ccer计量wooldrige第七讲多元回归分析

Multiple

Regression

Analysis多元回归分析y

=

b0

+

b1x1

+

b2x2

+

.

.

.

bkxk

+

u6.Heteroskedasticity

(HSK)异方差Intermediate

Econometrics,Yan

Shen1Chapter

Outline

本章提要Intermediate

Econometrics,Yan

Shen2Consequences

of

HSK

of

OLSOLS中异方差的影响HSK-Robust

Inference

after

OLSestimationOLS估计后“对异方差稳健”的统计推断Testing

for

HSK检验异方差Weighted

Least

Squares

Estimation加权最小二乘估计Lecture

Outline

本课提要Intermediate

Econometrics,Yan

Shen3What

is

HSK什么是异方差Consequences

of

HSK异方差的影响HSK-Robust

Inference

after

OLS

estimationOLS估计后的“对异方差稳健”统计推断HSK-robust

standard

errorHSK－异方差稳健标准差HSK-robust

t,F,LM

statisticsHSK－异方差稳健t,

F,

LM统计量What

is

Heteroskedasticity

(HSK)什么是异方差Intermediate

Econometrics,Yan

Shen4

Recalltheassumptionofhomoskedasticity

impliedthatconditional

on

the

explanatory

variables,

the

variance

oftheunobserved

error,u,

was

constant同方差假定意味着条件于解释变量，不可观测误差的方差为常数

Ifthis

isnottrue,

thatisif the

variance

of

u

is

different

fordifferent

values

of

the

x’s,

then

the

errors

are

heteroskedastic如果u

的方差随x变化，那么误差是异方差的。

Example:

estimatingreturns

to

education

and

ability

isunobservable,

and

thinkthe

variance

in

ability

differs

byeducational

attainment例子：估计教育回报并且能力不可观测，认为能力的方差随教育水平变化。Education

levelprimary.secondaryf(y|x)Illustration

of

Heteroskedasticity异方差图示college..E(y|x)

=

b0

+

b1xwageIntermediate

Econometrics,Yan

Shen5Intermediate

Econometrics,Yan

Shen6A

specific

example:

histograms

of wage

rates

for

each

educationdegree,

from

only

educated

1

year

to

18

years.一个具体例子：每一个教育年限（1－18年）对应人群的工资直方图0.1

.2

.30.1.2

.30.1

.2

.30.1

.2

.3051015

200510

15

2005101520

0510

15

20

020123456789101112131415161718Fraction5

10

15

wagesGraphs

by

gradeChecking

the

Existence

of HSK:

plottingthe

residuals

against

the

fitted

values-1.5-10.51Residuals-.54.54.64.84.9Intermediate

Econometrics,Yan

Shen74.7Linear

predictionWhen

there

is

heteroskedasticity…当存在异方差时…Intermediate

Econometrics,Yan

Shen8

OLS

isstillunbiased

and

consistent.OLS无偏且一致R-squared

or

adjusted

R-squared

are

stillfine

goodness-of-fitmeasures.R平方和调整后的R平方仍可以很好地度量拟合优度。They

are

estimates

of

the

population

R-squared,

1

–

[Var(u)/Var(y)],where

the

variances

are

the

unconditional

variances

in

thepopulation.它们是对总体R平方1

–[Var(u)/Var(y)]的估计，其中的方差是总体中的“非条件”方差。They

consistently

estimate

the

population

R-squared,

whether

or

notVar(u|x)

