4 第四章 习题 参考答案

第四章习题参考答案P 135

7. 1）用OLS法建立居民人均消费支出与可支配收入的线性模型。create u 20；data consump income；

ls consump c income

Dependent Variable: CONSUMP

Method: Least Squares

Sample: 1 20

Included observations: 20

t-Statistic Prob.

Variable Coefficient|

Std. Error

C

INCOME&

R-squared Mean dependent var

Adjusted R-squared. dependent var#

. of regression Akaike info criterion

Sum squared resid Schwarz criterion

Log likelihood/ F-statistic

Durbin-Watson stat Prob(F-statistic)

线性模型如下：

CONSUMP = 5389 + *INCOME

2）检验模型是否存在异方差性

（

图：是否有明显的散点扩大/缩小/复杂型趋势

i) X Y

scat income consump

ii)解释变量—残差图：是否形成一条斜率为0的直线

scat income resid^2 或者

genr ei2=resid^2；scat income ei2

由两个图形，均可判定存在递增型异方差。

还可以用帕克检验，戈里瑟检验，戈德菲尔德-匡特检验，怀

特检验等方法。

iii) 戈德菲尔德-匡特检验：共有20个样本，去掉中间1/4个样本（4

个），剩余大样本、小样本各8个。

Sort income ； smpl 1 8； ls consump C income

]

Smpl 13 20； ls consump C income

21

0.050.05615472.0126528.3

4.86

(,)(81,81) 4.28

11

811811

1111RSS RSS F F F n k n k n k n k =

==--=>=

--------------，存在异方差。

iV)怀特检验：因为只有一个变量，故是否含有交叉项是一样的。

2220112231425122

2

01234522011212

2

012:0,(),:0,(),i

i i i i i i i i

i i i

e a a X a X a X a X a X X v H a a a a a nR

q q e

a a X a X v H a a nR

q q χχ=++++++======+++==变量个数

变量个数

View ＼residual test ＼white heteroskedastcity

(cross terms / no cross terms )

White Heteroskedasticity Test: F-statistic

Probability

：

Obs*R-squared

Probability

Dependent Variable: RESID^2 Method: Least Squares

Sample: 1 20

Included observations: 20

Variable

！

Coefficient

Std. Error t-Statistic Prob.

C

INCOME

.

INCOME^2

R-squared

$

Mean dependent var

Adjusted R-squared . dependent var

. of regression

Akaike info criterion

~

Sum squared resid +10 Schwarz criterion

Log likelihood F-statistic

Durbin-Watson stat

~

Prob(F-statistic)

2

2220.05(),212.65213(2) 5.99n R q q n R χχ===>，存在异方差。

还可以通过概率()220.0017890.05P n R χ>=<判定存在异方差。

3）若存在异方差，用适当的方法估计模型对数(加权最小二乘法) ls consump C income；genr eijdz=abs(resid)

ls(w=1/eijdz) consump C income

Dependent Variable: CONSUMP

%

Method: Least Squares

Sample: 1 20

Included observations: 20

Weighting series: 1/EIJDZ

Variable Coefficient Std. Error t-Statistic Prob.

C，

INCOME

Weighted Statistics

《

Mean dependent var

R-squared

Adjusted R-squared. dependent var

. of regression】

Akaike info criterion

Sum squared resid Schwarz criterion

Log likelihood F-statistic

?

Prob(F-statistic)

Durbin-Watson stat

Unweighted Statistics

R-squared Mean dependent var

Adjusted R-squared ·

. dependent var . of regression Sum squared resid

Durbin-Watson stat

"

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

Probability

【

Test Equation:

Dependent Variable: STD_RESID^2 Method: Least Squares Sample: 1 20

Included observations: 20

Variable

Coefficient

Std. Error

t-Statistic

Prob.

