二級數(shù)量分析知識框架圖

FrameworkSS3R9

Correlation

and

regressionR10

Multiple

regression

and

issues

in

regression

ysisR11

Time-series

ysisR12

Excerpt

from

“Probabilistic

Approaches:

Scenarioysis,

Decision

Trees,

and

Simulation”2-23R9Correlation

and

regression3-23Correlation

ysisSignificance

test(Hypothesis

test)計算H0

:

0;t

test

RejectH0

RejectH0α/2

95%

α/2t

r

n

2

,df

n

21

r2

-t

critical

+t

critical結(jié)論:例如reject

the

null

hypothesis

->the

correlationcoefficient

between

X

and

Y

is

significantly

different

from

zero.Limitation①Outliers(異常值)：few

extreme

values②

Spurious

correlation：no

economic

explanation③

Nonlinear

relationships：Correlation

onlymeasures

the

linear

relationship4-23Simple

Linear

Regression建模ANOVA

Table分析檢驗?zāi)Ｐ徒

i

b0

b1Xi

i,i

1,,

nYj

=

dependent

variable,explained

variable,predicted

variableXj

=independent

variable,explanatory

variable,predictingvariableAssumption①

A

linear

relationship

exists

between

X

and

Y②

X

is

uncorrelated

with

the

error

term.i.e.,

E(

)

0③

The

expected

value

of

the

error

term

is

zeroi④The

variance

of

the

error

term

is

constant

(homoskedastic)⑤The

error

term

is

uncorrelated

across

observations(E(εiεj)=O

for

all

i≠j)⑥The

error

term

is

normally

distributed.CoefficientEstimation解釋：An

estimated

slope

coefficient

of

2:Y

will

change

two

units

for

every1

unitchange

in

X.Intercept

term

of

2%:the

X

iszero,Y

is

2%.計算b1

Cov(

X

,Y

)Var(

X

)b0

Y

b1

X5-23ANOVATable分析dfSSMSSRegressionk=1RSSMSR=RSS/kErrorn-k-1SSEMSE=SSE(n-k-1)Totaln-1SST-CoefficientDetermination(R2)解釋：R2of

0.90

indicates

that

the

variation

of

the

independent

variableexplains

90%of

the

variation

in

the

dependent

variable.SEE計算：性質(zhì)：The

SEE

gauges

the

"fit"

of

the

regression

line.

The

smaller

thestandard

error,the

betler

the

fit.The

SEE

is

the

standard

deviation

of

the

error

terms

in

the

regression.結(jié)

論計算：R2

SSR

1

SSESST

SSTYY?R2

r

2XY（多元都成立）>一元：R2

r

2SSE6-23

MSESEE

n

k

1Simple

Linear

Regression模型檢驗：回歸分析相當(dāng)于抽樣估計考試時給定條件CoefficientStandard

deviationt-statisticp-valueInterceptb?0S

?b0？0.18Slopeb?1S

?b1？<0.001參數(shù)估計(confidenceinterval)b?

t

s

t

：查表

t分布1

c

b?

c1

Confidence

level

(置信度)As

SEE

rises,

Sb?

a|so

incmses1假設(shè)檢驗(significancelevel)H0:b1=0(沒有特殊說明，題目中假設(shè)檢驗都是檢驗是否為0)t

b?

b

b

1 1

?S

0

df=n-2b?0Decision

rule:

reject

HO

if

+t

critical<t,

or

t

<

t

criticalRejection

of

the

null

means

that

the

slope

coefficient

isdifferent

from

zeroSimple

Linear

Regression7-23（Predicted

Value

of

Y)Point

estimateY?

b?

b?

Xo

1Confidence

interval

estimateY?