=

Var(y|x)

depends

on

x

.无论Var(u|x)=Var(y|x)是否依赖于x，它们都可以一致地估计总体R平方。Why

do

we

care?为何关心异方差？Intermediate

Econometrics,Yan

Shen9The

standard

errors

of

the

estimates

are

biasedif

we

have

heteroskedasticity.如果存在异方差，那么估计值的标准差是有偏的。

If

the

standard

errors

are

biased,

we

can

notuse

the

usual

t

statistics

or

F

statistics

or

LMstatistics

for

drawing

inferences.如果标准差有偏，我们就不能应用通常的t统计量或F统计量来进行统计推断。What

to

do?怎么办？Intermediate

Econometrics,Yan

Shen10Econometricians

have

learned

how

to

adjust

standard errors,

t,

F,

and

LM

statistics

so

that

they

are

valid

in the

presence

of

heteroskedasticity

of

unknown

form.计量经济学家已经知道如何调整标准差，t，F，LM量，使得它们当未知形式的异方差存在时仍然有效。White

(1980)

shows

that

the

variances,

Var(bˆj

)

,

canbe

estimated

in

the

presence

of

heteroskedasticity.White（1980）指出，在存在异方差时，方差Var(bˆj

)也是可以估计的。Variance

with

Heteroskedasticity异方差存在时的方差((

)(

)(

)(

)1122

211

x1121ˆ,

so

conditioning

on

x,ˆ2ix

ixiiSST

2xFor

the

simple

case,

bVar(bˆ

)

reduces

to

s

2/SST under

Homoskedasticity.x

-

x

u=

b

+x

-

xx

-

x

sVar

(bˆ

)=x

-

x

u=

b

+x

-

xVar

(bˆ

)=

i

i

i

i,

where

SST

=

(x

-

x

)

.

i

一个简单情况是b，所以对于给定的，(

)2Intermediate

Econometrics,Yan

Shen11

2xixSST

2x

-

x

s

i=

(x

-

x

)

.2i

，其中SST当同方差成立时Var(bˆ

)退化为s

2/SST

。1

xVariance

with

Heteroskedasticity异方差存在时的方差(

)(

)2Intermediate

Econometrics,Yan

Shen12222ˆWhite

shows

thatˆˆi

ii

iSST

2SST

2xis

a

valid

estimatorfor

Var(bˆ

j

),where

uˆi

are

the

OLS

residuals.x

-

x

ux

-

x

u

White指出，是Var(bj

)的一个x合适的估计量，其中uˆi

是OLS

残差.Variance

with

Heteroskedasticity异方差存在时的方差2

2ij

irˆ

uˆ,

where

rˆij

is

the

ith

residual

fromSSR2For

the

general

multiple

regression

model,

a

validestimator

of

Var(bˆ

j

)

with

heteroskedasticity

isVar(b

j

)

=

ˆ

2

2Intermediate

Econometrics,Yan

Shen13

ˆrˆ

uˆSSR2

ijjregressing

x

j

on

all

other

independent

variables,

andSSR

jis

the

sum

ofsquared

residuals

from

theauxiliary

regression.对于多元回归，当异方差存在时，Var(bˆ

j

)的一个合适的估计量是Var(b

j

)=i

，其中rˆij是将x

j对其它解释变j量回归时第i个观察值对应的残差，SSR

j是辅助方程中的残差平方和。Variance

with

Heteroskedasticity异方差存在时的方差Intermediate

Econometrics,Yan

Shen14The

square

root

of is

called:开平方被称为Heteroskedasticity-robust

standard

error,

or对异方差稳健的标准差，或White

standard

error,

orWhite标准差，或Huber

standard

error,

orHuber标准差，或Eicker

standard

errors,

orEicker

标准差

ˆVar(b

j

)

ˆVar(b

j

)Robust

Standard

Errors稳健标准差Intermediate

Econometrics,Yan

Shen15

Now

the

robust

standard

errors can

be

used

forinference稳健标准差可以用来进行推断。Sometimes

the

estimated

variance

is

correctedfor

degrees

of

freedom

by

multiplying

by

n/(n

–k

–

1)有时可以将估计的方差乘以n/(n–k–1)来修正自由度As

n

→

∞

it’s

all

the

same,

though.当n

→∞时，没有区别。Example:

robust

se

versus

usual

se例子：稳健标准差与常规标准差Intermediate

Econometrics,Yan

Shen16(0.1)(0.109)(0.055)(0.057)(0.058)(0.058)(0.056)(0.057)(0.067)(0.074)(0.055)(0.051)l

og(wage)

