应用计量经济学原书

Chapter

8Perfect

Multicollinearity

Perfect

multicollinearity

is

a

violation

of

ClassicaAssumption

VI.

It

is

the

case

where

the

variation

in

one

explanatoryvariable

can

be

completely

explained

by

movements

inanother

explanatory

variable.

Suchacase

between

two

independent

variables

wouldbe:Where

the

X’s

are

independent

variables

in:Perfect

Multicollinearity

(continued)Other

examples

of

perfect

linear

relationships:Real

world

examples?-Distance

between

two

cities.-Percent

of

voters

voting

in

favor

and

against

aproposition.Perfect

Multicollinearity

(continued)Perfect

Multicollinearity

(continued)

OLS

is

incapable

of

generating

estimates

of

regressioncoefficients

where

perfect

multicollinearity

is

prese

You

cannot

“hold

all

the

other

independent

variablesthe

equation

constant.”A

special

case

related

to

perfect

multicollinearity

isdominant

variable.

A

dominant

variable

is

so

highly

correlated

with

thedependent

variable

that

it

masks

the

effects

of

otherindependent

variables.

Don’t

confuse

dominant

variables

with

highly

signifivariables.Imperfect

Multicollinearity

Imperfect

multicollinearity:

linear

functionalrelationship

between

two

or

more

independent

variablesso

strong

that

it

can

significantly

affect

the

estimatiof

coefficients.

It

occurs

when

two

(or

more)independent

variables

areimperfectly

linearly

related,

as

in:Note

ui,

a

stochastic

error

term

in

Equation

(8.7)Imperfect

Multicollinearity

(continued)The

Consequences

of

MulticollinearityThemajorconsequencesofmulticollinearityare:1.Estimateswillremainunbiased.2.Thevariancesandstandarderrorsoftheestimate willincrease.3.Thecomputedt-scoreswillfall.4.Estimateswill esensitivetochangesin specification.5.Theoverallfitoftheequationandestimationof coefficientsofnonmulticollinearvariableswillunaffected.The

Consequences

ofMulticollinearity

(continued)Estimates

will

remain

unbiased.Even

if

an

equation

has

significant

multicollinearitthe

estimates

of

β

will

be

unbiased

if

first

sixClassical

Assumptions

hold.The

variances

and

standard

errors

of

the

estimateswill

increase.With

multicollinearity,

it

es

difficult

to

precisidentify

the

separate

effects

of

multicollinearvariables.OLS

is

still

BLUE

with

multicollinearity.But

the

“minimum

variances”

can

be

fairly

large.The

Consequences

ofMulticollinearity

(continued)The

Consequences

ofMulticollinearity

(continued)The

computed

t-scores

will

fall.

Multicollinearity

tends

to

decrease

t-scores

mainlybecause

of

the

formula

for

the

t-statistic.If

standard

error

increases,

t-score

must

fall.

Confidence

intervals

also

increase

because

standarderrors

increase.The

Consequences

ofMulticollinearity

(continued)4.Estimateswill esensitivetochangesinspecification.Adding/droppingvariablesand/orobservationswilloftencausemajorchangesinβestimateswhensignificantmulticollinearityexists.ThisoccursbecausewithseveremulticollinearityOLSisforcedtoemphasizesmalldifferencesbetweenvariablesinordertodistinguishtheeffectonemulticollinearvariable.The

Consequences

ofMulticollinearity

(continued)The

overall

fit

of

the

equation

and

estimation

of

tcoefficients

of

nonmulticollinear

variables

will

belargely

unaffected.

will

not

fall

much,

if

at

all,

with

significantmulticollinearity.

Combination

of

high

and

no

statistically

significanvariables

is

an

indication

of

multicollinearity.

