term paper_group 6

8/9/2019 Term Paper_Group 6

1/28

A v e r a g e ( A R I M A ) M o d e l f o r f o r

FinancialEngineering

Term Paper

Submitted to:

Dr. Nalini P.

Group – 6:Shradha Saraogi – 2012PP0!2

Dee"ak P – 2012PP0!#


2/28

FINANCIAL ENGINEERING – TERMPAPER

GROUP - 6

Contents

Introduction2

%iterature Revie2

*+,ectives of the Stud-.

/i'e Series Modeling.

ARIMA (" d ) Model

Data and Methodolog-!

3'"irical Anal-sis4

MA3 to assess the 5orecasting Accurac-1.

MAP3 to assess the 5orecasting Poer1.

%,ung67o8 /est or 9hite :oise /est1

AR;< /est 1

;oncluding *+servations 1!

%i'itations of the Stud-1!

References 1=

Appendices 1#

Dee"ak > Prasanna > Ra' Prasad > Shradha1 > P a g e


3/28


GROUP - 6

Introduction /he use of intelligent s-ste's and advanced techniues for 'arket "rediction has

+een idel- esta+lished /he develo"'ent of accurate techniues is critical to

econo'ists investors and anal-sts /he traditional statistical 'odels used in the

recent -ears for "redicting ?nancial 'arkets fail to ca"ture the inter relationshi"s

+eteen 'arket varia+les


4/28


GROUP - 6

Percentage 3rror (MAP3) ere used as factors for evaluating the "erfor'ance of the

'odels Dunis et al (2012) in his "a"er co'"ared the "erfor'ance of A::

'odel and the traditional ARMA 'odel in forecasting 3HRHSD e8change rate

during the ?nancial crisis of 2004 and found that A:: 'odel as far su"eriorto the traditional ARMA 'odel

$ariations of ARMA 'odel such as the vector ARMA for forecasting of /reasur- +ill

rates and changes in 'one- su""l- have +een discussed +- Aksu et all in 1JJ1 and

seasonal fractionall- dierenced ARMA 'odel for long range forecasting of revenue

of I7M +- Ra- in 1JJ. Siss :ational 7ank (S:7) uses ARIMA 'odel for forecasting

the inKation over the short ter' "eriod and literature regarding the 'odel

e'"lo-ed and factors considered regarding this has also +een studied A"art fro'

this ARIMA has +een used in forecasting a ide variet- of ite's 5or instance %isa

7ianchi (1JJ#) used ARIMA to "redict arrivals in a call center in his "a"er

FI'"roving forecasting for tele'arketing centers +- ARIMA 'odeling ithinterventionG /here have +een atte'"ts to co'+ine the dierent 'odels to for' a

h-+rid 'odel Lhang Peter (200.) F/i'e Series 5orecasting Hsing a Shradha. > P a g e


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GROUP - 6

• Stationary – /hose ti'e series those hose statistical "ro"erties re'ain

constant over ti'e /his consistenc- 'a- range even u"6to the fourth 'o'ents

(e8"ectations variance ske6ness and kurtosis)

MC !po" In#e$ %MCCOM&E'

M;C 5uture Inde8 (M;C;*MD3C) is a signi?cant +aro'eter for the "erfor'ance

of co''odities 'arket and ould +e an ideal invest'ent tool in

co''odities 'arket over a "eriod of ti'e M;C also co'"utes the dail- S"ot

Inde8 value (M;C;*MD3C) for its M;C ;*MD3C M;C ARI M;C M3/A%

M;C 3:3RE +- using the current s"ot "rices of the res"ective co''odities

vis a vis their s"ot "rices in the sa'e +ase "eriod of average of 2001 /he

M;C ;*MD3C is the si'"le eighted average of the three grou" indices 6

M;C ARI M;C M3/A% M;C 3:3RE /he grou" indices are co'"uted

+ased on eo'etric Mean

Gol# ( !il)er !po"

In India old and silver "rices generall- rise hen senti'ents on the econo'- and

the ?nancial 'arkets are +earish or there is uncertaint- over future trends Riding

the rall- gold e8change6traded funds (3/5s) in India gave their highest ever

'onthl- return of 1!N in August 2011 All this ca'e ith increased volatilit-O there

ere 'an- 'onths hen +oth gold and silver gave negative returns old and silver

follo an al'ost si'ilar "attern and historicall- the- have 'aintained a ratio that

has Kuctuated idel- +eteen 1! and 100 since the 1J40s

CN NIFT* /he ;:C :ift- also called the :ift- !0 or si'"l- the :ift- is :ational Stock

