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Lecture11

SimpleRegression

Statisticsfor

Manag

ement

Decisions

#

Regres

sion

Regressionanalysisenables

ustoestimate

thestrengthanddirectionof

relations

betweenvariables

Spec

ificallybetweendependent(Y)and

independentvariables(x1,x2,

etc.)

Forexample:

Theeffectofyearsofeducatio

nonincome

Theeffectofenginesizeonegasmileage

Theeffectofhousesizeonprice

First

Covarian

ceandCorrelation

(toseew

hetherarelationship

EXISTS)

Statisticsfor

Management

Decisions

#

Example

Considerthefollo

wingexamplecompar

ing

thereturnsofConsolidatedMoosePasture

stock(CMP)and

theTSX300Index

Thenextslideshows25monthlyreturns

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Statisticsfor

Manag

ement

Decisions

#

Examp

leData

TSX

CMP

TSX

CMP

TS

X

CMP

x

y

x

y

x

y

3

4

-4

-3

2

4

-1

-2

-1

0

-1

1

2

-2

0

-2

4

3

4

2

1

0

-2

-1

5

3

0

0

1

2

-3

-5

-3

1

-3

-4

-5

-2

-3

-2

2

1

1

2

1

3

-2

-2

2

-1

Statisticsfor

Manag

ement

Decisions

#

Examp

le

Fromt

hedata,

itappearstha

tapositive

relationshipmayexist

Most

ofthetimewhentheTSX

isup,

CMPis

up

Likew

ise,whentheTSXisdow

n,

CMPisdown

mostofthetime

Sometimes,

theymoveinoppositedirections

Letsg

raphthisdata

Statisticsfor

Management

Decisions

#

GraphOfData

-6-4-20246

-6

-4

-2

0

2

4

6

TSX

C

MP

Statisticsfor

Management

Decisions

#

GraphOfData

-6-4-20246

-6

-4

-2

0

2

4

6

TSE

C

MP

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Statisticsfor

Manag

ement

Decisions

#

ExampleSummaryStatis

tics

Thedata

doappeartobepositive

lyrelated

Letsderivesomesummarystatisticsaboutthese

data:

Mean

s2

s

TSX

0.0

0

7.2

5

2.6

9

CMP

0.0

0

6.2

5

2.5

0

Statisticsfor

Manag

ement

Decisions

#

Observ

ations

Bothhavemeansofzeroand

standard

deviationsjustunder3

Howev

er,eachdatapointdoesnothave

simply

onedeviationfromthe

mean,

it

deviatesfromb

othmeans

Consid

erPointsA,

B,

CandDonthenext

graph

Statisticsfor

Management

Decisions

#

GraphofData

-6-4-20246

-6

-4

-2

0

2

4

6

TSX

CMP

Statisticsfor

Management

Decisions

#

Implications

Whenpointsinth

eupperrightandlowe

r

leftquadrantsdominate,

thenthesums

of

theproductsofth

edeviationswillb

e

positive

Whenpointsinth

elowerrightanduppe

r

leftquadrantsdominate,

thenthesums

of

theproductsofth

edeviationswillb

e

negative

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Statisticsfor

Manag

ement

Decisions

#

AnImp

ortantObservation

Thesu

msoftheproductsofthedeviations

willgiv

eustheappropriatesignofthe

slopeofourrelationship

Statisticsfor

Manag

ement

Decisions

#

Covariance

population

sample

Statisticsfor

Management

Decisions

#

Covariance

Inthesameun

its

as

Variance

(ifbo

th

varia

blesare

inthesameun

it),i.e.

units

square

d

Very

importan

telemen

to

fmeasuring

port

folioris

kinfin

ance

Statisticsfor

Management

Decisions

#

UsingCovarian

ce

VeryusefulinFin

anceformeasuring

portfoliorisk

Unfortunately,

itishardtointerpretfortwo

reasons:

Whatdoesthem

agnitude/sizeimply?

