第五章-异方差性(作业任务)

2023-12-31 来源：我们爱旅游

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5.3为了研究中国出口商品总额 2015年相关的指标数据，如表

EXPORT对国内生产总值 GDP的影响，搜集了 1990

5.3所示。

表3中国出口商品总额与国内生产总值时间出口商品总额国内生产总值时间 (单位：亿元) 出口商品总额国内生产总值 EXPORT 1991 1992 1993 1994 1995 1996 1997 1998 1999 3827.1 4676.3 5284.8 10421.8 12451.8 12576.4 15160.7 15223.6 16159.8 20634.4 2000 2001 2002 2003 22024.4 26947.9 36287.9 GDP 22005.6 27194.5 35673.2 48637.5 61339.9 71813.6 79715.0 85195.5 90564.4 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 100280.1 110863.1 121717.4 EXPORT 49103.3 62648.1 77597.2 93627.1 100394.9 82029.7 107022.8 123240.6 129359.3 137131.4 143883.7 141166.8 GDP 161840.2 187318.9 219438.5 270232.3 319515.5 349081.4 413030.3 489300.6 540367.4 595244.4 643974.0 685505.8 2014 2015 137422.0 资料来源：《国家统计局网站》 (1) 根据以上数据，建立适当线性回归模型。

(2) 试分别用White检验法与ARCH检验法检验模型是否存在异方差? (3)

如果存在异方差，用适当方法加以修正。

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解：（1）

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 15:38 Sample: 1991 2015 Included observations: 25

Variable C X

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) 模型回归的结果：

Coefficient -673.0863 4.061131 0.946323 0.943990

49784.06 5.70E+10 -304.8174 405.4924

Std. Error 15354.24 0.201677

t-Statistic -0.043837 20.13684

Prob. 0.9654

0.0000 234690.8 210356.7 0 24.545424.642924.57240.36622

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

1 4 8

0.000000

Y 673.0863 4.0611Xi

t ( 0.0438 )(20.1368)

* *

R2 0.9463, n 25

(2) white:该模型存在异方差

Heteroskedasticity Test: White F-statistic Obs*R-squared Scaled explained SS

4.493068 7.250127 8.361541

Prob. F(2,22) Prob. Chi-Square(2) Prob. Chi-Square(2)

0.0231 0.0266 0.0153

Test Equation:

Dependent Variable: RESIDE Method: Least Squares Date: 04/18/20 Time: 17:45 Sample: 1991 2015 Included observations: 25

Variable C XA2 X

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient -1.00E+09 -0.455420 102226.2 0.290005 0.225460 3.38E+09 2.51E+20 -582.4119 4.493068

Std. Error 1.43E+09 0.420966 60664.19

t-Statistic -0.700378 -1.081847 1.685117

Prob. 0.4910

0.2910 1 2.28E+09 3.84E+09 46.832946.979246.87350.749885 2 2 6

0.106

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

0.023110

ARCH检验：该模型存在异方差

Heteroskedasticity Test: ARCH F-statistic Obs*R-squared 18.70391 11.02827 Prob. F(1,22) Prob. Chi-Square(1) 0.0003 0.0009 Test Equation:

Dependent Variable: RESIDA2 Method: Least Squares Date: 04/18/20 Time: 19:55 Sample (adjusted): 1992 2015

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Included observations: 24 after adjustments

Variable

RESIDA2(-1) R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 8.66E+08 0.817146 0.459511 0.434944 2.93E+09 1.89E+20 -556.1552 18.70391

Std. Error 6.92E+08 0.188944

t-Statistic 1.251684 4.324802

Prob. 0.2238

0.0003 2.37E+09 3.90E+09

46.51293 46.61110 46.53898 0.888067

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

0.000273

(3)修正：加权最小二乘法修正

却 WF Woricflil-ri U

TLECi id tl e^cJ\\ 1i « T t\"l t-|H

L日芦£臼电*电引 OdiJ 1 0*左(■ 20 3>5

r^lucilifl -MI^I TGR 1 Z7Q I S w= — T ,皿”=

「「」工口工『日匕日吝口口「cDK E Ba^-oa 山口 fE=-UH a P-OE= -口曰 3.2 1 且-口9 I B之与尸-口口 ti .3-Z2E-DO 出q,峙尸・C旦( 4.3-1 E-O^

3 0 3IE 09 N.HMU O-QI 立o右匚

Included observations: 25 Weighting series: W2

Weight type: Inverse variance (average scaling)

Variable

Coefficient

Std. Error

t-Statistic

Prob.

