第五章：异方差性(作业)

表3 中国出口商品总额与国内生产总值 （单位：亿元） 时间 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 出口商品总额 国内生产总值 EXPORT GDP 时间 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 出口商品总额 EXPORT 国内生产总值 GDP 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。 解：（1）

700,000600,000500,000400,000Y300,000200,000100,0000020,00060,000X

Dependent Variable: Y Method: Least Squares Date: 04/18/20 Time: 15:38

100,000140,000

Sample: 1991 2015

Included observations: 25

Variable

CX

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

模型回归的结果：

Coefficient

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

+10 Schwarz criterion

Hannan-Quinn criter.

Durbin-Watson stat

Yi673.08634.0611X

i^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

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 04/18/20 Time: 17:45 Sample: 1991 2015

Included observations: 25

Variable

C X^2

Coefficient

+09

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

Std. Error t-Statistic Prob.

+09

X

R-squared

+09 +09

Mean dependent var . dependent var Akaike info +09 criterion

+20 Schwarz criterion

Hannan-Quinn criter.

Durbin-Watson stat

Prob. F(1,22) Prob. Chi-Square(1)

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

ARCH检验：该模型存在异方差

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

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 04/18/20 Time: 19:55 Sample (adjusted): 1992 2015

Included observations: 24 after adjustments

Variable

CRESID^2(-1) R-squared Adjusted R-squared . of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)

Coefficient

+08

Std. Error t-Statistic Prob.

+08

+09 +09

Mean dependent var . dependent var Akaike info +09 criterion

+20 Schwarz criterion

Hannan-Quinn criter.

Durbin-Watson stat

（3）修正：加权最小二乘法修正

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

Included observations: 25 Weighting series: W2

Weight type: Inverse variance (average scaling)

Variable

CX

R-squared

Coefficient

Weighted Statistics

Std. Error t-Statistic Prob.

+10

Mean dependent var . dependent var Akaike info criterion

+09 Schwarz criterion

Hannan-Quinn criter.

Durbin-Watson stat Weighted mean dep.

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

R-squared Adjusted R-squared . of regression

Unweighted Statistics

Mean dependent var . dependent var Sum squared resid

修正后进行white检验：

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

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

Scaled explained SS

Test Equation:

Dependent Variable: WGT_RESID^2 Method: Least Squares Date: 04/18/20 Time: 20:41 Sample: 1991 2015

Included observations: 25

Prob. Chi-Square(2)

Collinear test regressors dropped from specification

Variable

CX*WGT^2 WGT^2

R-squared

Coefficient

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

+16 Schwarz criterion

Hannan-Quinn criter.

Durbin-Watson stat

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

修正后的模型为

^

Yi10781.173.931606X

it(4.925821)(20.47667)

R20.9480,n25

表的数据是2011年各地区建筑业总产值（X）和建筑业企业利润总额（Y）。

表 各地区建筑业总产值（X）和建筑业企业利润总额（Y） (单位：亿元) 地 区 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 湖 北 湖 南 广 东 广 西 海 南 地 区 建筑业总产值X 建筑业企业利润总额Y 辽 宁 吉 林 黑龙江 上 海 江 苏 浙 江 安 徽 福 建 江 西 山 东 河 南 重 庆 四 川 贵 州 云 南 西 藏 陕 西 甘 肃 青 海 宁 夏 新 疆 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，并判断模型是否存在异方差，如果有异方差，选用最简单的方法加以修正。

解：散点图：

700600500400Y300200100002,0006,000X10,00014,000

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

Coefficient

Std. Error t-Statistic Prob.

R-squared

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat

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

white检验：

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

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 04/18/20 Time: 21:19 Sample: 1 31

Included observations: 31

Variable

CX^2 X

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

Coefficient

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

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat

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

CX

R-squared

Coefficient

Weighted Statistics

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat Weighted mean dep.

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

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

Unweighted Statistics

Mean dependent var . dependent var Sum squared resid

加权后进行white检验：

Heteroskedasticity Test: White

F-statistic Obs*R-squared Scaled explained SS

Test Equation:

Dependent Variable: WGT_RESID^2 Method: Least Squares Date: 04/18/20 Time: 21:25

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

Sample: 1 31

Included observations: 31

Collinear test regressors dropped from specification

Variable

CX*WGT^2 WGT^2

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

Coefficient

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat

修正成功，修正后的模型为：

Yi0.0207340.034505X

i^t(0.015338)(14.11049)

R20.8729,n31

表是2015年中国各地区人均可支配收入（X）与居民每百户汽车拥有量（Y）的数据。 表 中国各地区人均可支配收入X与居民每百户汽车拥有量 时间 人均可支配收入（元） X 居民每百户汽车拥有量（辆）Y 时间 人均可支配收入（元） X 居民每百户汽车拥有量（辆）Y 北 京 天 津 河 北 山 西 内蒙古 辽 宁 吉 林 黑龙江 上 海 江 苏 湖 北 湖 南 广 东 广 西 海 南 重 庆 四 川 贵 州 云 南 西 藏 浙 江 安 徽 福 建 江 西 山 东 河 南 陕 西 甘 肃 青 海 宁 夏 新 疆 (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：

50454035Y302520151010,00020,00030,000X40,00050,00060,000

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

Variable

C X

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

Coefficient

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat

Yi8.9202360.000612X

i^t(2.738133)(4.441620)

R20.4049,n31

（2）模型检验：white检验

Heteroskedasticity Test: White

F-statistic Obs*R-squared Scaled explained SS

Test Equation:

Dependent Variable: RESID^2 Method: Least Squares Date: 04/18/20 Time: 21:35 Sample: 1 31

Included observations: 31

Variable

C X^2 X

R-squared

Coefficient

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

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat

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

nR27.7319，由White检验知，

在0.05下，查 分布表 分布表得临界值 20.052(2)5.9915

2nR2χ(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

CX

R-squared

Coefficient

Weighted Statistics

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat Weighted mean dep.

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

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

Heteroskedasticity Test: White

F-statistic Obs*R-squared Scaled explained SS

Unweighted Statistics

Mean dependent var . dependent var Sum squared resid

white检验

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

Test Equation:

Dependent Variable: WGT_RESID^2 Method: Least Squares Date: 04/18/20 Time: 21:54 Sample: 1 31

Included observations: 31

Collinear test regressors dropped from specification

Variable

CX*WGT^2 WGT^2

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

^

Coefficient

Std. Error t-Statistic Prob.

Mean dependent var . dependent var Akaike info criterion

Schwarz criterion Hannan-Quinn criter.

Durbin-Watson stat

Yi0.0207340.034505X

it(0.015338)(14.11049)

R20.8729,n31