It

is

possible

for

an

F-test

of

overall

significance

tothe

null

even

though

none

of

the

individual

t-tests

do.Two

Examples

of

the

Consequencesof

MulticollinearityExample:

Student

consumption

functionwhere:COiithroom=

annual

consumption

expenditures

of

thestudent

on

items

other

than

tuition

andand

board.Ydi=

annual

disposable e

(including

gifts)

of

thstudentLAi=

liquid

assets

(savings,

etc.)

of

theithstudeεi

=

stochastic

error

termTwo

Examples

of

the

Consequencesof

Multicollinearity

(continued)Estimate

Equation

8.9

with

OLS:Including

only

disposable

e:Two

Examples

of

the

Consequencesof

Multicollinearity

(continued)Example:

Demand

for

gasoline

by

statewhere:PCONi=

petroleum

consumption

in

the

ith

state(trillions

of

BTUs)=

urban

highway

miles

within

the

ith

state=

gasoline

tax

in

the

ith

state

(cents

pergallon)=

motor

vehicle

registrations

in

the

ith

stat(thousands)UHMi

TAXiREGiTwo

Examples

of

the

Consequencesof

Multicollinearity

(continued)Estimate

Equation

8.12

with

OLS:If

you

drop

UHM:The

Detection

of

MulticollinearityMulticollinearityexistsineveryequation.Importantquestionishowmuchexists.Theseveritycanchangefromsampletosample. Therearenogenerallyaccepted,truestatisticaltestmulticollinearity. Researchersdevelopageneralfeelingfortheseveritymulticollinearitybyexamininganumberofcharacteristics.Twocommononesare:SimplecorrelationcoefficientVarianceinflationfactorsHigh

Simple

Correlation

Coefficients

The

simple

correlation

coefficient,

r,

is

a

measure

othe

strength

and

direction

of

the

linear

relationship

ovariables.Range

of

r

is

+1

to

-1.Sign

of

r

indicates

the

direction

of

the

correlation.

If

r,

in

absolute

value,

is

high,

then

the

two

variablequite

correlated

and

multicollinearity

is

a

potentialproblem.High

Simple

CorrelationCoefficients

(continued)How

high

is

high?Some

researchers

select

arbitrary

number,

such

as

0.80

Better

answer

might

be

r

is

high

if

it

causesunacceptable

large

variances.

The

use

of

r

to

detect

multicollinearity

has

amajorlimitation:

groups

of

variables

acting

together

can

camulticollinearity

without

any

single

simple

correlaticoefficient

being

high.High

Variance

Inflation

Factors

(VIFs)

Variance

inflation

factor

(VIF)

isa

method

of

detectthe

severity

of

multicollinearity

by

looking

at

the

extto

which

a

given

explanatory

variable

can

be

explainedby

all

other

explanatory

variables

in

an

equation.

Suppose

the

following

model

with

K

independentvariables:

Need

to

calculate

a

VIF

for

each

of

the

K

independentvariables.High

Variance

InflationFactors

(VIFs)

(continued)To

calculate

VIFs:Run

an

OLS

regression

that

has

Xi

as

a

function

ofall

the

other

explanatory

variables

in

the

equatio2.CalculatethevarianceinflationfactorforHigh

Variance

InflationFactors

(VIFs)

(continued)

The

higher

the

VIF,

the

more

severe

the

effects

ofmulticollinearity.But,

there

are

no

formal

critical

VIF

values.

A

common

rule

of

thumb:

if

VIF

>

5,

multicollinearity

isevere.

It’s

possible

to

have

large

multicollinearity

effecthaving

a

large

VIF.Remedies

for

MulticollinearityRemedy1:Donothing Existenceofmulticollinearitymightnotmeananythin(i.e.coefficientsstillsignificantandmeetexpectat Ifyoudeleteamulticollinearvariablethatbelongsimodel,youcausespecificationbias. Everytimearegressionisrerun,weriskencounteringspecificationthataccidentlyworksonthespecificsample.Remedies

for

Multicollinearity

(continueRemedy

2:

Drop

a

redundant

variable

Two

or

more

variables

in

an

equation

measuringessentially

the

same

thing

might

be

called

redundant.

Dropping

redundant

variable

is

nothing

more

thanmaking

up

for

a

specification

error.

In

case

of

severe

multicollinearity,

it

makes

no

statidifference

which

variable

is

dropped.

The

theoretical

underpinnings

of

model

should

be

thebasis

for

dropping

a

redundant

variable.Remedies

for

Multicollinearity

(continueExample:

Student

consumption

function:Remedies

for

Multicolli