P U 7 nQ r U J-4m y Q M-n!R-0 ;a 9m 0: B oaooom 1 m = 2 QH rK > - nO

4 TDE--W Z.&15^=- DC

1 hi-tiE - \"IIIJ i. um r ci Q SJ^F -iii i旦日二-①口

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 20:46 Sample: 1991 2015

C X

10781.17 3.931606

2188.706 0.192004

4.925821 20.47667

0.0001 0.0000

Weighted Statistics

R-squared

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

0.947998 0.945737 8420.515 1.63E+09

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion

51703.40 11816.72 20.99135 21.08886

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Log likelihood F-statistic Prob(F-statistic)

-260.3919 419.2938 0.000000

Hannan-Quinn criter. Durbin-Watson stat Weighted mean dep.

9 3 0

21.01830.5398639406.3

Unweighted Statistics

R-squared

Adjusted R-squared S.E. of regression

0.944994 0.942602 50396.82

Mean dependent var S.D. dependent var Sum squared resid

7 0

234690.8

210356.5.84E+1

修正后进行white检验:

Heteroskedasticity Test: White F-statistic Obs*R-squared Scaled explained SS

0.261901 0.581387 0.211737

Prob. F(2,22) Prob. Chi-Square(2) Prob. Chi-Square(2)

0.7720 0.7477 0.8995

Test Equation:

Dependent Variable: WGT_RESIDA2 Method: Least Squares Date: 04/18/20 Time: 20:41 Sample: 1991 2015 Included observations: 25

Collinear test regressors dropped from specification

Variable C X*WGTA2 WGTA2

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 71441488 -2711.961 13536351 0.023255 -0.065539 63753972 8.94E+16 -483.1391 0.261901

Std. Error 22046212 5055.773 20714871

t-Statistic 3.240534 -0.536409 0.653461

Prob. 0.0038

0.5971 2

65232673 61762160 38.89113 39.037338.93170.898909 0 7

0.520

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

0.771953

修正后的模型为

Y 10781.17 3.931606Xi

t (4.925821 )(20.47667)

R2 0.9480, n 25

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5.4 表5.4的数据是2011年各地区建筑业总产值（X）和建筑业企业利润总额（Y）。

表5.4 地区各地区建筑业总产值（X）和建筑业企业利润总额（

建筑业总产值X 建筑业企业利润总额Y 地区 Y）（单位：亿元）建筑业总产值X 建筑业企业利润总额Y 北京天津河北山西内蒙古辽宁吉林黑龙江上海江苏浙江安徽福建江西山东河南 6046.22 2986.45 3972.66 2324.91 1394.68 6217.52 1626.65 2029.16 4586.28 15122.85 14907.42 3597.26 3692.62 2095.47 6482.90 5279.36 216.78 79.54 127.00 49.22 105.37 224.31 89.03 58.92 166.69 595.87 411.57 127.12 126.47 62.37 291.77 湖北湖南广东广西海南重庆四川贵州云南西藏陕西甘肃青海宁夏新疆 5586.45 3915.02 5774.01 1553.07 255.47 3328.83 5256.65 824.72 1868.40 124.47 3216.63 925.84 319.42 427.92 1320.37 231.46 124.77 251.69 26.24 6.44 155.34 177.19 14.39 61.88 5.75 104.38 29.33 8.35 11.25 27.60 200.09 数据来源：国家统计局网站

根据样本资料建立回归模型，分析建筑业企业利润总额与建筑业总产值的关系，存在异方差，如果有异方差，选用最简单的方法加以修正。

并判断模型是否

解：散点图:

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建立线性回归模型:

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 21:16 Sample: 1 31

Included observations: 31

Variable

C X

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

i Coefficient 2.368138 0.034980 0.932055 0.929712 34.33673 34191.33 -152.5761 397.8152

