Multiple

Regression

ysis:Estimation(1)多元回歸分析：估計(jì)（1）y

=

0

+

1x1

+

2x2

+

.

.

.kxk

+u1Chapter

Outline

本章大綱Motivation

for

Multiple

Regression使用多元回歸的動(dòng)因Mechanics

andInterpretation

of

Ordinary

Least

Squares普通最小二乘法的操作和解釋The

Expected

Values

of

the

OLS

EstimatorsOLS估計(jì)量的期望值The

Variance

of

the

OLSEstimatorsOLS估計(jì)量的方差Efficiency

of

OLS:

The

Gauss-Markov

TheoremOLS的有效性：

－

定理2Lecture

Outline

課堂大綱Motivation

for

multivariate

ysis

使用多元回歸的動(dòng)因The

Model

模型The

Estimation

估計(jì)Propertiesof

the

OLS

estimates

OLS估計(jì)的性質(zhì)The

Partialling

out

Interpretation

對(duì)“排除其它變量影響”的解釋Simple

versus

multiple

regressions

比較簡(jiǎn)單回歸模型與多元回歸模型Goodness

of

Fit

擬合優(yōu)度3Motivation:

Advantage動(dòng)因：優(yōu)點(diǎn)The

primary

drawback

of

the

simple

regression ysis

forempirical

workis

that

it

is

very

difficult

to

draw

ceteris

paribusconclusions

about

how

x

affects

y.在實(shí)證工作中使用簡(jiǎn)單回歸模型的主要缺陷是：要得到在其它條件不變的情況下，x對(duì)y的影響非常。Whether

the

ceteris

paribus

effects

are

reliable

or

not

depends

onwhether

the

conditional

mean

assumption

is

realistic.在其它條件不變情況假定下 估計(jì)出的x對(duì)y的影響值是否

依賴，完全取決于條件均值零值假設(shè)是否現(xiàn)實(shí)。If

other

factors

that

affecting

y

are

not

correlated

with

x,

changingx

can

ensure

thatu

is

not

changed,and

the

effect

of

x

ony

can

beidentified.如果影響y的其它因素與x不相關(guān)，則改變x可以保證u不變，從而x對(duì)y的影響可以被識(shí)別出來(lái)。4Motivation

:

Advantage動(dòng)因：優(yōu)點(diǎn)Multiple

regression ysis

is

more

amenable

to

ceteris

paribusysis

because

it

allows

us

to

explicitly

control

for

many

otherfactors

that

simultaneously

affect

the

dependent

variable.多元回歸分析更適合于其它條件不變情況下的分析，因?yàn)槎嘣貧w分析允許

明確地控制許多其它也同時(shí)影響因變量的因素。Multiple

regression

models

can modate

manyexplanatoryvariables

that

may

be

correlated.多元回歸模型能容許很多解釋變量，而這些變量可以是相關(guān)的。Important

for

drawing

inference

about

causal

relations

betweeny

and

explanatory

variables

when

using

non-experimentaldata.在使用非實(shí)驗(yàn)數(shù)據(jù)時(shí)，多元回歸模型對(duì)推斷y與解釋變量間的因果關(guān)系很重要。5Motivation

:

Advantage動(dòng)因：優(yōu)點(diǎn)It

can

explain

more

of

the

variation

in

thedependent

variable.它可以解釋

的因變量變動(dòng)。It

can

incorporate

more

general

functional

form.它可以表現(xiàn)更一般的函數(shù)形式。The

multiple

regression

model

is

the

most

widelyused

vehicle

for

empirical

ysis.多元回歸模型是實(shí)證分析中最廣泛使用的工具。6Motivation:

AnExample動(dòng)因：一個(gè)例子7Consider

a

simple

version

of

the

wage

equation

forobtaining

the

effect

of

education

on

hourly

wage:考慮一個(gè)簡(jiǎn)單版本的解釋教育對(duì)小時(shí)工資影響的工資方程。exper:

years

of

labor

marketexperienceexper：在勞動(dòng)力市場(chǎng)上的經(jīng)歷，用年衡量wage

0

1educ

2

exp

er

uIn

this

example

experience

is

explicitly

taken

out

ofthe

error

term.在這個(gè)例子中，“在勞動(dòng)力市場(chǎng)上的經(jīng)歷”被明確地從誤差項(xiàng)中提出。Motivation:

