791 274.631 315.617 383.275 466.255 469.394 480.886 477.053 468.741 468.679 470.933 449.902 527.849 2005 521.20096 表33 601.733 ˆˆXβ(附：点预测公式为：Yff)

②、区间预测 A、平均值区间预测

平均值Yf的置信度为1的预测区间为：ˆtˆXfXXXf Yf/214627.166ˆ在本题中t/23.145，330.5119,利用EXECL软件nk14进行计算可得预测区间为：1494.054-290.588，2ei

B、个别值区间预测

个别值Yf的置信度为1的预测区间为：ˆtˆ1XfXXXf Yf/214627.166在本题中t/2330.5119,利用EXECL软件nk14进行计算可得预测区间为：1971.665-768.199，eˆ3.145，2i

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