Std. Error 9.049371 0.001754

t-Statistic 0.261691 19.94530

Prob. 0.7954

0.0000 134.4574 129.5145 9 9.9726410.065110.00282.57284

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

6 1 1

0.000000

white检验：

Heteroskedasticity Test: White

* *

0.000

F-statistic Obs*R-squared

26.00369 20.15100

Prob. F(2,28) Prob. Chi-Square(2)

0.0000

* *

Scaled explained SS 40.83473 Prob. Chi-Square(2) 0.0000

Test Equation:

Dependent Variable: RESIDA2 Method: Least Squares Date: 04/18/20 Time: 21:19 Sample: 1 31

Included observations: 31

Variable C XA2 X

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 498.3340 4.51E-05 -0.158176 0.650032 0.625035 1477.458 61120730 -268.6499 26.00369

Std. Error 559.4185 1.45E-05 0.221918

t-Statistic 0.890807 3.110610 -0.712768

Prob. 0.3806

0.0043 9 1102.946 2412.791 17.52580 17.664517.57102.732317 4 8

0.481

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

0.000000

模型存在异方差

模型修正：加权最小二乘法

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 21:24 Sample: 1 31

Included observations: 31 Weighting series: W2

Weight type: Inverse variance (average scaling)

Variable

C X

Coefficient 0.020734 0.034505

Std. Error t-Statistic 1.351842 0.002445

0.015338 14.11049

Prob. 0.9879 0.0000

Weighted Statistics

R-squared

0.872866 Mean dependent var

19.08548

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Prob(F-statistic)

0.800417 0.868482 6.525709 1234.962 -101.1016 199.1059 0.000000

S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat Weighted mean dep.

6.416052 6.651717 6.744233 6.681875 2.201198 9.525906

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Unweighted Statistics

R-squared

Adjusted R-squared S.E. of regression Durbin-Watson stat

0.930826 0.928441 34.64582 2.531761

Mean dependent var S.D. dependent var Sum squared resid

74 45 66

134.45129.5134809.

加权后进行white检验:

Heteroskedasticity Test: White F-statistic Obs*R-squared Scaled explained SS

0.224402 0.489051 1.141138

Prob. F(2,28) Prob. Chi-Square(2) Prob. Chi-Square(2)

0.8004 0.7831 0.5652

Test Equation:

Dependent Variable: WGT_RESIDA2 Method: Least Squares Date: 04/18/20 Time: 21:25 Sample: 1 31

Included observations: 31

Collinear test regressors dropped from specification

Variable C X*WGTA2 WGTA2

R-squared

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

Coefficient 28.90647 0.074634 -9.628706 0.015776 -0.054526 96.03099 258214.7 -183.9143 0.224402

Std. Error 24.35074 0.111471 15.02003

t-Statistic 1.187088 0.669539 -0.641058

Prob. 0.2452

0.5086 7 39.83747 93.51533 8

Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

6 2 3 12.058912.197712.10422.053490.526

Mean dependent var S.D. dependent var Akaike info criterion

修正成功，修正后的模型为：

* *

Y 0.020734 0.03450X

t （0.015338）（14.11049）

R 0.8729, n 31

5.5表5.5是2015年中国各地区人均可支配收入（

的数据。

X）与居民每百户汽车拥有量（ Y）

表5.5 中国各地区人均可支配收入居民每百户人均可支配时间收入（元）汽车拥有量 X与居民每百户汽车拥有量人均可支配时间收入（元）汽车拥有量居民每百户 X （辆）Y X （辆）Y 北京 48458.0 31291.4 18118.1 17853.7 22310.1 24575.6 18683.7 18592.7 49867.2 45.0 37.7 30.4 20.7 28.5 18.1 19.3 10.7 24.3 湖北 20025.6 19317.5 27858.9 16873.4 18979.0 20110.1 17221.0 13696.6 15222.6 12.9 16.4 24.6 18.0 15.5 15.4 15.3 15.0 24.2 天津湖南河北广东山西广西内家古海南辽宁重庆吉林四川黑龙江贵州上海云南

* *

Prob(F-statistic)

0.800417

江苏 29538.9 35537.1 18362.6 25404.4 18437.1 22703.2 17124.8 31.3 39.8 14.8 22.3 16.7 37.8 西藏 12254.3 17395.0 13466.6 15812.7 17329.1 16859.1 21.5 15.8 13.7 24.3 25.2 19.9 浙江陕西安徽甘肃福建青海江西宁夏山东新疆河南 17.4 (1) 试根据上述数据建立各地区人均可支配收入与各地区居民每百户汽车拥有量的线性回归模型。