=

0.321+

0.213

x1

-

0.198

x2

-

0.11

x3+0.0789edu+

0.0268exp

er

+

...Numbers

in

the

1st

row

with

brackets

are

usualstandard

errors,

and

those

in

the

2nd

are

robust

se.第一行括号中是常规标准差，第二行为稳健

标准差Example:

robust

se

versus

usual

se例子：稳健标准差与常规标准差Intermediate

Econometrics,Yan

Shen17What

do

we

learn?我们学到了什么？Robust

standard

errors

can

be

eitherlarger

or

smaller

than

theusual

standarderrors.稳健标准差可能比常规标准差大，也可能小。But

empirically

the

robust

standard

errors

are

often

found

to

belarger

than

the

standard

errors.但是实证中常常发现稳健标准差要大些。If

the

differences

between

these

two

errors

arelarge,then

theconclusions

for

statisticalinference

can

beverydifferent.如果这两种标准差的差异很大，那么统计推断的结论可能有很大差异。Now,

why

care

about

the

usual

se?为何要考虑常规标准差？Intermediate

Econometrics,Yan

Shen18Given

that

robust

standard

errors

are

validwhether

or

not

heteroskedasticity

is

present,then

why

do

we

still

need

the

usual

standarderror?如果稳健标准差无论异方差存在与否都是适用的，为什么我们还需要常规标准差？Notice

that

Robust

standard

errors

are

justifiedonly

when

the

sample

size

is

large.我们应当注意到，稳健标准差的适用性依赖于大样本。Robust

Standard

Errors稳健标准差Intermediate

Econometrics,Yan

Shen19

Whenthesamplesizeissmall

and

the

homoskedasticyassumptionactually

holds,

theusualtstatistics

haveexactt distribution,

butthis

willnot

bethe

casefor

robust

standard

errors,

henceinferences

may

notbe

correct如果是小样本同方差情形，那么常规的t统计量精确地服从t分布，但是这并不适用于稳健标准差，因此，在这种情况下使用稳健标准差就可能导致推断错误。

When

the

sample

size

is

large,

reporting

robust

standard

errors

(ortogether

with

the

usualstandard

errors)

are

mended,

esp.

inusingcross-sectional

data.在大样本情形下，特别是应用截面数据的时候，我们推荐报告稳健标准差（或同时报告常规的标准差）。Heteroskedasticy

(HSK)-robust

Inference

after

OLSestimationOLS估计后的HSK-稳健推断Intermediate

Econometrics,Yan

Shen20Let

rse

denote

HSK-robust

standard

errorstrse=(estimate-hypothesized

value)/(rse)记rse为对异方差稳健的标准差trse=(估计值-假设值)/(异方差稳健的标准差)(1

-a

)%C.I

.

=

[bˆ

j

-

c

rse,

bˆ

j

+

c

rse]The

HSK-Robust

F

statistic对异方差稳健F统计量With

HSK

the

usual

F

statistic

is

no

longer

F

distributed.在异方差下，常规F统计量不再服从F分布。The

HSK-Robust

F

statistic

is

also

called

Wald

statisticHSK－稳健F统计量也称为Wald统计量Stata

automatically

calculate

it

after

robust

regressionStata在稳健回归后自动计算Example:

use

birth.dta,

compare

the

usual

and

robustregressions:

the

usual

regressions例子：比较常规回归和稳健回归：常规回归use

birth.dta.

reg

lbwght parity

male

white

cigs

motheduc

lfamincSource

|

SS

df

MS+-Number

of

obs

=F(

6, 1184)

=11919.96Model

|2.020604016.336767335Prob

>

F=0.0000Residual

|40.04446411184.033821338R-squared=0.0480+-Total

|

42.0650681

1190

.035348797Adj

R-squared

=Root

MSE

=0.0432.18391lbwght

|Coef.Std.