AnExample動(dòng)因：一個(gè)例子8Consider

a

model

that

says

family

consumptionis

a

quadratic

function

of

family

e:考慮一個(gè)模型：家庭消費(fèi)是家庭收入的二次方程。Cons

=

0

+

1

inc+2

inc2

+uNow

the

marginal

propensity

to

consume

isapproximated

by現(xiàn)在，邊際消費(fèi)傾向可以近似為MPC=

1

+22The

Model

with

kIndependentVariables含有k個(gè)自變量的模型T eral

multiple

linearregression

model

can

be

written

as一般的多元線性回歸模型可以寫為y

0

1x1

2

x2

k

xk

u9Parallels

with

Simple

Regression類似于簡(jiǎn)單回歸模型0

is

still

the

intercept

0仍是截距1

to

k

all

called

slope

parameters1到k都稱為斜率參數(shù)u

is

still

the

error

term(or

disturbance)

u仍是誤差項(xiàng)（或干擾項(xiàng)）Still

need

to

make

a

zero

conditional

mean

assumption,

so

nowassume

that

仍需作零條件期望的假設(shè)，所以現(xiàn)在假設(shè)E(u|x1,x2,

…,xk)

=

0Still

minimizing

the

sum

of

squaredresiduals,

so

have

k+1order

conditions

仍然最小化殘差平方和，所以得到k+1個(gè)一階條件10Obtaining

the

OLS

Estimates11如何得到OLS估計(jì)值The

method

of

ordinary

least

squareschooses

the

estimates

to

minimizethe

sum

of

squared

residuals,普通最小二乘法選擇能最小化殘差平方和的估計(jì)值，01

i1ni(

y

i

1Obtaining

the

OLS

Estimates如何得到OLS估計(jì)值The

k+1

order

conditions

arek+1

個(gè)一階條件是i

2ikni1ni1ni1ni1xi1

(

yi

?

?

x

?

x0

1

i1

k

ikx

(

yi

?

?

x

?

x0

1

i1

k

ik(

yi

?

?

x

?

x

)

00

1

i1

k

ik)

0)

0x

(

yi

?

?

x

?

x

)

00

1

i1

k

ik...12Obtaining

the

OLS

Estimates如何得到OLS估計(jì)值The order

conditions

are

also

the

samplecounterparts

of

the

related

population

moments.一階條件也是相關(guān)的總體矩在樣本中的對(duì)應(yīng)。After

estimationwe

obtain

the

OLS

regressionline,

or

the

sample

regression

function

(SRF)得到OLS回歸線，或稱為樣本回歸k

ik101i

...

?xx

??在估計(jì)之后，方程（SRF）?i

13Interpreting

Multiple

Regression對(duì)多元回歸的解釋141

1

2

2

k

ky?

?

?

x

?

x

...

?

x

,

so0 1

1

2

2

k

ky?

?

x

?

x

...

?

x

,y?

?

x

,

that

is

each

hasso

holding

x2

,...,

xk

fixed

implies

that所以，保持

x2

,...,xk

不變意味著1

1a

ceteris

paribus

interpretation即，每一個(gè)

都有一個(gè)局部效應(yīng)，或其它情況不變效應(yīng),的解釋Example:Determinants

of

College

GPA例子：大學(xué)GPA的決定因素Two-independent-variable

regression兩個(gè)解釋變量的回歸pcolGPA:

predicted

values

of

college

grade

point

averagepcolGPA:大學(xué)績(jī)點(diǎn)

值hsGPAhsGPAACTACT:

high

school

GPA:高中績(jī)點(diǎn):

achievement

test

score:成績(jī)測(cè)驗(yàn)分?jǐn)?shù)pcolGPA

=

1.29

+

0.453hsGPA+0.0094ACT15Example:Determinants

of

College

GPA例子：大學(xué)GPA的決定因素16One-independent-variable

regression一個(gè)解釋變量的回歸pcolGPA

=

2.4

+0.0271ACTThe

coefficients

on

ACT

is

three

times

larger.ACT的系數(shù)大三倍。If

these

two

regressions

were

both

true,

they

can

beconsidered

as

the

results

of

two

differentexperiments.如果這兩個(gè)回歸都是對(duì)的，它們可以被認(rèn)為是兩個(gè)不同實(shí)驗(yàn)的結(jié)果。Holding

other

factors

fixed“保持其它因素不變”的含義The

power

of

multiple

regression ysis

isthat

it

allowsus

to n

non-experimentalenvironments

what

natural

scientists

are

able

ton

a

controlled

laboratory

setting:

keep

otherfactors

fixed.多元回歸分析的優(yōu)勢(shì)在于它使能在非實(shí)驗(yàn)環(huán)境中去做自然科學(xué)家在受控實(shí)驗(yàn)中所能做的事情：保持其它因素不變。17Properties