Theunitsarecon

fusing

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Statisticsfor

Manag

ement

Decisions

#

AMore

UsefulStatistic

Wecansimultaneouslyadjus

tforbothof

theseshortcomingsbydividingthe

covariancebythetworelevantstandard

deviations

Thisop

eration

Removestheimpactofsize&

scale

Eliminatestheunits

Statisticsfor

Manag

ement

Decisions

#

Correla

tion

Correlationindicatesapositiv

e/negative

relationbetweentwovariable

s

Both

variablesmovetogether,

eitherinthe

samedirectionorinopposited

irections

E.g.whenonegoesupsodoe

stheother

Statisticsfor

Management

Decisions

#

TheCorrelation

Coefficient

Thecorrelationcoe

fficientmeasuresthestrength

ofthelinearrelation

shipbetweentwovariables.

Coefficient=-1

pe

rfectnegative

Coefficient=0

no

relation

Coefficient=1

per

fectpositive

Statisticsfor

Management

Decisions

#

CalculatingCor

relation

population

sample

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Statisticsfor

Manag

ement

Decisions

#

Examp

le

X

Y

20

16

18

12

24

18

20

17

22

21

14

10

18

10

Create

ascatterplot,whattypeo

frelationship

exists?

Computethecorrelationcoefficie

nt

Testthesignificanceofthecorre

lationcoefficient

atthe0

.05level

Statisticsfor

Manag

ement

Decisions

#

Scatterplot

0510

15

20

25

0

5

10

15

20

25

30

X

Statisticsfor

Management

Decisions

#

Correlationcoefficient

X

Y

X2

Y2

XY

20

16

400

256

320

18

12

324

144

216

24

18

576

324

432

20

17

400

289

340

22

21

484

441

462

14

10

196

100

140

18

10

324

100

180

136

104

2704

1654

2090

Statisticsfor

Management

Decisions

#

Correlationcoefficient

InExce

l:use

the

CORRELfunc

tion

,=

CORREL(A2:A

8,B

2:B

8)

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Statisticsfor

Manag

ement

Decisions

#

Signific

ance

Hypo

thes

istes

ton

the

truepopu

lationparame

ter

(rho,

r)

H0:r=0

HA:r

0

Tes

tsta

tis

tic

(n-2

degreeso

ffree

dom

):

estima

tedfromt

he

sample

Statisticsfor

Manag

ement

Decisions

#

Signific

ancetest

3.5

63>

2.5

71(tcriticalvalue,

5d

f)

Rejectthenullh

ypothesisandco

ncludethatthe

correlationcoefficientissignificant(significantly

differentthan0)

Statisticsfor

Management

Decisions

#

Correlationvs.Regression

Correlationindicate

sarelationbetweentwo

variables

Regressionindicate

scausality

betweenan

independentanda

dependentvariable.

Changesintheindependentvariablesarethose

causingthechange

inthedependentvariable

Wellstartwithsimp

leregressiononeindependent

variableandthenlookatmultiplevariables.

Simple

regression

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Statisticsfor

Manag

ement

Decisions

#

Ascatterplot

0

500

1000

1500

2000

2500

3000

3500

4000

0

2000

4000

6000

8000

10000

12000

Income

Theinco

me/consumptionexample

Depen

dentvariable(y)consumption

Indepe

ndentvariable(x)income

Statisticsfor

Manag

ement

Decisions

#

Thesto

ry

Wethin

kthatincomeaffectsconsumption

Them

oreyoumakethemoreyoubuy

Weare

lookingtostudythisrelat

ionshipinmore

depth

Isthereindeedasignificanteffect?

What

isthemagnitudeofthiseffect?

(Welimitourdiscussiontolineareffects)

Wellcreateandtestaregressio

nmodelofthe

relation

shipbetweenconsumptio

nandincome

Statisticsfor

Management

Decisions

#

Modelingthelin

earrelationships

Premise:thereisatruer

elationshipbetweenincome

and

consumption.

