4第四章习题参

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

Variable

Coefficient

C INCOME R-squared

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

Std. Error

t-Statistic

Prob.

Mean dependent

criterion

Sum squared resid

Schwarz

criterion

1

Log likelihood Durbin-Watson stat

F-statistic

Prob(F-statistic)

线性模型如下：

CONSUMP = 53 + *INCOME 2）检验模型是否存在异方差性

i) XY图：是否有明显的散点扩大/缩小/复杂型趋势

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。

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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

FRSS2RSS1nk1nk1615472.0126528.38118114.86，存在异方差。

F0.05(nk1,nk1)F0.05(811,811)4.28iV)怀特检验：因为只有一个变量，故是否含有交叉项是一样的。

ea0a1X1ia2X2ia3X12ia4X22ia5X1iX2iviH0:a1a2a3a4a50,nR e2i22i(q),q变量个数

2a0a1X1ia2X12ivi2H0:a1a20,nR(q),q变量个数2View＼residual test＼white heteroskedastcity

(cross terms / no cross terms )

White Heteroskedasticity Test: F-statistic

Obs*R-squared

Probability

Probability

3

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

Included observations: 20

Variable

Coefficient

C INCOME INCOME^2 R-squared

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

Std. Error

Mean dependent

t-Statistic

Prob.

criterion

Sum squared resid

Log likelihood Durbin-Watson stat

+10

Schwarz

criterion

F-statistic

Prob(F-statistic)

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nR22(q),q2nR212.652130.052(2)5.99，存在异

方差。还可以通过概率P2nR20.00170.05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

C INCOME

Weighted Statistics R-squared

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

Mean dependent

Std. Error

t-Statistic

Prob.

criterion

5

Sum squared resid

Log likelihood Durbin-Watson stat

Unweighted Statistics R-squared

Schwarz

criterion

F-statistic

Prob(F-statistic)

var

Mean dependent

Adjusted R-squared

. of regression

. dependent var

Sum squared

resid

Durbin-Watson stat

White Heteroskedasticity Test: F-statistic

Obs*R-squared

Test Equation:

Dependent Variable: STD_RESID^2 Method: Least Squares

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Probability

Probability

Sample: 1 20

Included observations: 20 Variable

Coefficie

nt

C INCOME

Std. Error

t-Statis

tic

Prob.

nR20.07200.052(q2)5.99或

P2nR20.9625110.05,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 = 53 + *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：

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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-statisti

c Obs*R-squa

red 模型取对数 F-statisti

c Obs*R-squa

red

模型不取对数，但加权 F-statisti

c Obs*R-squa

red

模型取对数，且加权 F-statisti

Probability

Probability

Probability

Probability

Probability

Probability

Probability

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c Obs*R-squa

red

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

C LOG(INVEST) R-squared

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

Std. Error

Mean dependent

t-Statistic

Prob.

9

criterion

Sum squared resid

Log likelihood Durbin-Watson stat

Schwarz

criterion

F-statistic

Prob(F-statistic)

LOG(INDUSTRY) = 1. + *LOG(INVEST) i) et1,et散点图 ii) et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。

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iv) 拉格朗日乘数检验

Breusch-Godfrey Serial Correlation LM Test: F-statistic Obs*R-squared

Variable

Coefficient

C LOG(INVEST) RESID(-1) R-squared Adjusted R-squared

Log likelihood Durbin-Watson stat

Probability Probability

Std. Error

t-Statistic

Prob.

Mean dependent var . dependent var

F-statistic Prob(F-statistic)

一阶LM Test：LM Test

20.05,LMnpR2(p1),npR2=9.836218>2(p)3.84

P2npR20.0017110.05

RESID(-1)的t统计量显著（P=<），至少存在一阶自相关。

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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：

YtYt10(1)1(XtXt1)t-t1Yt0(1)Yt11Xt1Xt1t-t1①

i.e.tOLS: ls y c y(-1) x x(-1)ls y c y(-1) x x(-1)

ˆ，代回差分模型①，再次进行OLS估计得y(-1)前面的系数：到

原

模

型

的

参

数

估

计

量

，

即

OLSˆ*ˆ,ˆ*,ˆˆ*(1ˆ①ˆ)。 11000

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

C Y(-1)

Std. Error

t-Statistic

Prob.

12

X X(-1)

R-squared

Adjusted R-squared . of regression

Mean dependent var . dependent var Akaike info

criterion

Sum squared resid Log likelihood Durbin-Watson stat

Schwarz criterion F-statistic 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

C *X(-1)

R-squared

var

13

Std. Error

t-Statistic

Prob.

Mean dependent

Adjusted R-squared . of regression

. dependent var Akaike info

criterion

Sum squared resid

Schwarz

criterion

Log likelihood Durbin-Watson stat

F-statistic

Prob(F-statistic)

.= 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

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Variable Coefficient

Std. Error

t-Statistic

Prob.