(tc

s

f

)了解Simple

Linear

Regression模型

：只要求掌握抽樣估計8-23R10Multiple

regression

and

issues

in

regression

ysis9-23Difference

compared

with

unit

regressionInterpreting

thecoefficientEach

slope

coefficient

is

the

estimated

change

in

Y

for

a

one

unit

change

inXj，holding

the

other

independent

varbia?

jbles

constant.單個檢驗

H

:

b

0

t

df

n

k1Significancetest(t-test):

0

jSb?j聯(lián)合檢驗(F-test):The

test

assesses

the

effectiveness

of

the

model

as

awhole

in

explaining

the

dependent

variableH0:b1=b2=b3=...=bk=0Ha:

atleast

one

b≠j0(j=1

to

k) F

MSR

SSR

/

k

reject

H0:

if

F(test-statistic)>Fc(critical

value)

MSE

SSE

/(n

k

1)The

F-test

hereis

always

a

one-tailed

testR2解釋：R2

of

0.90

indicates

that

the

model,

as

a

whole,explains

90%of

thevariation

in

the

dependent

variable.缺點：R2

almost

always

increases

as

variable

are

added

to

the

model,

even

if

themarginal

contribution

of

the

new

variables

is

not

statistically

significant.adjusted

R2

1

1

R2

adjusted

R2

1

n

1

n

k

1SSE/n

k

1

SST/n

1R2r2YY?R2

r2XYHereoskedasticityImpactUnconditional:

no

major

problemsConditional:significant

problemsNot

affect

the

consistency

of

parameter

estimatorsCoefficient

estimates

are

not

affectedStandard

errors

are

usually

unrelliable

estimatestoo

small Type

?

errortoo

large Type

∥

errorDctectionBreush-Pagen

2

testHo:

No

hereoskedasticityBP=n×Rr

2,

df=k,

one-tailed

testesidualCorrectionrobust,

or

White-corrected

standard

eroorsgeneralized

least

squaresMultiple

Regression

Assumption

Violations11-23Multiple

Regression

Assumption

ViolationsSerial

correlation(autocorrelation)ImpactSerial

correlation

is

often

found

in

time

series

dataNot

affect

the

consistency

of

estimated

regression

coefficients

and

coefficientestimatesPositive

serial

correlation

is

much

common:

Positive

serial

correlation→

coefficient

standard

errors

that

are

too

small

→Type

?

error

&

F-test

unreliableDetectionDurbin-Watson

test

(看下圖)→Ho:No

serial

correlation,

DW≈2×(1-r)Reject

H0,

Not

reject

HInconclusive

RejectH0,positive

serial

Inconclusive

0

positive

serialcorrelation

correlation0

d1

du

2

4-du

4-d1

4Correctionadjusting

the

coefficient

standard

errors(e.g.，Hansen

method):

the

Hansenmethod

also

corrects

for

conditional

heteroskedaticity

.incorporate

thetime-seriesnature12-23Multiple

Regression

Assumption

ViolationsMulticollinearityImpactCoefficient

estimates

are

imprecise

and

unreliable;

not

affectconsistency;

inflate

standard

error->

type

∥

errorDetection①t-test都丌通過，即單個檢驗的b=0;F-test顯著;R2

is

high。以上同時出現(xiàn)，一定有Multicollinearity。②

?rx1x2?>0.7CorrectionRemove

one

or

more

independent

variables13-23Dummy

variablesModelMisspecification模型Qualitative

variable:

0

and

1n

categories->n-1

dummy

variables例:EPSt

=b0

+b1Q1t

+b2Q2t

+b3Q3t

+εtEPSt

=

a

quarterly

observation

of

earnings

pershareQ1t

=1

if

period

t

is

the

ftrst

quarter，Q1t

=0

otherwiseQ2t

=1

if

periodt

is

thesecond

quarter，Q2t

=0

otherwiseQ3t

=1

ifperiodt

is

the

third

quarter，Q3t

=0otherwiseInterpreting

thecoefficientsb0:

average

value

of

EPS

for

the

fourth

quarterSlope

coefficient:

difference

in

EPS(o age)