性質(zhì)The

sample

average

of

the

residuals

is

zero.殘差項(xiàng)的樣本平均值為零The

sample

covariance

between

each

independentvariable

and

the

OSL

residuals

is

zero.每個(gè)自變量和OLS協(xié)殘差之間的樣本協(xié)方差為零。The

point

(x1,

x2

, ,

xk

,

y)

isalways

on

the

OLSregression

line.點(diǎn)(x1,

x2

, ,

xk

,

y)

總位于OLS回歸線上。18A

“Partialling

Out”

Interpretation19對(duì)“排除其它變量影響”的解釋Consider

regression

line

of

考慮回歸線1One

way

to

express

?

is0

1

1

2

2??iy

?

x

?

x1??i1

iri1

i

(1?

的一種表達(dá)是r?i1is

obtained

in

the

following

way:r?i1

由以下方式得出：A

“Partialling

Out”

Interpretation對(duì)“排除其它變量影響”的解釋In

other

words, is

the

residual

from

the

regression然后，將y向

進(jìn)行簡(jiǎn)單回歸得到。r11obtain

?

.1r1?Regress

our independent

variable

x1

on

oursecond

independentvariable

x2

,and

then

obtainthe

residualr1

.將第一個(gè)自變量對(duì)第二個(gè)自變量進(jìn)行回歸，然后得到殘差r1

。x?1

?0

?1x?2換句話說(shuō)，r1

是由回歸

x?1

?0

?1x?2得到的殘差。Then,

do

a

simple

regression

of

y

on

r1

to20“Partialling

Out”

continued“排除其它變量影響”（續(xù)）

Previous

equation

implies

that

regressingy

on

x1and

x2

gives

same

effect

of

x1

as

regressing

y

onresiduals

from

a

regression

of

x1

on

x2上述方程意味著：將y同時(shí)對(duì)x1和x2回歸得出的x1的影響與先將x1對(duì)x2回歸得到殘差，再將y對(duì)此殘差回歸得到的x1的影響相同。

This

meansonly

the

part

of

x1

that

is

uncorrelatedwith

x2

are

being

related

to

y

,

so

we’re

estimatingthe

effect

ofx1

on

y

after

x2

has

been

“partialled

out”這意味著只有x1中與x2不相關(guān)的部分與y有關(guān)，所以在x2被“排除影響”之后，

再估計(jì)x1對(duì)y的影響。21“Partialling

Out”

continued“排除其它變量影響”（續(xù)）In

t eral

model

with

k

explanatoryequationcomes

from

the

regression

of

x1

on

x2…

,

xk.的回歸。Thusmeasures

the

effect

of

x1

on

y

afterx2,…

,

xk.has

been

partialled

out.x1對(duì)y的影響。?i1

ir?i1

i

(1

,

but

the

residual

r11variables,

?

can

still

be

written

asin1?1在一個(gè)含有k個(gè)解釋變量的一般模型中，?

仍然可以?i1

ir?i1

i

(1

1寫成

，但殘差

r

來(lái)自x1對(duì)x2…

,xk1于是?

度量的是，在排除x2…,xk等變量的影響之后，22Simple

vs

Multiple

Regression

Estimates比較簡(jiǎn)單回歸和多元回歸估計(jì)值11Generally,

1

?

unless:Compare

the

simple

regression

y

0

1

x1比較簡(jiǎn)單回歸y

0

1

x1with

the

multiple

regression

y?

?

?

x

?

x0

1

1

2

2與多元回歸

y?

?

?

x

?

x0

1

1

2

223一般來(lái)說(shuō)，1

?

，除非：

0

(i.e.

no

partial

effect

of

x2

)

OR?2?2

（0

也就是x2對(duì)y沒(méi)有局部效應(yīng)），或x1

and

x2

are

uncorrelated

in

the

sample在樣本中x1和x2不相關(guān)Simple

vs

Multiple

Regression

Estimates比較簡(jiǎn)單回歸和多元回歸估計(jì)值24regression

of

x2

on

x1

. The

proof.This

is

because

there

existsa

simple

relationship這是因?yàn)榇嬖谝粋€(gè)簡(jiǎn)單的關(guān)系~

?