Thisrelationshipcanbe

describedinalinearform:

Ormoregenerally:

Statisticsfor

Management

Decisions

#

SimpleLinearR

egressionModel

Notethatboth

and

arepopulation

parameterswhich

areusuallyunknowna

nd

henceestimatedfrom

thedata.

y

x

run

rise

=slope

(=rise/run

)

=y-intercept

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Statisticsfor

Manag

ement

Decisions

#

Modelingthelinearrelationships

Withsim

plelinear

regressionwetryto

capturethetrue

relationshipbetweenthe

twovaria

blewithasingle

line.

Theestim

atedregression

modelis:

0

500

1000

1500

2000

2500

3000

3500

4000

0

2000

4000

6000

8000

10000

12000

Income

Statisticsfor

Manag

ement

Decisions

#

Unders

tandingtheerrort

erm

Noline

canhitallthepointsinthescatter

plot,orevenmostofthepoin

ts

Theam

ountwemissbyisca

llederroror

residu

al.

Itisthedifferencebetweenthe

predictedvalue

(from

theregressionline)and

thetruevalue

Statisticsfor

Management

Decisions

#

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0

5000

10000

15000

20000

25000

30000

35000

Population

(thousands)

error(residual,deviation)

Agoodregressionline

willbetheonethatminimizesthe

totalofthesquarede

rrors(SSE).

Morefo

rmally

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Statisticsfor

Manag

ement

Decisions

#

Regres

sionAnalysis

Astatisticaltechniquefordeterminingthe

bestfitlinethroughaseriesofdata

There

gressionlineistheuni

quelinethat

minimizesthetotalofthesqu

ared

deviations(orerrors).

Thestatisticaltermi

sSumo

f

SquaredErrors

orSS

E

This

lineiscalledtheleastsquares

line

Statisticsfor

Manag

ement

Decisions

#

RequiredConditions-e

Thepr

obabilitydistributionof

ei

snormal

E(e)=

0

sei

sconstantandindepende

ntofx,

the

indepe

ndentvariable

Theva

lueofea

ssociatedwithany

particu

larvalueofyisindepe

ndentofthe

valueofea

ssociatedwithan

yothervalue

ofy

Statisticsfor

Management

Decisions

#

Findingtheline

equation

Theequation:

Where:

estimated

b1=

XY-

X

Y

n

Statisticsfor

Management

Decisions

#

Example

Obs.

#o

fs

ites

(X)

Capac

ity

(Y)

XY

X2

1

13

81.8

2

1063.6

6

169

2

10

81.8

2

818.2

100

3

13

58.1

8

756.3

4

169

4

8

43.6

4

349.1

2

64

5

5

40

200

25

6

7

36.3

6

254.5

2

49

7

4

34.5

5

138.2

16

8

8

32.7

3

261.8

4

64

9

7

29.0

9

203.6

3

49

10

3

25.4

5

76.3

5

9

Total

SX=

78

SY=

463.6

4

SXY=

4121.8

6

SX2

=714

calculated

Dataon

refining

capacityof

anoil

companys

sites

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Statisticsfor

Manag

ement

Decisions

#

The

lea

stsquares

line

Meaning:capacitywill

increaseby

4.7

87unitsforevery

siteadded

Understandingand

assessingtheregression

model

Statisticsfor

Management

Decisions

#

Example

TheHarrisCorporation

hasrecentlydoneastu

dy

ofhomesthathavesol

din

theDetroitareawithinthe

past18months.Data

wererecordedforthe

askingprice(x)andthe

numberofweeks(y)each

homewasonthemark

et

beforeitsold.

Weekson

theMarket

Asking

Price

23

$76,5

00

48

$102,0

00

9

$53,0

00

26

$84,2

00

20

$73,0

00

40

$125,0

00

51

$109,0

00

18

$60,0

00

25

$87,0

00

62

$94,0

00

33

$76,0

00

11

$90,0

00

15

$61,0

00

26

$86,0

00

27

$70,0

00

56

$133,0

00

12

$93,0

00

Statisticsfor

Management

Decisions

#

Themodeltobeestimated

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Statisticsfor

Manag

ement

Decisions

#

Excel

TheRegressionTool

Tools

Data

Analysis

ChooseRegressionfromthedialogueboxmenu.