C *X(-1) RESID(-1) R-squared

var

Mean dependent

Adjusted R-squared

. of regression

. dependent var

Akaike info

criterion

Sum squared resid

Log likelihood Durbin-Watson stat

20.05,LMnpR2=1.553708(p1)3.84

Schwarz

criterion

F-statistic

Prob(F-statistic)

P2npR20.21250.05

拉格朗日乘数检验：D-W stat： > ，不存在序列相关性。

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ˆ*ˆ0.90350211ˆ*0.415361所以 0

*ˆˆˆ1.1283001ˆ0.631866矫正后的模型：

LOG(INDUSTRY) = + *LOG(INVEST) 原模型：

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

广义差分法

..1.348513(dL,dU) ls y c x ar(1) dL1.20,dU1.41,DW（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

C X

Std. Error

t-Statistic

Prob.

16

AR(1) R-squared

var

Mean dependent

Adjusted R-squared

. of regression

. dependent var

Akaike info

criterion

Sum squared resid

Log likelihood Durbin-Watson stat

Schwarz

criterion

F-statistic

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

ity

Obs*R-squared

Probability

Dependent Variable: RESID

17

Probabil

Variable Coefficient

Std. Error

t-Statis

tic

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 1212132232212E()n1n2n31n2n2nˆ(X1X)1X1Y 2若仅存在一阶自相关

tt1vt

ls0.688217,21.4733

2n3n1n21121,11201018

1n1

100n200 1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

C

RESID_CORR(-1)

R-squared

var

Durbin-Watson stat

1Std. Error

t-Statis

tic

Prob.

Mean dependent

Prob(F-statistic)

11ˆ(XX)XY1.7539,0.8386 直接计算 模型为LOG(INDUSTRY)=+*LOG(INVEST)，误差偏大。

**XXX,Y3）采用差分形式ttt1tYtYt1，估计原模型

Yt*Xt*t。

ls D(industry) C D(invest) ＯＲ ＣＯＭＭＡＮＤ ls industry–industry(-1) C invest–invest(-1)

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

Included observations: 20 after adjusting endpoints

Variable

Coefficient

C D(INVEST) R-squared

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

Std. Error

Mean dependent

t-Statistic

Prob.

criterion

Sum squared resid

Schwarz

criterion

Log likelihood Durbin-Watson stat

F-statistic

Prob(F-statistic)

Breusch-Godfrey Serial Correlation LM Test:

20

F-statistic

Obs*R-squared

Test Equation:

Dependent Variable: RESID Method: Least Squares

Variable

Coefficient

C D(INVEST) RESID(-1) R-squared

Probability

Probability

Std. Error

t-Statistic

Prob.

Mean dependent

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

criterion

Sum squared resid

Schwarz

criterion

21

Log likelihood Durbin-Watson stat

F-statistic

Prob(F-statistic)

原模型存在一阶正自相关，但经过一阶自相关差分处理后不存在序列相关性（.= > 或Ｐ=>）。

模型为：D(INDUSTRY) = + *D(INVEST)

说明：在有的方法不能判别自相关性时，可以用其他方法测试。

9. 说明下述回归模型是否可靠

Y01X1tincome2X2twealthtconsump

Ls CONSUMP C INCOME WEALTH

Dependent Variable: CONSUMP Method: Least Squares Sample: 1 10

Included observations: 10

Variable

Coefficient

C INCOME WEALTH

22

Std. Error

t-Statistic

Prob.

R-squared var

Mean dependent

Adjusted R-squared

. of regression

. dependent var

Akaike info

criterion

Sum squared resid

Log likelihood Durbin-Watson stat

Schwarz

criterion

F-statistic

Prob(F-statistic)

Dependent Variable: CONSUMP

Variable

Coefficient

INCOME C

Std. Error

t-Statistic

Prob.

Dependent Variable: CONSUMP

Variable

Coefficient

23

Std. Error

t-Statistic

Prob.

WEALTH C

Dependent Variable: INCOME

Variable

Coefficient

WEALTH C

R-squared

var

Adjusted R-squared

. of regression

Akaike info

. dependent var

Std. Error

Mean dependent

t-Statistic

Prob.

criterion

Sum squared resid

Log likelihood Durbin-Watson stat

Schwarz

criterion

F-statistic

Prob(F-statistic)

1) 财富变量WEALTH前的符号是负号，与现实相反，经济意义

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不合理。

2) 拟合优度值较高，收入与财富可以解释消费支出的96%，然而在5%的显著水平下，二者的t检验都不显著。

3）分别做一元回归时，两个变量均显著，说明二者对消费支出确实有影响。但财富变量WEALTH前的符号改变了，且一元回归和二元回归的参数的方差变化很大。

4) F统计量的值很大，说明方程总体上线性显著成立。

5）做两个变量的回归，发现拟合优度很高（）。

6）收入INCOME和财富WEALTH的相关系数很大，相关系数阵为：

综上，收入与财富间可能存在着较高的相关性（共线性

0.998577），正是二者的高度相关使得各自对消费的影响无法分

辨。所以这个回归结果是不可靠的，应该首先消除共线性，之后再做OLS回归。

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