?

~1

1

2

1~where

1

is

theslope

coefficient

from

thesimple這里，1是x2對(duì)x1的簡(jiǎn)單回歸得到的斜率系數(shù)。證明如下。~1

125211

11

111

11121

1

1

2

20

1

1

2

2?~

?

?

~1

2

1

?

?

(x

x

)2(x1

x1

)(x2

x2

)(x

x

)2

x2

)]

x

)[

(x

x1

)

?2

(x2

(x1(x

x

)2(x

x1

)(

y

y)

x

),

thereforey

y

?

(x

x

)

?

(x

u?

so

thatBecause

y

?

?

x

?

xSimple

vs

Multiple

Regression

Estimates簡(jiǎn)單回歸和多元回歸估計(jì)值的比較Let

β?j

,

j

0,1,...,

k

be

the

OLS

estimators

from

theregression

using

full

set

of

explanatory

variables.令β?j

,j

0,1,...,k為用全部解釋變量回歸的OLS估計(jì)量。

Let

βj

,j

0,1,...,k

1be

the

OLS

estimators

fromthe

regression

that

leaves

out

xk

.令βj

,j

0,1,...,k－1為用除xk

外的解釋變量回歸的OLS估計(jì)量。

Let

δj

be

the

slope

coefficient

on

xj

in

the

regressionof

xk

on

x1

,...,

xk-1.Then令δj為xk向x1

,...,xk-1回歸中x

j的斜率系數(shù)。那么βj

β?j

β?k

δj

.26Simple

vs

Multiple

Regression

Estimates簡(jiǎn)單回歸和多元回歸估計(jì)值的比較27In

the

case

with

k

independentvariables,

thesimple

regression and

the

multiple

regressionproduce

identical

estimate

for

x1

only

if在k個(gè)自變量的情況下，簡(jiǎn)單回歸和多元回歸只有在以下條件下才能得到對(duì)x1相同的估計(jì)(1)

the

OLS

coefficients

on

x2

through

xk

are

allzero,or對(duì)從x2到xk的OLS系數(shù)都為零，或(2)

x1

isuncorrelated

with

eachof

x2…

,

xk.（2）

x1與x2…,xk中的每一個(gè)都不相關(guān)。Summary

總結(jié)In

this

lecture

we

introduce

the

multiple

regression.在本次課中，

介紹了多元回歸。Important

concepts:重要概念：Interpreting

the

meaning

of

OLS

estimates

in

multipleregression解釋多元回歸中OLS估計(jì)值的意義Partialling

effect局部效應(yīng)（其它情況不變效應(yīng)）Properties

of

OLSOLS的性質(zhì)When

will

the

estimates

from

simple

and

multipleregression

to

be

identical什么時(shí)候簡(jiǎn)單回歸和多元回歸的估計(jì)值相同2829Multiple

Regression ysis:

Estimation(2)多元回歸分析：估計(jì)（2）y

=

0

+

1x1

+

2x2

+

.

.

.kxk

+

uChapter

Outline

本章大綱Motivation

for

Multiple

Regression使用多元回歸的動(dòng)因Mechanics

and

Interpretation

of

Ordinary

Least

Squares普通最小二乘法的操作和解釋The

Expected

Values

of

the

OLS

EstimatorsOLS估計(jì)量的期望值The

Variance

of

the

OLS

EstimatorsOLS估計(jì)量的方差Efficiency

of

OLS:

The

Gauss-MarkovTheoremOLS的有效性：

－

定理30Lecture

Outline

課堂大綱31The

MLR.1–

MLR.4

Assumptions假定MLR.1–MLR.4The

Unbiasedness

of

the

OLS

estimatesOLS估計(jì)值的無(wú)偏性O(shè)ver

or

Under

specification

of

models模型設(shè)定不足或過(guò)度設(shè)定Omitted

VariableBias遺漏變量的偏誤Sampling

Variance

of

the

OLS

slope

estimatesOLS斜率估計(jì)量的抽樣方差The

expected

value

of

the

OLS

estimatorsOLS估計(jì)量的期望值We

now

turn

to

the

statistical

propertiesof

OLSforestimating

the

parameters

in

an

underlyingpopulation

model.現(xiàn)在轉(zhuǎn)向OLS的統(tǒng)計(jì)特性，而 知道OLS是估計(jì)潛在的總體模型參數(shù)的。Statistical

properties

are

the

properties

ofestimators

when

random

sampling

is

donerepeatedly.