Statisticsfor

Manag

ement

Decisions

#

The

regression

output(Excel)

SUMMARYOUTPUT

RegressionStatistics

MultipleR

0.705948422

R

Square

0.498363174

AdjustedRS

quare

0.464920719

StandardError

11.96417889

Observations

17

ANOVA

df

SS

MS

F

SignificanceF

Regression

1

2133.111647

2133.1116471

4.90211089

0.001541086

Residual

15

2147.123648

143.1415765

Total

16

4280.235294

Coefficients

StandardError

tStat

P-value

Intercept

-16.22506178

12.20252667

-1.3296477220

.203501866

AskingPrice

0.000528163

0.000136818

3.8603252320

.001541086

Statisticsfor

Management

Decisions

#

Theestimatedmodel

=

-16.2

251+0.00053x

Intercept(b0):-16.2

251

Slope(b1):0.0

00

53

Whatisthemean

ingof-16.2

251?

Istheeffectofas

kingpriceonnumbero

f

weekssignificant

?

lil

.

.

.

.

i

i

ii

i

.

.

.

.

i

l

.

.

l

.

Coefficients

Standard

Error

tStat

P-value

Intercept

-16.22506178

12.202

52667

-1.329647722

0.203501866

AskingPrice

0.000528163

0.0001

36818

3.860325232

0.001541086

Statisticsfor

Management

Decisions

#

Test1:TestingtheSlope

Thehypotheses:

HO:b1=0

HA:b10

Wefollowat-test:

Thestandard

error

oftheestimat

e

e

stimated

true

OR

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Statisticsfor

Manag

ement

Decisions

#

Thesta

ndarderrorofthe

estimate

Thesta

ndarderroroftheestimat

e(Se

orSEE)

measureshowthedatavariesar

oundthe

regress

ionline

Simila

rtotheconceptofstandarddeviation

WewouldlikeSet

obesmall

thesmalleritisthe

larger

thet-statisticisandthemore

likelyweareto

reject

thenullhypothesisthattheslo

peiszero

k=1fo

rsimpleregression

Statisticsfor

Manag

ement

Decisions

#

Whatd

owehaveinthetables?

SUMMARYOUTPUT

RegressionStatistics

MultipleR

0.705948422

R

Square

0.498363174

AdjustedRS

quare

0.464920719

StandardError

11.96417889

Observations

17

ANOVA

df

SS

MS

F

SignificanceF

Regression

1

2133.111647

2133.1116471

4.90211089

0.001541086

Residual

15

2147.123648

143.1415765

Total

16

4280.235294

Coefficients

StandardError

tStat

P-value

Intercept

-16.22506178

12.20252667

-1.3296477220

.203501866

AskingPrice

0.000528163

0.000136818

3.8603252320

.001541086

S=SEE

SSE

Sb1

b1

Statisticsfor

Management

Decisions

#

Test1:Testing

the

Slope

lil

.

.

.

.

i

i

ii

i

.

.

.

.

i

l

.

.

l

.

Coefficients

Standard

Error

tStat

P-value

Intercept

-16.22506178

12.202

52667

-1.329647722

0.203501866

AskingPrice

0.000528163

0.0001

36818

3.860325232

0.001541086

t=

0.0

00528163-

0

0.0

00136818

=>

Canconc

lude

tha

tthes

lope

isdifferen

t

from

zero

[

/2

,(n

-2)df]

Statisticsfor

Management

Decisions

#

TestingtheSlope

Ifwewishtotestforpositive

ornegative

linea

r

relationshipswecond

uctone-tailtests,

i.e.our

researchhypothesisb

ecome:

H1:10

(testing

forapositiveslope)

Ofcourse,

thenullhy

pothesisremains:H0:1

=0.

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Statisticsfor

Manag

ement

Decisions

#

Isthet

estdifferentfrom1

?