Err.tP>|t|[95%

Conf.

Interval]+parity|.0174315.00613232.840.005.0054002.0294628male|.0344839.01070143.220.001.0134881.0554797white|.0463804.01507043.080.002.0168127.0759482cigs|-.0052704.001026-5.140.000-.0072834-.0032573motheduc|-.0008691.0024551-0.350.723-.0056859.0039478lfaminc|.0131714.00837081.570.116-.0032519.0295947_cons|4.659946.0377218123.530.0004.5859374.733955Intermediate

Econometrics,Yan

Shen21Example:

use

birth.dta,

compare

the

usual

and

robustregressions:

the

robust

regressions例子：比较常规回归和稳健回归：稳健回归.

reg

lbwght parity

male

white

cigs

motheduc

lfaminc,

robustRegression

with

robust

standard

errors Number

of

obs

=F(

6,Prob

>

FR-squaredRoot

MSE11911184)

=

11.17=

0.0000=

0.0480=

.18391|lbwght

|Coef.RobustStd.

Err.tP>|t|[95%

Conf.

Interval]+parity|.0174315.00589482.960.003.0058661.0289969male|.0344839.01066353.230.001.0135624.0554054white|.0463804.017912.590.010.0112416.0815193cigs|-.0052704.0010249-5.140.000-.0072812-.0032595motheduc|-.0008691.002227-0.390.696-.0052384.0035002lfaminc|.0131714.00778591.690.091-.0021043.0284471_cons|4.659946.038248121.840.0004.5849044.734987Intermediate

Econometrics,Yan

Shen22Example:

use

birth.dta,

F

statistic

for

the

usual

regression例子：应用birth.dta，常规回归的F统计量Intermediate

Econometrics,Yan

Shen23To

test

whether

the

variable

measuring

mother’s

education(motheduc)and

whether

log

family

income

(lfaminc)

jointly

have

statisticallysignificantimpacts,

just

type

in

STATA为了检验度量母亲教育水平的变量（motheduc）和对数家庭收入（lfaminc）是否联合显著，在STATA

中输入.

test motheduc

lfamincthen

STATA

returns(

1) motheduc

=

0(

2) lfaminc

=

0F(

2, 1184)

=Prob

>

F

=1.270.2807Example:

use

birth.dta,

F

statistic

for

the

robustregression例子：应用birth.dta,稳健回归的F统计量Intermediate

Econometrics,Yan

Shen24For

the

robust

regression,

the

F

statistic

is

now对于稳健回归，F

统计量为.

test motheduc

lfaminc(

1) motheduc

=

0(

2) lfaminc

=

0F(

2, 1184)

=Prob

>

F

=1.450.2361A

Robust

LM

Statistic稳健的LM统计量Intermediate

Econometrics,Yan

Shen25RunOLS

onthe

restrictedmodeland

savethe

residuals

ŭ在有限制模型下进行OLS，保存残差ŭ

Regress

each

ofthe

excluded

variables

on

allof

the

includedvariables

(q

different

regressions)

and

saveeach

setofresiduals

ř1,ř2,

…,řq将每一个排除变量对全部未排除变量进行回归（q个回归）并将每一组残差ř1,ř2,…,řq保存

Regress

avariable

definedtobe

=1

on

ř1

ŭ,ř2

ŭ,…,řq

ŭ,

withno

intercept将1向量对ř1

ŭ,ř2

ŭ,…,řq

ŭ进行无截矩回归。

The

LM

statistic

is

n

–

SSR1,

where

SSR1

is

thesum

ofsquaredresiduals

from

this

finalregressionLM定义为n

–SSR1其中SSR1

为最后一次回归的残差平方和。Intermediate

Econometrics,Yan

Shen26Example

birth.dta

:

the

LM

for

the

usual

regression

(1)例子birth.dta：常规回归的LMWe

first

estimate

the

restricted

model

and

save

the

residuals首先估计限制回归，保存残差.

reg

lbwght parity

male

white

cigsSource

|

SS

df

MS Number

of

obs

=-+----------

F(

4,Model

|Residual

|1.93640869

4

.484102172 Prob

>

F40.1376213

1187

.03381434R-squared

=-+----------Total

|

42.07403

1191

.035326641Adj

R-squared

=Root

MSE

=11921187)

=

14.32=

0.00000.04600.0428.18389lbwght|Coef.Std.