We

do

not

care

about

how

an

estimatordoes

in

a

specific

sample.統(tǒng)計(jì)性質(zhì)是估計(jì)量在隨機(jī)抽樣不斷重復(fù)時(shí)的性質(zhì)。并不關(guān)心在某一特定樣本中估計(jì)量如何。32Assumption

MLR.1

(Linear

in

Parameters)假定MLR.1（對(duì)參數(shù)而言為線性）33In

the

population

model

(or

the

true

model),

thedependent

variable

y

is

related

to

the

independentvariable

x

and

the

error

u

as在總體模型(或稱真實(shí)模型）中，因變量y與自變量x和誤差項(xiàng)u關(guān)系如下y=

0+

1x1+

2x2+

…+kxk+uwhere

1,

2

…,

k

are

the

unknown

parametersof

interest,and

u

is

an

unobservable

random

error

or

randomdisturbance

term.其中，1,2

…,k

為所關(guān)心的未知參數(shù)，u為不可觀測(cè)的隨機(jī)誤差項(xiàng)或隨機(jī)干擾項(xiàng)。Assumption

MLR.2

(Random

Sampling)假定MLR.2（隨機(jī)抽樣性）We

can

use

a

random

sampleof

size

n

from

thepopulation,{(xi1,可以使用總體的一個(gè)容量為n的隨機(jī)樣本xi2…,

xik;

yi):

i=1,…,n}，wherei

denotesobservation,and

j=

1,…,k

denotesthe

jth

regressor.其中i

代表觀察，j=1,…,k代表第j個(gè)回歸元Sometimes

we

write

有時(shí) 模型寫為yi=

0+

1xi1+

2xi2+

…+kxik+ui34Assumptions

MLR.3

假定MLR.3MLR.3(Zero

Conditional

Mean)

（零條件均值）

:E(u|

xi1,

xi2…,xik)=0.When

this

assumption

holds,

we

say

all

of

theexplanatory

variables

are

exogenous;

when

it

fails,

wesay

that

the

explanatory

variables

are

endogenous.當(dāng)該假定成立時(shí)，稱所有解釋變量均為外生的；否則，則稱解釋變量為內(nèi)生的。We

will

pay

particular

attention

to

the

case

thatassumption

3

fails

because

of

omitted

variables.特別注意當(dāng)重要變量缺省時(shí)導(dǎo)致假定3不成立的情況。35Assumption

MLR.4

假定MLR.4MLR.4(No

perfect

collinearity)

（不存在完全共線性）

:In

the

sample,none

of

the

independent

variables

is

constant,

and

there

are

noexactlinearrelationshipsamongtheindependentvariables.在樣本中，沒(méi)有一個(gè)自變量是常數(shù)，自變量之間也不存在嚴(yán)格的線性關(guān)系。When

one

regressor

is

an

exact

linear

combination

of

the

other

regressor(s),wesaythemodelsuffersfromperfectcollinearity.當(dāng)一個(gè)自變量是其它解釋變量的嚴(yán)格線性組合時(shí)，說(shuō)此模型有嚴(yán)格共線性。Examples

of

perfect

collinearity：完全共線性的例子：y=0+

1x1+

2x2+

3x3+u，

x2

=

3x3，y=

0+

1log(inc)+

2log(inc2

)+uy=

0+

1x1+

2x2+

3x3+

4x4

u，x1+x2

+x3+

x4

=1.Perfect

collinearity

also

happenswhen

y=0+1x1+2x2+3x3+u,n<(k+1).當(dāng)y=0+1x1+2x2+3x3+u,n<(k+1)也發(fā)生完全共線性的情況。The

denominator

of

the

OLS

estimator

is

0

when

there

is

perfect

collinearity,hence

the

OLS

estimator

cannot

be

performed.You

can

check

this

by

looking

atthe

formula

of

the

estimator

for

2

in

the

session

discussing

the

partialling-outeffect.在完全共線性情況下，OLS估計(jì)量的分母為零，因此OLS估計(jì)量不能得到。你可以回顧“排除其它變量影響”部分中的2估計(jì)量的式子，來(lái)檢驗(yàn)這一點(diǎn)。36Theorem