Thehypotheses:

HO:b1=1

HA:b11

t=

0.0

00528163-1

0.0

00136818

Statisticsfor

Manag

ement

Decisions

#

Test2:

modelfit

Testing

theoverallsignificanceo

fthemodel

H0

:b1=b2=b3==

0

H1

:atleastonebisdifferentthanzero

Weneedtoseethatatleastoneofourindependent

variab

leshasasignificantaffect

Note:

weonlyhaveb1sothistests

houldgiveusthe

same

resultsastheprevioust-test(

andwellseethatit

does)

Thetes

tstatisticisanF-ratio

WellhaveanANOVAtable(fromExcel)

Statisticsfor

Management

Decisions

#

The

regression

output(Excel)

SUMMARYOUTPUT

RegressionStatistics

MultipleR

0.705948422

RSquare

0.498363174

AdjustedR

Square

0.464920719

StandardError

11.96417889

Observations

17

ANOVA

df

SS

MS

F

SignificanceF

Regression

1

2133.111647

2133.111647

14.90211089

0.001541086

Residual

15

2147.123648

143.1415765

Total

16

4280.235294

Coefficients

Sta

ndardError

tStat

P-value

Intercept

-16.22506178

12.20252667

-1.329647722

0.203501866

AskingPrice

0.000528163

0.000136818

3.860325232

0.001541086

Statisticsfor

Management

Decisions

#

FRatio

MeanSq

uares=SS/df

Significantmo

del

ANOVA

df

SS

MS

F

Significa

nceF

Regression

1

21

33.111647

2133.111647

14.90211089

0.001541086

Residual

15

21

47.123648

143.1415765

Total

16

42

80.235294

Degreesof

Freedom

Sumof

Squares

Mean

Square

F-Statistic

Regression

1

SSR

MSR

=SSR/1

F=MSR/MSE

Residual

n-2

SSE

MSE=SSE/(n-2)

Total

n-1

SST

Thegeneralformf

orasimple

regression:

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Statisticsfor

Manag

ement

Decisions

#

SymmetryinTesting

SUMMARYOUTPUT

Regr

essionStatistics

MultipleR

0.705948422

R

Square

0.498363174

AdjustedRS

quare

0.464920719

StandardError

11.96417889

Observations

17

ANOVA

df

SS

MS

F

SignificanceF

Regression

1

2133.111647

2133.111647

14.90211089

0.001541086

Residual

15

2147.123648

143.1415765

Total

16

4280.235294

Coefficients

StandardError

tStat

P-value

Intercept

-16.22506178

12.20252667

-1.329647722

0.203501866

AskingPrice

0.000528163

0.000136818

3.860325232

0.001541086

Statisticsfor

Manag

ement

Decisions

#

Test3:

R2-CoefficientofD

etermination

TheR2t

ellsoftheproportionofthevariabilityinthe

depende

ntvariableisexplainedbythe

independent

variable

Wewo

uldliketoseehighvalues(1isthehighest)

Note:forsimpleregression,

R-squared

isthesquareofthe

correlationcoefficient(r):R2=(r)2.

Statisticsfor

Management

Decisions

#

CoefficientofD

etermination

Aswedidwithanalysisofvariance,wecanpartitionthe

variationinyintotwoparts:

SST=Variationiny=SSE+SSR

SSESum

ofSquaresE

rrormeasurestheamountof

variationinythatremains

unexplained(i.e.

duetoerror

)

SSRSum

ofSquaresR

egressionmeasurestheamount

ofvariationinyexplained

byvariationintheindepende

nt

variablex.