Err.tP>|t|[95%

Conf.Interval]+parity|.0172523.00610722.820.005.0052701.0292345male|.0335047.01067193.140.002.0125669.0544425white|.0515271.01470233.500.000.0226818.0803725cigs|-.0054452.0009989-5.450.000-.007405-.0034853_cons|4.688392.018168258.060.0004.6527474.724037.

predict

uhat,

residIntermediate

Econometrics,Example

birth.dta

:

the

LMfor

the

usualregression

(2)例子birth.dta:常规回归的LM.

reg uhat

parity

male

white

cigs

motheduc

lfamincSource

|

SS

df

MSNumber

of

obs=1191-+-------F(

6,

1184)=0.42Model

|

.086037667

6

.014339611Prob

>

F=0.8633Residual

|

40.0444636

1184

.033821337R-squared=0.0021-+-------Adj

R-squared=-0.0029Total

|

40.1305012

1190

.03372311Root

MSE=.18391[95%

Conf.

Interval]Now

the

LM

statistic

=

1191

*

0.0021

=

2.501uhat

|

Coef. Std.

Err.

t

P>|t|+parity|.0001792.00613230.030.977-.0118521.0122105male|.0009792.01070140.090.927-.0200166.0219749white|-.0051467.0150704-0.340.733-.0347144.024421cigs|.0001748.0010260.170.865-.0018382.0021878motheduc|-.0008691.0024551-0.350.723-.0056859.0039478lfaminc|.0131714.00837081.570.116-.0032519.0295947_cons|-.0284458.0377218-0.750.451-.1024549.0455633Yan

Shen27Intermediate

Econometrics,Examplebirth.dta:

the

robust

LM

statistic

(1)例子birth.dta:稳健LM统计量（1）First,regress

motheduc

on

all

variables

not

with

exclusion

restrictions,save

the

residual:首先，将motheduc对所有未排除变量回归，保存残差。.

reg motheduc

parity

male

white

cigsSource

|

SS

df

MSNumber

of

obs=1191+--F(

4,

1186)=18.46Model

|

407.507715

4

101.876929Prob

>

F=0.0000Residual

|

6546.85165

1186

5.520111R-squared=0.0586+--Adj

R-squared=0.0554Total

|

6954.35936

1190

5.84399946Root

MSE=2.3495motheduc

|Coef.Std.

Err.tP>|t|[95%

Conf.

Interval]+parity|-.2326565.0780469-2.980.003-.3857819-.0795311male|-.0364812.1364159-0.270.789-.3041246.2311622white|.4204476.18785952.240.025.0518737.7890215cigs|-.0969509.0127639-7.600.000-.1219931-.0719086_cons|13.33622.232187257.440.00012.8806813.79176.

save

umother,

residYan

Shen28Intermediate

Econometrics,Yan

Shen29Examplebirth.dta:

the

robust

LM

statistic

(2)例子birth.dta:稳健LM统计量（2）Second,

regress

the

log

of

family

income

on

all

variables

without

exclusion

conditions

and

save

theresiduals:然后，将lfaminc

对所有未排除变量进行回归，保存残差。.

reg

lfaminc

parity

male

white

cigsSource

|

SS

df

MSNumber

of

obs=1192-+-----F(

4,

1187)=23.64Model

|

45.0196703

4

11.2549176Prob

>

F=0.0000Residual

|

565.198789

1187

.476157362R-squared=0.0738-+-----Adj

R-squared=0.0707Total

|

610.218459

1191

.512358068Root

MSE=.69004lfaminc

|Coef.Std.

Err.tP>|t|[95%

Conf.