between

therespective

quarter

(i.e.,quarter

1,2,or

3)and

the

omitled

quarter.比如，b1=EPS1-EPS4①

The

functional

form

can

be

misspecified.Important

variables

are

omitled.Variables

should

be

transformed.Data

is

improperly

pooled.②Time

series

misspecification

.A

lagged

Y

is

used

as

an

X

with

serially

correlated

errors.A

function

ofthe

Y

is

used

as

an

X

(forecastingthe

past).Independent

variables

are

measured

with

error.③Time-series

data:

nonstationarity.14-23R11Time-Series

ysis15-23Trend

Models每期增長量是constant

amount每期的增長率是constant

rateLinear

trendLog-linear

trend

model用DWtest檢驗ε

是否有serialcorrelationNo使用trend

modelYesARModel以AR(1)開始模型的估計AR(P)

xt

b0

b1xt1

b2

xt2...

bp

xt

p

tChain

rule

of

forecastingx?t

1

b?

b?x

計算0

1

tAssumption(具體看后面)No

autocorrelationNo

Conditional

HeteroskedasticityCovariance-stationary

series檢驗是否有Seasonality(類似autocorrelation)xt

b0

b1xt1

b2

xt

4

tCompareforecasting

powersmallest

RMSE

forout-of-sample

一〉最好RMSE計算Yt

b0

b1Xt

tXt,Yt

都是time

series

dataRegression

with

More

Than

One

Time

Series具體看后面16-23AR

Model

assumption1、No

autocorrelation：針對residual

termDetectionH0

:

,0No

autocorrelationt

tkt

statistics

,

t

tk1/n

standard

error=1/

nRejectH0:

t>

+

t

critical,or

t

<

-

t

criticalCorrectionReject

Ho:(add

lagged

values)AR(1)

→AR(2)考試時給的表格17-231-0.15380.0528-2.914220.10970.05282.078230.06570.05281.244240.09200.05281.7434Autocorrelations

of

tte

ResidualLag

Autocorrelation

StandardErrort-Statistic2、No

Conditional

Heteroskedasticity：針對residual

term(用ARCH)含義Heteroskedasticity

refers

to

the

situation

that

the

variance

of

the

errorterm

is

notconstantDetectionTest

Conditional

Heteroskedasticity

=

Test

whether

a

time

series

is

ARCH(1)

2

a

a

2

ut

0 1

t

1

ta1

is

significantly

different

from

0一>Conditional

Heteroskedasticity

existCorrectionGeneralized

least

squares含義3、Covariance-stationary

series:針對Xtxt

b0

b1xt1

tCovariance-stationaryMean

reversionMean-reverting

:x

>mean->x

+1<xt

txt<mean->xt+1>xt

b

01

b1t相反Simplerandomwalk:

Xt

=Xt-1+εtRandom

walkwith

a

drift:①Constant

and

finite

expected

value

of

thetimeseries②Constant

and

finite

variance

of

the

timeseries③Constant

and

finite

covariance

withleading

or

lagged

valuesRandom

walk b1=1(undefinedmean)unit

root

nonstationary檢驗修正Define

yt

as

yt=xt-xt-1->

AR(1)

model

yt=bo+b1yt-1+utThe

unit

root

test

of

nonstationarity:

Dickey-Fuller

test(DF

test)xt

b0

b1

xt

1

xt

1

t

xt

b0

gxt

1

t

(

g

b1

1)H0:g=0

(has

a

unit

root

and

is

nonstationary);Ha:g<0Reject

H0->the

time

series

does

not

have

a

unit

root

and

is

stationarydifferencingxt

b0

b1xt

1

t

(b0

0)18-23Regression

with

More

Than

One

Time

SeriesScenarios是否可做多元回歸None

of

the

timeseries

has

a

unit

rootAt

least

onetime

series

has

a

unit

root

while

at

least

one

timeseriesdoes

not×Each

time

series

has

a

unit

root:whetherthe

time

series

are

cointegrated

?conintegrated×nocointegrationTest

the

cointegration:Dickey-Fuller

Engle-Granger

test

(DF-EG

test)Ho:

no

cointegration Ha:cointegrationIf

we

can