;

C

INCOME

，

220.050.076420(2) 5.99n R q χ===<或()220.9625110.05P n R χ>=>,均

可判定加权处理后的模型不存在异方差。

模型经取对数或加权处理都可以一定程度地消除异方差性。

ls log(consump) C log(income)； genr eijdz=abs(resid)； ls(w=1/eijdz) log(Consump) C log(Income)

普通最小二乘模型

CONSUMP = 5389 + *INCOME

加权最小二乘模型

CONSUMP = + *INCOME

~

对数模型：

LOG(CONSUMP)=+*LOG(INCOME)

加权对数模型：

LOG(CONSUMP)=+ *LOG(INCOME)

对各种模型的White检验结果，综合如下

模型不取对数

| F-statistic Probability

Obs*R-squared Probability

模型取对数

F-statistic Probability

~

Probability

Obs*R-squared

模型不取对数，但加权

F-statistic Probability

Obs*R-squared 、

Probability

模型取对数，且加权

F-statistic Probability

Obs*R-squared

&

Probability

可见，各种方法都可以起到抑制异方差的效果。

8. 1）若采用对数模型，是否存在序列相关性

ls log(industry) C log(invest)

Dependent Variable: LOG(INDUSTRY)

Method: Least Squares

Sample: 1901 1921

Included observations: 21

—

Variable

Coefficient Std. Error t-Statistic Prob.

C

|

LOG(INVEST)

R-squared Mean dependent var

Adjusted R-squared

"

. dependent var

. of regression Akaike info criterion

Sum squared resid

Schwarz criterion

}

Log likelihood F-statistic Durbin-Watson stat

Prob(F-statistic)

LOG(INDUSTRY) = 1. + *LOG(INVEST)

|

i) 1,t t e e 散点图

ii)

t e 随t 变化的散点图

由两个图形，均可判定存在正序列相关。

还可以利用回归检验法，D -W 检验，拉格朗日乘数检验等方法。

iii) D -W检验(DL(21, =, DU(21, =

.= < DL(21, 2，=，至少存在一阶正自相关；但.只适用判别一阶自相关。

iv) 拉格朗日乘数检验

Breusch-Godfrey Serial Correlation LM Test:

F-statistic# Probability

Obs*R-squared Probability

Prob.

Variable Coefficient Std. Error。

t-Statistic

C

LOG(INVEST)\

RESID(-1)

R-squared Mean dependent var: Adjusted R-squared. dependent var

Log likelihood F-statistic

Durbin-Watson stat| Prob(F-statistic)

一阶LM Test ：LM Test

()()22

220.05,(1),

=(9.836218>) 3.84LM n p R p n p R p αααχχ==->=-=

()()220.0017110.05P n p R χα>-=<=

RESID(-1)的t 统计量显著（P=<），至少存在一阶自相关。

2）按照一阶自相关，用杜宾两步法和广义最小二乘法估计原模型。

；

杜宾两步法：

{}

10111101111

(1)()(1)..t t t t t t t t t t t t t OLS : ls y c y(-1) Y Y X X Y Y X X x x -)

e (1i ββρβρρμρμβρρρβρνμμ---------?-+=++=+-+①

--

ls y c y(-1) x x(-1)

y(-1)前面的系数：?ρ

，代回差分模型①，再次进行OLS 估计得到原模型的参数估计量，即 ***11000

???????,,(1)OLS ρβββββρ???→==-→①。

genr y = log(industry)； genr x = log(invest)； Step 1: ls y c y(-1) x x(-1)

Dependent Variable: Y Method: Least Squares Sample(adjusted): 1981 2000

{

Included observations: 20 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C—

Y(-1)

X:

X(-1)

R-squared Mean dependent var

. dependent var

·

Adjusted R-squared

. of regression Akaike info criterion

Sum squared resid<

Schwarz criterion

Log likelihood F-statistic

Durbin-Watson stat Prob(F-statistic)

{

Step 2: ls y - * y(-1) c x - * x(-1)

Dependent Variable: *Y(-1)

Method: Least Squares

Sample(adjusted): 1981 2000

Included observations: 20 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

，

C

*X(-1)

Mean dependent var

&

R-squared

Adjusted R-squared. dependent var

. of regression]

Akaike info criterion

Sum squared resid Schwarz criterion

Log likelihood F-statistic

Prob(F-statistic)

；

Durbin-Watson stat

.=介于DL(21-1, 2，=与DU(21-1, 2，=之间，不能判别是否存在一阶正自相关，但可由拉格朗日乘数法判断，此时不存在序列相关性。Breusch-Godfrey Serial Correlation LM Test:

F-statistic Probability>

Obs*R-squared Probability

Test Equation:

Dependent Variable: RESID

Method: Least Squares

t-Statistic Prob.

Variable Coefficient]

Std. Error

C

*X(-1):

RESID(-1)

R-squared>

Mean dependent var

Adjusted R-squared. dependent var

. of regression Akaike info criterion

Schwarz criterion

?