Interval]+parity|-.0322678.0229176-1.410.159-.0772314.0126958male|-.0855547.0400465-2.140.033-.1641246-.0069848white|.422755.05517087.660.000.3145117.5309982cigs|-.0199343.0037485-5.320.000-.0272888-.0125798_cons|3.05056.068176144.750.0002.9168013.184319.

predict

ufamin,

residExamplebirth.dta:

the

robust

LM

statistic

(3)例子birth.dta:稳健LM统计量（3）Now

we

wish

to

calculate

the

robust

LM

statistic.Why

it

doesnot

work?现在我们想计算稳健的LM统计量，为何不成功？.

gen

v1=uhat*umother.

gen

v2=uhat*ufamin(

a

column

with1s

is

addedinto

the

data

and

named

constant).

regconstant

v1

v2Source

|

SS

df

MS Number

of

obs

=

1190-+----F(2,1187)=.Model

|

0

2

0Prob>F=.Residual

|

0

1187

0-+----Total

|

0

1189

0R-squared

=Adj

R-squared

=Root

MSE

=..0constant|+Coef.Std.Err.v1|0.v2|0._cons|1.t...P>|t|[95%Conf.Interval].........Intermediate

Econometrics,Yan

Shen30Examplebirth.dta:

the

robust

LM

statistic

(4)例子birth.dta:稳健LM统计量（4）the

robust

LM

statistic.reg

constant

v1

v2,

noconstantv1

|-.0374009.0800843-0.470.641-.1945234.1197216v2

|.4501214.2670281.690.092-.0737776.9740205Source

|

SS

df

MSNumber

of

obs=1190+-F(

2,

1188)=1.46Model

|

2.91720215

2

1.45860108Prob

>

F=0.2327Residual

|

1187.0828

1188

.999227944R-squared=0.0025+-Adj

R-squared=0.0008Total

|

1190

1190

1 Root

MSE

=constant

|

Coef. Std.

Err.

t

P>|t| [95%

Conf.

Int+-.

disp

1190-1187.08282.9172

<5.99(卡方分布，自由度为2的5％显著性水平临界值).99961erval]-Intermediate

Econometrics,Yan

Shen31Chapter

Outline

本章提要Intermediate

Econometrics,Yan

Shen32Consequences

of

HSK

of

OLSOLS中异方差的影响HSK-Robust

Inference

after

OLS

estimationOLS估计后“异方差－稳健”的统计推断Testing

for

HSK检验异方差Weighted

Least

Squares

Estimation加权最小二乘估计Lecture

Outline

本课提要Intermediate

Econometrics,Yan

Shen33Testing

for

HSK

检验异方差The

Breuschn

TestB-P

检验The

WhiteTestWhite检验Weighted

Least

squares

加权最小二乘法WLS

whenHSKisknown

uptoamultiplicativeconstant当在比例意义上已知异方差时的加权最小二乘法WLS

when

HSK

is

of

unknown

form:

the

feasible

GLS当异方差具有未知形式时的加权最小二乘法：可行GLSTesting

for

HSK检验异方差Intermediate

Econometrics,Yan

Shen34Though

we

have

methods

of

computing

HSK-robust

t,

F

andLM

statistics,

there

arestillreasons

for

having

simple

tests

that

candetectthe

presence

of

heteroskedasticity.虽然我们有办法计算HSK－稳健的t,F和LM统计量，我们仍然有理由去寻找可以识别异方差的简单检验。Testing

for

HSK检验异方差Intermediate

Econometrics,Yan

Shen35Reason

No.

1:

We

may

prefer

to

see

the

usual

OLSstandard

errors

and

test

statistics

reported

unlessthere

is

evidence

of

heteroskedasticity.理由1：除非有证据显示异方差存在，我们仍会偏好于常规OLS的标准差及检验统计量。Reason

No.