Statisticsfor

Management

Decisions

#

In

the

Exceloutput

SUMMARYOUTPUT

RegressionStatistics

MultipleR

0.7

05948422

R

Square

0.4

98363174

AdjustedR

Square

0.4

64920719

StandardError

11.9

6417889

Observations

17

ANOVA

df

SS

MS

F

Significan

ceF

Regression

1

21

33.1

11647

2133.1

11647

14.9

0211089

0.0

01541086

Residual

15

21

47.1

23648

143.1

415765

Total

16

42

80.2

35294

Coefficients

StandardError

tStat

P-value

Intercept

-16.2

2506178

12

.20252667

-1.3

29647722

0.2

03501866

AskingPrice

0.0

00528163

0.000136818

3.8

60325232

0.0

01541086

2133.1

11647

4280.2

35294

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Statisticsfor

Manag

ement

Decisions

#

Coe

ffic

ien

to

fDe

term

inat

ion

R2hasava

lueo

f.4

984

.Thismeans

49.8

4%

ofthevaria

tion

intheweek

sonmarke

t(y)isexp

laine

dby

thevaria

tion

inthe

as

kingprice

(x).Therema

ining

50

.16%

is

unexplained

,i.e.

due

toerror.

Un

like

theva

lueo

fa

tes

ts

tatis

tic,

thecoe

fficientof

determination

doesnothaveacriticalvalue

tha

tena

bles

us

todraw

conc

lus

ions.

Ingenera

lthe

higher

theva

lueo

fR2

,the

better

themo

de

l

fitsthe

data.

R2=

1:

Perfec

tma

tchbe

tween

the

linean

dthe

da

tapo

ints

.

R2=

0:

Thereareno

linearre

lations

hipbe

tweenxan

dy.

Statisticsfor

Manag

ement

Decisions

#

Summa

ryofsimpleregress

ionoutput

SUMMARYOUTPU

T

Regression

Statistics

MultipleR

0.7

05948422Correlationcoefficinetbetweenxandy

R

Square

0.4

98363174Coefficientofdetermination

AdjustedRSquare

0.4

64920719

StandardError

11.9

6417889S

Observations

17N

ANOVA

df

SS

MS

F

SignificanceF

Regression

1

2133.1

11647

2133.1

11647

14.9

02

11089

0.0

01541086

Residual

15

2147.1

23648

143.1

415765

Total

16

4280.2

35294

Coefficients

StandardError

tStat

P-value

Intercept

-16.2

2506178

12.2

0252667

-1.3

29647722

0.2

035

01866

AskingPrice

0.0

00528163

0.0

00136818

3.8

60325232

0.0

015

41086

b0a

ndb1

Sb0

andSb1

Learnthe

relationships

betweenthethree

tablescomponents

Confidenceandprediction

intervals

Statisticsfor

Management

Decisions

#

Prediction

Supposeyouwantedto

knowhowmanyweeksitwou

ld

taketosellahousepricedat$100,0

00

Theregressionequationwas:=

-16.2

251+0.0

0053

x

Substitutex=100,0

00

y=-16.2

251+0.00053*(100,0

00)=36.7

749

Importantsidenote:payattentiontotheunitsofmeasurementin

thedata

y=36.7

749isapointe

stimateofthenumberofweek

s

Pointestimatesaresubjecttoerrors

whatisthetrueprice?

8/3/2019 700579308

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Statisticsfor

Manag

ement

Decisions

#

Scatterplot

010

20

30

40

50

60

70

$50,

000

$60,

000

$70,

000

$80,0

00

$90,0

00

$100,0

00

$110,

000

$120,0

00$

130,0

00

$140,0

00

Needto

constructapredictioni

ntervalarou

ndthisestimate

Statisticsfor

Manag

ement

Decisions

#

Prediction

interval

Xp=100000,

y=36.5

9126539

-40

-200

20

40

60

80

100 5

0000

62500

75000

87500

100000

112500

125000

Price

f

Predictioninterval

Statisticsfor

Management

Decisions

#

Prediction

Interval

y

ta/2,n-2

se

1+

1 n

OR

y

ta/2,n-2se

1+

1 n

(textboo

k)

derived

Statisticsfor

Management

Decisions

#

Inourexample

8/3/2019 700579308

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Statisticsfor

Manag

ement

Decisions

#

Adiffer

entquestion

Suppos

eIownseveralpropertiesinDetroitand

priceth

emallat$100,0

00.

What

istheexpected

numbe

rofweeksforsellingthesehomes?