(2) 选用适当方法检验模型是否存在异方差，并说明存在异方差的理由。

(3) 如果存在异方差，用适当方法修正。

解:( 1)

散点图:

50 10 15

10,000 45 40 35

丫 30

25 20

20,000

30,000

40,000 50,000

60,000

建立线性回归模型

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 21:32 Sample: 1 31

Included observations: 31

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Variable

C X

Coefficient 8.920236 0.000612 0.40480.38436.78771336.122

-102.3220 19.72799 19

0.0001

Std. Error 3.257781 0.000138

t-Statistic 2.738133 4.441620

Prob. 0.0100.0001

22.33871 8.650729

6.73046.82296.76061.488249 64 07 28

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

59 37 23

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

Y 8.920236 0.000612Xi

t (2.738133)(4.441620)

R 0.4049, n 31

(2)模型检验：white 检验

Heteroskedasticity Test: White F-statistic Obs*R-squared Scaled explained SS

4.652107 7.731852 5.765810

Prob. F(2,28) Prob. Chi-Square(2) Prob. Chi-Square(2)

0.0180 0.0209 0.0560

Test Equation:

Dependent Variable: RESIDA2 Method: Least Squares Date: 04/18/20 Time: 21:35 Sample: 1 31

Included observations: 31

Variable C XA2 X

R-squared

Adjusted R-squared S.E. of regression

Coefficient 4.550271 3.94E-08 0.000756 0.249415 0.195801 51.29246

Std. Error 76.95546 9.43E-08 0.005825

t-Statistic

. 0.059129 0.417357 0.129842

Prob0.9533 0.6796 0.8976 43.10072 57.19682 10.80473

Mean dependent var S.D. dependent var Akaike info criterion

* *

Sum squared resid Log likelihood F-statistic

Prob(F-statistic)

73665.66 -164.4733 4.652107

Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

50 97 88

10.94310.8492.1007

0.018014

nR 7.7319，由 White 检验知，

在

0.05下，查

分布表

分布表得临界值

0.05

(2) 5.9915

nR2 X(2)

存在异方差

(3) 模型修正：加权最小二乘法

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 21:54 Sample: 1 31

Included observations: 31 Weighting series: W2

Weight type: Inverse variance (average scaling)

Variable C X

Coefficient 0.020734 0.034505

Std. Error 1.351842 0.002445

t-Statistic 0.015338 14.11049

Prob. 0.9879

0.0000

Weighted Statistics

R-squared

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

0.872866 0.868482 6.525709 1234.962 -101.1016 199.1059 0.000000

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat Weighted mean dep.

19.08548 6.416052 7 3 5 8 6 6.651716.744236.681872.201199.52590

Prob(F-statistic)

Unweighted Statistics

R-squared

Adjusted R-squared

0.930826 0.928441

Mean dependent var S.D. dependent var

134.4574

129.5145

* *

S.E. of regression

Durbin-Watson stat

34.64582 2.531761

Sum squared resid

34809.66

white检验

Heteroskedasticity Test: White F-statistic Obs*R-squared Scaled explained SS

0.224402 0.489051 1.141138

Prob. F(2,28) Prob. Chi-Square(2) Prob. Chi-Square(2)

0.8004 0.7831 0.5652

Test Equation:

Dependent Variable: WGT_RESIDA2 Method: Least Squares Date: 04/18/20 Time: 21:54 Sample: 1 31

Included observations: 31

Collinear test regressors dropped from specification

Variable C X*WGTA2 WGTA2

R-squared

Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient 28.90647 0.074634 -9.628706 0.015776 -0.054526 96.03099 258214.7 -183.9143 0.224402

Std. Error 24.35074 0.111471 15.02003

t-Statistic 1.187088 0.669539 -0.641058

Prob. 0.2452

0.5086 7 39.83747 93.51533 8 12.058912.197712.10422 2.053493

0.526

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat

0.800417

Y 0.020734 0.03450%

t (0.015338)(14.11049) R 0.8729 ,n 31

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第五章-异方差性(作业任务)