Sum squared resid

Log likelihood F-statistic

Durbin-Watson stat…

Prob(F-statistic)

()22

1.5537080.05,=(1) 3.84LM n p R p ααχ<==-==

()()220.2125890.05P n p R χα>-=>=

拉格朗日乘数检验：D-W stat ： > ，不存在序列相关性。

所以 ()*11

**0

00??0.903502?0.415361???1 1.1283?0.631866βββββρ

ρ==?=??=-=?=??

矫正后的模型：

LOG(INDUSTRY) = + *LOG(INVEST)

\

原模型：

LOG(INDUSTRY) = 1. + *LOG(INVEST)

广义差分法

ls y c x ar(1) 1.20, 1.41,.. 1.348513(,)L U L U d d DW

d d ===∈ （不能判定是否存在一阶自相关）

Dependent Variable: Y Method: Least Squares Sample(adjusted): 1981 2000

Included observations: 20 after adjusting endpoints

|

Convergence achieved after 15 iterations

Variable Coefficient Std. Error t-Statistic Prob.

C￥

X

AR(1)&

R-squared Mean dependent var

Adjusted R-squared. dependent var

. of regression? Akaike info criterion

Sum squared resid Schwarz criterion

Log likelihood F-statistic]

Durbin-Watson stat Prob(F-statistic)

但由LM检验：概率为>，故此时不存在序列相关性。因此模型只存在一阶自相关性。

Breusch-Godfrey Serial Correlation LM Test:

F-statistic Probability！

Obs*R-squared

Probability

Dependent Variable: RESID

Variable

Coefficient

Std. Error

t-Statistic

*

Prob.

C

X

；

AR(1)

RESID(-1)

,

Durbin-Watson stat Prob(F-statistic)

模型为 Y = + *X + * AR(1) 与杜宾两步法矫正的模型：LOG(INDUSTRY) = + *LOG(INVEST) 非常接近。

广义最小二乘法 211213

1222111212

2322123

?()()n n n n n n E X X X Y σσσσσσσσμμσ

σβ

σσσσ---?? ? ?

'''=Ω?=ΩΩ ?

???

若仅存在一阶自相关

210.688217, 1.473643t t t

v ls μρμρρ-=+?==

。

21221212

3

1

011011,10

00

11n n n n n ρρ

ρρρρρρρρρρρ

ρρ-------??

?? ? ?-+- ?

?Ω≈Ω≈

? ?-- ?

???

??

ls log(industry) C log(invest) genr resid_corr = resid ls resid_corr Ｃ resid_corr(-1) 注：resid 是内置变量；

Dependent Variable: RESID_CORR Method: Least Squares

Variable

Coefficient

Std. Error

%

t-Statistic

Prob.

C

RESID_CORR(-1)

>

R-squared Mean dependent var Durbin-Watson stat

Prob(F-statistic)

/

直接计算

()111?() 1.7539,0.8386X X X Y β---'''=ΩΩ=

模型为LOG(INDUSTRY)=+*LOG(INVEST)，误差偏大。

3）采用差分形式**

11,t t t t t t X X X Y Y Y --=-=-，估计原模型

*

*

t t

t Y X αβν=++。

ls D(industry) C D(invest) ＯＲ ＣＯＭＭＡＮＤ

ls industry–industry(-1) C invest–invest(-1)

Dependent Variable: D(INDUSTRY)

Method: Least Squares

Sample(adjusted): 1981 2000

Included observations: 20 after adjusting endpoints

Variable)

Std. Error t-Statistic Prob.

Coefficient

C

D(INVEST)》

R-squared Mean dependent var

Adjusted R-squared#

. dependent var

. of regression Akaike info criterion

Sum squared resid Schwarz criterion

F-statistic

（

Log likelihood

Durbin-Watson stat Prob(F-statistic)

Breusch-Godfrey Serial Correlation LM Test:

F-statistic$ Probability

Obs*R-squared Probability

Test Equation:

Dependent Variable: RESID

Method: Least Squares

Coefficient Std. Error t-Statistic Prob.

！

Variable

C

"

D(INVEST)

RESID(-1)

Mean dependent var

~

R-squared

Adjusted R-squared. dependent var

. of regression\

Akaike info criterion

Sum squared resid Schwarz criterion

Log likelihood F-statistic

Prob(F-statistic)

】

Durbin-Watson stat