Instead

ofpredictinganindividua

lvalue,

Iam

askingforanexpectedvalue(i.e

.themean

numberofweek)

Wecanuseaconfidenceintervalfortheestimationof

them

ean.

Thed

istinctionbetweenconfidenceintervaland

predic

tionintervalissimilartothedifferencebetween

theCIofthemeanvs.

theCIofanindividualvalue

Statisticsfor

Manag

ement

Decisions

#

ConfidenceInterval

Narrowerthantheprediction

interval

-40

-200

20

40

60

80

100

50000

62500

75000

87500

100000

112500

125000

Statisticsfor

Management

Decisions

#

ConfidenceInte

rval

y

ta/2,n-2

se

1 nOR

y

ta/2,n-2se

1 n

(textboo

k)

derived

Statisticsfor

Management

Decisions

#

Inourexample

Note:Point,PredictionandCo

nfidenceintervalsinExcelareobtainedby

Add-

Ins>DataAnalysisPlus>PredictionInterval

8/3/2019 700579308

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Statisticsfor

Manag

ement

Decisions

#

Thecu

rve

Bothin

tervalsarecurved,becoming

narrow

eraroundtheaverage

valueofx(x-

bar).

ThecloserXgistoX-barthe

betterour

estima

teandthusthenarrow

erthe

interva

l.

Statisticsfor

Manag

ement

Decisions

#

Examp

le

Thefollowingsummarystatistics

wereobtained

fromaregressionanalysis:

Provide

a90%C

Ifortheaveragey,givenxg=80

Statisticsfor

Management

Decisions

#

Solution

y

ta/2,n-2

se

1 n

Needto

compute

using

SSE

80

67.2

0

9,7

84345.5

0*80=-17,8

56

a=0.1

n-

2=18

Statisticsfor

Management

Decisions

#

Solution

Computingthestandarderrorofthe

estimate

Fromt

hettable

:t0.0

5,1

8=1.7

34

8/3/2019 700579308

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Statisticsfor

Manag

ement

Decisions

#

Solutio

n

y

ta

/2,n-

2se

1 n

Statisticsfor

Manag

ement

Decisions

#

Regres

sionDiagnostics

Thereare

threeconditionsthatarerequiredinorderto

perform

aregressionanalysis.Theseare:

Theerrorvariablemustbenormally

distributed,

Theerrorvariablemusthaveaconstantvariance,&

Theerrorsmustbeindependentofeachother.

Howcanw

ediagnoseviolationsoftheseconditions?

ResidualAnalysis,thatis,examine

thedifferences

between

theactualdatapointsandthosepredictedby

thelinea

requation

Statisticsfor

Management

Decisions

#

Res

idua

lAnalys

is

Reca

llthe

dev

iations

be

tween

theac

tua

lda

tapo

intsan

dthe

regress

ion

linewerecalle

dresiduals

.Exce

lca

lcu

lates

res

idua

lsasparto

fitsregress

ionana

lys

is:

Wecanuse

theseres

iduals

tode

term

inew

he

ther

theerror

varia

bleisnonnorma

l,w

hether

theerrorvariance

iscons

tant,

an

dw

he

ther

theerrorsarein

depen

den

t

X

Y

Fitted

Res

idua

l

St.res

id

76500

23

24

.17942851

-

1.1

79428507

-

0.1

02346097

102000

48

37

.64759194

10

.35240806

0.9

06924006

53000

9

11

.76759162

-

2.7

67591621

-

0.2

59720474

84200

26

28

.2462857

-

2.2

46285699

-

0.1

93608631

73000

20

22

.33085706

-

2.3

30857056

-

0.2

03458396

125000

40

49

.79534718

-

9.7

95347185

-

0.9

46239962

109000

51

41

.34473484

9.6

55265163

0.8

62374431

60000

18

15

.46473452

2.5

35265477

0.2

30053966

87000

25

29

.72514286

-4.7

2514286

-

0.4

07099584

94000

62

33

.42228576

28

.57771424

2.4

71464614

76000

33

23

.91534687

9.0

84653129

0.7

88907244

90000

11

31

.30963267

-

20

.30963267

-

1.7

51163484

-

-

Statisticsfor

Management

Decisions

#

Nonnorma

lity

Wecan

take

theres

idua

lsan

dpu

tthem

intoa

histog

ram

tov

isua

llyc

hec

kforn

orma

lity

we

re

loo

king

fora

bells

hape

dhistogram

()w

iththe

mean

close

tozero

().