2:

If

heteroskedasticity

is

present,

theOLS

estimator

is

no

longer

the

BLUE,

then

it

ispossible

to

obtain

a

better

estimator

than

OLS.理由2：如果异方差存在，OLS不再是BLUE，那么就有可能得到比OLS更好的估计量。The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差Intermediate

Econometrics,Yan

Shen36

Essentially

we

want

to

test

H0:Var(u|x1,x2,…,xk)=s2,which

is

equivalent

to

H0:E(u2|x1,x2,…,xk)=E(u2)=s2本质上，我们想检验H0:Var(u|x1,x2,…,xk)=s2

这等价于检验H0:E(u2|x1,x2,…,xk)=E(u2)=s2If

we

assume

the

relationship

between

u2

and

xj

will

belinear,

can

test

it

as

a

set

of

linear

restrictions如果我们假设u2和xj之间具有线性关系，则可以通过一组线性约束来完成检验。

So,

for

u2

=

d0

+

d1x1

+…+

dk

xk

+

v

this

means

testingH0:

d1

=

d2

=

…

=

dk

=

0所以,对于u2

=d0

+d1x1

+…+dk

xk

+v

这意味着检验H0:d1

=

d2

=

…

=

dk

=

0The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差Intermediate

Econometrics,Yan

Shen37Under

the

null

hypothesis,

it

is

often

reasonable

toassume

that

the

error

v

is

independent

of

x1

,…,

xk

.在零假设下，通常可以假定误差v与x1

,…,xk独立

Then

either

F

or

LM

statistics

for

overall

significance

ofthe

independent

variables

in

explaining

u2

can

be

usedto

test

HSK.那么，如果将u2视为被解释变量，检验全部解释变量显著性的F

或LM

统计量就可以用来检验异方差。They

are

asymptotically

valid

test

since

u2

is

notnormally

distributed

in

the

sample.由于u2在样本中不是正态分布，这些统计量只在渐近的意义下适用。The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差Intermediate

Econometrics,Yan

Shen38The

error

cannot

be

observed

by

can

beestimated

from

OLS

residuals.不可观测的误差可以通过OLS残差进行估计。

After

regressing

the

residuals

squared

on

all

ofthe

x’s,

can

use

the

R2

to

form

an

F

or

LM

test.将残差平方对所有的x回归之后，可以通过R2构造F

或LM

检验。The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差uˆIntermediate

Econometrics,Yan

Shen39uˆand

keep

the

R

squared

from

it,

R2

.Estimate

the

model

估计模型y=b0

+

b1x1

+...

+

bk

xk

+

uand

save

the

residuals,

uˆ保存残差uˆDo

the

second

regression

进行第二个回归uˆ2

=

d0

+

d1x1

+

...

+

dk

xk

+

v,保存R2

.The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差Intermediate

Econometrics,Yan

Shen40R2

/

kF

=

uˆ

,(1

-

R2

)

/(n

-

k

-1)3.Calculate

the

F

statistic

计算F统计量uˆwhich

has

an(approximately)

Fk,n-k-1

distributionunder

the

null

hypothesis.在零假设下渐近服从Fk,n-k-1分布The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差Intermediate

Econometrics,Yan

Shen41uˆUnder

the

null

hypothesis,

LM

is

distributedkcalled

the

Breusch-Pagan

test

for

HSK.uˆasymptotically

as

c2

.

The

LM

version

is

typically4.