3.1

(Unbiasedness

of

OLS)37定理3.1（OLS的無(wú)偏性）Under

assumptions

MLR.1

throughMLR.4,

the

OLS

estimators

areunbiased

estimator

of

thepopulation

parameters,

that

is在假定MLR.1～MLR.4下，OLS估計(jì)量是總體參數(shù)的無(wú)偏估計(jì)量，即E(

j

)

j

,

j

1,2,...,kTheorem

3.1

(Unbiasedness

of

OLS)定理3.1（OLS的無(wú)偏性）Unbiasedness

is

the

property

of

an

estimator,thatis,

the

procedure

that

can

produce

an

estimate

fora

specific

sample,

not

an

estimate.無(wú)偏性是估計(jì)量的特性，而不是估計(jì)值的特性。估計(jì)量是（過(guò)程），該方法使得給定一個(gè)樣本，法可以得到一組估計(jì)值。 評(píng)價(jià)的是方法的優(yōu)劣。Not

correct

to

say“5

percent

is

anunbiasedestimate

of

the

return

of

education”.不正確的說(shuō)法:“5%是教育匯報(bào)率的無(wú)偏估計(jì)值?！?8Too

Many

or

TooFew

Variables變量太多還是太少了？What

happens

if

we

include

variables

in

our

specificationthat

don’t

belong?如果

在設(shè)定中包含了不屬于真實(shí)模型的變量會(huì)怎樣？A

model

is

overspecifed

when

one

or

more

of

theindependent

variablesis

included

in

the

model

even

thoughit

has

no

partial

effect

on

y

in

the

population盡管一個(gè)（或多個(gè)）自變量在總體中對(duì)y沒(méi)有局部效應(yīng)，但卻被放到了模型中，則此模型被過(guò)度設(shè)定。

There

is

no

effect

on

our

parameter

estimate,

and

OLSremains

unbiased.

But

it

can

have

undesirable

effects

on

thevariances

of

the

OLS

estimators.過(guò)度設(shè)定對(duì)

的參數(shù)估計(jì)沒(méi)有影響，OLS仍然是無(wú)偏的。但它對(duì)OLS估計(jì)量的方差有不利影響。39Too

Many

or

TooFew

Variables變量太多還是太少了？What

if

we

exclude

a

variable

from

ourspecification

that

doesbelong?如果 在設(shè)定中排除了一個(gè)本屬于真實(shí)模型的變量會(huì)如何？If

a

variable

th tually

belongs

in

the

true

model

is

omitted,

wesay

the

modelis

underspecified.如果一個(gè)實(shí)際上屬于真實(shí)模型的變量被遺漏，說(shuō)此模型設(shè)定不足。OLS

will

usually

be

biased.此時(shí)OLS通常有偏。Deriving

the

bias

caused

by

omitting

animportant

variable

isanexample

ofmisspecification

ysis.推導(dǎo)由遺漏重要變量所造成的偏誤，是模型設(shè)定分析的一個(gè)例子。40Omitted

Variable

Bias遺漏變量的偏誤

121i11

ii1x

x

y

x

xSuppose

the

true

model

is

given

as假定真實(shí)模型如下y

0

1

x1

2

x2

u,but

we

estimate

y

0

1

x1

u,

then但

估計(jì)的是

y

0

1

x1

u，有41Omitted

Variable

Bias

(cont)遺漏變量的偏誤（續(xù)）24221

i1

1

1

xi1

x1

0

x

x回想真實(shí)模型，有

yi

0

1xi1

2

xso

the

numerator

be所以分子為Omitted

Variable

Bias

(cont)遺漏變量的偏誤（續(xù)）

1222111121i1i1i11

i

2i1E

x

x

x

x

xx

xx

x

x

x

i1 1

i

2

xi1

x1

ui

2

x

since

E(ui

)

0,taking

expectations

we

have由于E(ui

)

0，取期望, 得到43

12i11

i

2i1

1x

x

xx2

0

1

x1

then

x

xso

E

1

1

21Omitted

Variable

Bias

(cont)遺漏變量的偏誤（續(xù)）Consider

the

regression

of

x2

on

x1考慮x2對(duì)x1的回歸44Omitted

Variable

Bias

Summary遺漏變量的偏誤 總結(jié)45Two

cases

where

biasisequal

to

zero

兩種偏誤為零的情形2

=

0,

that

isx2

doesn’t

really

belongin

model2

=0，也就是，x2實(shí)際上不屬于模型x1

and

x2

are

uncorrelated

inthe

sample樣本中x1與x2不相關(guān)