8/3/2019 700579308

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Statisticsfor

Manag

ement

Decisions

#

Heteroscedasticity

Whenther

equirementofaconstantvaria

nceisviolated,we

haveaconditionofheteroscedasticity.

Wecandia

gnoseheteroscedasticitybyplottingtheresidual

againstthe

predictedy.

Statisticsfor

Manag

ement

Decisions

#

He

terosce

das

tic

ity

Ifthevarian

ceo

ftheerrorvaria

ble(

)is

no

tcons

tan

t,then

we

have

heteroscedasticity

.Here

s

thep

loto

fthe

res

idual

aga

ins

tthepre

dictedva

lueofy:

ther

e

doesntappearto

be

a

chan

ge

in

the

spread

ofthe

plotted

points,

therefore

no

heteroscedasticity

Statisticsfor

Management

Decisions

#

Nonindependence

oftheErrorVariable(f

or

timeseriesdatanotinthiscourse)

Ifweweretoobservethenu

mberofweekshousesstayonth

e

marketformanyweeksfor,say,ayear,thatwouldconstitu

tea

timeseries.

Whenthedataaretimeseries,theerrorsoftenarecorrelated.

Errortermsthatarecorrelatedovertimearesaidtobe

autocorrelatedorseriallyc

orrelated.

Wecanoftendetectautocorrelationbygraphingthe

residualsagainstthetime

periods.Ifapatternemerges,itis

likelythattheindependence

requirementisviolated.

Statisticsfor

Management

Decisions

#

Nonindependen

ceoftheError

Variable

Patternsintheappearanceoftheresidualsovertime

indicatesthatautocorrelationexists:

Note

the

runs

ofpositive

residu

als,

replaced

by

runs

ofnegative

residuals

Note

the

oscillating

behavior

ofthe

residuals

around

zero.

8/3/2019 700579308

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Statisticsfor

Manag

ement

Decisions

#

Outliers

Anoutlie

risanobservationthat

isunusually

smallorunusuallylarge.

E.g.

inou

rhousesexamplethep

ricesrange

from$53,0

00to$133,0

00.

Sup

posewehave

avalue

of$1,0

00,0

00t

hispointisan

outlier

.

Statisticsfor

Manag

ement

Decisions

#

Outliers

Possiblere

asonsfortheexistenceofoutliersinclude:

Therewasanerrorinrecordingthevalue

Thepoin

tshouldnothavebeeninclude

dinthesample

Perhaps

theobservationisindeedvalid

.

Outlierscanbeeasilyidentifiedfrom

ascatterplot.

Iftheabsolutevalueofthestandardresid

ualis>2,we

suspectthepointmaybeanoutlierand

investigatefurther.

Theyneed

tobedealtwithsincetheycaneasily

influenc

etheleastsquaresline

Statisticsfor

Management

Decisions

#

Outliersoure

xample

OlgaKaminer,2009

Statisticsfor

Management

Decisions

#

rocedureforReg

ressionDiagnostics

Developamodelthathasatheoreticalbasis.

Gatherdataforthetwovariablesinthemodel.

Drawthescatterdia

gram

todeterminewhetheralinear

modelappearstobeappropriate.

Identifypossible

outliers.

Determinetheregre

ssionequation.

Calculatetheresidu

alsandchecktherequired

conditions(normality,

homoscedasticity,

independen

ce)

Assessthemodels

fit(t-testfortheslope,

theove

rall

F-ratio,

R2)

Ifthemodelfitsth

edata,usetheregression

equationtopredicta

particularvalue

(confidence/prediction

intervals)ofthedependent

variable.