Or,

calculate

the

LM

statistic

as

LM

=

nR2

.或者，通过LM

=

nR2计算LM统计量，在零假设下，LM渐近服从c2。LM形式的检验k通常称为Breusch

-Pagan异方差检验The

Breusch-Pagen

Test

for

HSK用B-P检验检验异方差Intermediate

Econometrics,Yan

Shen42If

we

suspect

that

HSK

depends

onlyupon

certain

regressors,

we

canmodifythe

BP

test

to

regress

residuals

from

step1

on

those

regressorsand

carry

out

theappropriate

F

or

LMtest.如果我们怀疑HSK仅依赖与某些特定的解释变量，我们可以做一些调整：将第一步的残差只对那些解释变量回归，并进行适当的F或LM检验。The

White

Test

for

HSK用White检验检验异方差Intermediate

Econometrics,Yan

Shen43

The

Breusch-Pagan

test

will

detect

any

linear

forms

ofheteroskedasticityB-P检验可以识别任意线性形式的异方差

The

White

test

allows

for

nonlinearities

by

usingsquares

and

cross

products

of

all

the

x’sWhite检验通过加入x平方项和交叉项引入了一定的非线性。

Still

just

using

an

F

or

LM

to

test

whether

all

the

xj,

xj2,and

xjxh

are

jointly

significant仍然是用F和LM检验来检验xj,xj2

,xjxh是否联合显著The

White

Test

for

HSK用White检验检验异方差Intermediate

Econometrics,Yan

Shen44This

can

get

to

be

unwieldy

pretty

quickly.这个办法很快就会显出其笨重之处。For

example,

if

we

have

three

explanatory

variables,x1

,x2

,

and

x3then

the

White

test

will

have

9

restrictions:3

on

levels,

3

on

squares,

and

3

on

cross-products.例如，如果我们有三个解释变量x1,x2,x3那么White检验有9个约束，三个对线性项，三个对平方项，三个对交叉项。With

small

samples,

degrees

of

freedom

will

soon

be

runout

with

more

regressors.在小样本情形，自由度将会随着解释变量数目增加而迅速减少。Alternate

form

of the

White

testWhite检验的变形Intermediate

Econometrics,Yan

Shen45Consider

that

the

fitted

values

from

OLS,ŷ,

are

a

function

of

all

the

x’s考虑到OLS的预测值ŷ是所有x的函数。

Thus,

ŷ2

will

be

a

function

of

the

squaresand

cross

products.

Therefore,

ŷ

and

ŷ2j

j

hcan

proxy

for

all

of

the

xj,

x

2,

andx

x

.因此，ŷ2是平方项和交叉项的函数。ŷ

和ŷ2可以用来替代所有的xj,x

2,x

xj

j

hAlternate

form

of the

White

testWhite检验的变形Intermediate

Econometrics,Yan

Shen46Regress

the

residuals

squared

on

ŷ

and

ŷ2

anduse

the

R2

to

form

an

F

or

LM

statistic,将残差平方对ŷ和ŷ2回归，用R2来构建F或LM统计量uˆ2

=

d0

+d1

yˆ

+d2

yˆ2

+

vNow

we

only

need

to

test

2

restrictions

now.现在只需要检验两个约束Example

birth.dta:

the

BP

testIntermediate

Econometrics,Yan

Shen47.The

first

step

regression:第一步回归reg

lbwght parity

male

white

cigs

motheduclfaminc(result

omitted,

same

as

before).

predict

uhat,

resid.gen

uhatsq=uhat*uhatExample

birth.dta:

the

BP

test

(1).reg uhatsq

parity

male

white

cigs

motheduclfamincSource

|

SS

df

MSNumber

of

obs=1191-+----F(

6,

1184)=0.84Model

|

.073684436

6

.012280739Prob

>

F=0.5382Residual

|

17.2902482

1184

.01460325R-squared=0.0042-+----AdjR-squared=-0.0008Total

|

17.3639327

1190

.01459154Root

MSE=.12084uhatsq

|Coef.Std.

Err.tP>|t|[95%

Conf.

Interval]+parity|-.004511.0040295-1.120.263-.0124167.0033948male|-.0035922.0070318-0.510.610-.0173885.0102041white|-.0179305.0099027-1.810.070-.0373594.0014983cigs|-.000018.0006742-0.030.979-.0013407.0013048motheduc|.0005157.00161320.320.749-.0026494.0036808lfaminc|-.0016011.0055004-0.290.771-.0123928.0091906_cons|.0564056.02478692.280.023.0077745.1050367Intermediate

Econometrics,Yan

Shen48Example

birth.dta:

the

BP