If

correlation

between

x2

,x1

and

x2

,y

isthe

same

direction,

bias

will

be

positive

如果x2與x1間相關(guān)性和x2與y間相關(guān)性同方向，偏誤為正。

If

correlation

between

x2

,x1

andx2

,y

is

theopposite

direction,

bias

will

be

negative

如果x2與x1間相關(guān)性和x2與y間相關(guān)性反方向，偏誤為負(fù)。Omitted

Variable

Bias

Summary遺漏變量的偏誤 總結(jié)When

E(1

)

1,we

say

that

1

hasupwardbias.當(dāng)E(1

)

1，

說(shuō)1上偏。When

E(1

)

1,

we

say

that1

hasdownwardbias.當(dāng)E(1

)

1，

說(shuō)1下偏。46Summaryof

Direction

ofBias偏誤方向總結(jié)47Corr(x1,

x2)

>

0Corr(x1,

x2)

<

02

>

0Positive

bias偏誤為正Negative

bias偏誤為負(fù)2

<

0Negative

bias偏誤為負(fù)Positive

bias偏誤為正Omitted-Variable

Bias

遺漏變量偏誤In

general

,

2

is

unknown;

and

when

a

variable

isomitted,

it

is

mainly

because

of

this

variable

isunobserved.

In

other

words,

we

do

not

know

thesign

of

Corr(x1,

x2).

Whatto

do?

但是，通常

不能觀測(cè)到2，而且，當(dāng)一個(gè)重要變量被缺省時(shí)，主要原因也是因?yàn)樵撟兞繜o(wú)法觀測(cè)，換句話說(shuō)，無(wú)法準(zhǔn)確知道Corr(x1,x2)的符號(hào)。怎么辦呢？We

rely

on

economic

theories

and

intuition

tomake

a

educated

guess

ofthesign.

依靠經(jīng)濟(jì)理論和 來(lái)幫助 對(duì)相應(yīng)符號(hào)做出較好的估計(jì)。48Example:

hourly

wage

equation例子：小時(shí)工資方程Suppose

the

model

log(wage)

=

0+1educ+

2abil

+uis

estimated

with

abil

omitted.

What

is

the

direction

of

biasfor

1?假定模型

log(wage)=0+1educ+2abil+u，在估計(jì)時(shí)遺漏了abil。1的偏誤方向如何？

Since

in

general

ability

has

positive

partial

effect

on

yandability

and

education

years

is

positive

corrected,

we

expect1

to

have

a

upwardbias.因?yàn)橐话銇?lái)說(shuō)ability對(duì)y有正的局部效應(yīng)，并且ability和education

years正相關(guān)，所以預(yù)期1上偏。49The

More

General

Case更一般的情形Technically,

it

is

more

difficult

to

derive

the

signof

omitted

variable

bias

with

multiple

regressors.從技術(shù)上講，要推出多元回歸下缺省一個(gè)變量時(shí)各個(gè)變量的偏誤方向更加

。

But

remember

that

if

an

omitted

variable

haspartial

effects

on

y

and

it

is

correlated

with

atleast

one

of

the

regressors,

then

the

OLSestimators

of

all

coefficients

will

be

biased.需要記住，若有一個(gè)對(duì)y有局部效應(yīng)的變量被缺省，且該變量至少和一個(gè)解釋變量相關(guān)，那么所有系數(shù)的OLS估計(jì)量都有偏。50The

More

General

Case更一般的情形ymodel2ytruey?model1

0

Suppose

corr(x1

,

x3It

is

notdifficult

to

beliestimator

of

2

.Will

1

be

un若corr(x1,x3

)

0,corr(x2

,x3

)

0很容易想到2是2的一個(gè)有偏估計(jì)量而1是有偏的嗎？51The

More

General

Case更一般的情形1

1 3

1

2

2?

??

,

1When

corr(x1

,

x2

)

0corr(x1

,

x3

)

0.There當(dāng)corr(x1

,x2

)

0，即

is

a

biasedestimato52—

有偏估計(jì)Variance

of

the

OLS

EstimatorsOLS估計(jì)量的方差Now

we

know

that

the

sampling

distribution

of

ourestimate

iscentered

around

the

true

parameter。現(xiàn)在

知道估計(jì)值的樣本分布是以真實(shí)參數(shù)為中心的。Want

to

think

about

how

spreadout

this

distribution

is還想知道這一分布的分散狀況。

Much

easier

to

think

about

this

variance

underanadditional

assumption,

so在一個(gè)新增假設(shè)下，度量這個(gè)方差就容易多了，有：53Assumption

MLR.5

(Homoskedasticity)假定MLR.5（同方差性）Assume

Homoskedasticity:同方差性假定：Var(u|x1,

x2,…,

xk)

=

2

.Means

that

the

variancein

the

errorterm,

u,conditionalontheexplanatorcombinations

ofbles,

is

the

same

for

alles

of

explanatory

variables.意思是，不管解釋變量出現(xiàn)怎樣的組合，誤差項(xiàng)u的條件方差都是一樣的。If

the

assumption

fails,

we

say

the

model

exhibitsheteroskedasticity.如果這個(gè)假定不成立， 說(shuō)模型存在異方差性。54Variance

of

OLS

(cont)OLS估計(jì)量的方差（續(xù)）Let

x

standfor(x1,x2,…xk)

用x表示(x1,x2,…xk)

Assuming

that

Var(u|x)=2

also

implies

thatVar(y|

x)=2

假定Var(u|x)=2，也就意味著

Var(y|

x)=2

Assumption

MLR.1-5

are

collectively

known

asthe

Gauss-Markov

assumptions.假定MLR.1-5共同被稱為

－

假定55

22?jj2jjjjij

jand

R2

is

the

R2from

regressing

xj

on

all

other

x's

2Var

,

whereSST

1

RSSTj

xij

xTheorem

3.2

(Sampling

Variances

of

the

OLS

SlopeEstimators)定理3.2（OLS斜率估計(jì)量的抽樣方差）Given

the

Gauss-Markov

Assumptions給定

－

假定其中，SST

x

x

，R2是x

向所有其它x回歸所得到的R2j

j56Interpreting

Theorem

3.2對(duì)定理3.2的解釋57

Theorem

3.2

shows

that

the

variances

of

theestimated

slope

coefficients

are

influenced

by

threefactors:定理3.2顯示：估計(jì)斜率系數(shù)的方差受到三個(gè)因素的影響：The

error

variance誤差項(xiàng)的方差The

total

sample

variation總的樣本變異Linear

relationships

among

the

independent

variables解釋變量之間的線性相關(guān)關(guān)系Interpreting

Theorem

3.2:

The

Error

Variance對(duì)定理3.2的解釋（1）：誤差項(xiàng)方差A(yù)

larger

2

implies

a

larger

variance

forthe

OLSestimators.更大的2意味著更大的OLS估計(jì)量方差。A

larger

2

means

more

noises

in

the

equation.更大的2意味著方程中的“噪音”越多。This

makes

it

more

difficult

toextract

theexact

partial

effectof

the

regressor

on

the

regressand.

這使得得到自變量對(duì)因變量的準(zhǔn)確局部效應(yīng)變得更加。Introducing

more

regressors

can

reduce

the

variance.

Butoften

this

is

not

possible,

neither

is

it

desirable.

引入 的解釋變量可以減小方差。但這樣做不僅不一定可能，而且也不一定總令人滿意。2

does

not

depends

on

sample

size.

2

不依賴于樣本大小58Interpreting

Theorem

3.2:

The

total

sample

variation對(duì)定理3.2的解釋（2）：總的樣本變異A

larger

SSTj

implies

a

smaller

variance

for

the

estimators,andvice

versa.更大的SSTj意味著更小的估計(jì)量方差，反之亦然。Everything

else

being

equal,more

sample

variation

in

x

is

always

preferred.其它條件不變情況下，x的樣本方差越大越好。One

way

to

gain

more

sample

variation

is

to

increase

thesample

size.

增加樣本方差的

法是增加樣本容量。This

components

of

parameter

variance

depends

on

thesample

size.

參數(shù)方差的這一組成部分依賴于樣本容量。59Interpreting

Theorem

3.2:

multicollinearity對(duì)定理3.1的解釋（3）：多重共線性60A

larger

R

2

implies

a

larger

variance

for

the

estimatorsjj更大的R

2意味著更大的估計(jì)量方差。jA

large

R

2

meansother

regressors

can

explain

much

of

the2variationsin

xj.如果Rj

較大，就說(shuō)明其它解釋變量解釋可以解釋較大部分的該變量。j

jWhen

R

2

is

very

close

to

1,

x

is

highly

correlated

with

other2regressors,

this

is

called

multicollinearity.

當(dāng)Rj

非常接近1時(shí)，xj與其它解釋變量高度相關(guān)，被稱為多重共線性。Severe

multicollinearity

means

the

variance