38change of Indias +ench'ark inde8 for Indian euit- 'arket ;:C in its na'e

stands for ;RISI% :S3 Inde8 /he ;:C :ift- covers 22 sectors of the Indian

econo'- and oers invest'ent 'anagers e8"osure to the Indian 'arket in one

"ortfolio /he to" to contri+utors (in ter's of eight) to ;:C :ift- !0 Inde8 include

?nancial services (2!2=N) and I/ sector (1=2N) *ther sectors like industrial

'anufacturing contri+ute ,ust 04.N eightage hile there is no eightage for

agricultural sector in the inde8 /he inde8 as initiall- calculated on full 'arket

ca"italiQation 'ethodolog- 5ro' une 2= 200J the co'"utation as changed to

free Koat 'ethodolog- /he +ase "eriod for the ;:C :ift- inde8 is :ove'+er .

1JJ!

ARIMA !" d" #$ Model /he data series that e are orking on are non6stationar- /herefore to +etter

understand the data or to "redict future "oints in the series e need a uni6variate

'odel that converts the non6stationar- data into a stationar- data *ne such 'odel

Dee"ak > Prasanna > Ra' Prasad > Shradha > P a g e


6/28


GROUP - 6

is ARIMA Autoregressive Integrated Moving Average hich is one of the 'ost

co''onl- used linear 'odels /he other 'odels include :on6linear 'odels like

dierent versions of Arti?cial :eural :etorks (A::) Model etc

7oth Autoregressive (AR) and Moving Average (MA) 'odels e8"ress dierent kindsof stochastic de"endence AR "rocesses enca"sulate a Markov6like ualit- here

the future de"ends on the "ast hereas MA "rocesses co'+ine ele'ents of

rando'ness fro' the "ast using a 'oving indo An o+vious ste" is to co'+ine

+oth t-"es of +ehavior into an ARMA(" ) 'odel hich is o+tained +- a si'"le

concatenation

Ct T1CtU1 V W W W V T"CtU" V Xt V Y1XtU1 V W W W V YXtU

Auto6regressive ele'ent " lingering eects of "receding scores

Integrated ele'ent d trends in the data

Moving Average ele'ent lingering eects of "receding rando'

shocks

/hese are the in"ut "ara'eters for ARIMA /herefore so'e of the uestions to +e

ansered +efore running the 'odel are 6 Shradha! > P a g e


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GROUP - 6

o+served si'"le line "lot of the data a 'ore a""ro"riate 'ethod ould +e to "lot

the autocorrelation function to o+serve if the lags are inKuencing the current value

of data or not If e ?nd that there is signi?cant autocorrelation "resent in the data

then it 'eans that data is not stationar- Data can +e transfor'ed into stationar-

data +- si'"le dierencing /he dierenced data is then checked for stationar-

nature if the ?rst dierence does give us the stationar- data then e 'ove on to

second dierenced data /his "rocess ill continue until e get transfor' the data

to +e stationar- In general e attain stationar- data ithin ?rst to dierenced

data "oints onl- *nce the stationar- data is o+tained note the degree of dierence

value /his ill serve as the value of FdG in ARIMA ("d)

/he second ste" in ARIMA i'"le'entation is "o i#en"i7 "3e au"ocorrela"ion

or#er 8p9 an# "3e or#er o7 mo)ing a)erage. 89 /hese values can +e

o+tained fro' the autocorrelation function (A;5) and "artial autocorrelation function

(PA;5) of the dierenced data 7- looking at the autocorrelation function of thedierenced data the order of 'oving average 'odel can +e "redicted /his is done

+- o+served hich of the lags is signi?cantl- correlated to the current value if the

second lag in A;5 is signi?cantl- correlated to the current value then e sa- that

the value is 2 Si'ilarl- +ased on the lag that is signi?cantl- correlated to the

current value in PA;5 e "redict the order of auto regressive 'odel

A;ai;e Shradha= > P a g e


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GROUP - 6

(Autoregressive Moving Average) ARIMA (Autoregressive Integrated Moving

Average) or SARIMA (Seasonal Autoregressive Integrated Moving Average)

A;ai;e in7orma"ion cri"erion %AIC'

It is a 'easure of the relative ualit- of a statistical 'odel for a given set of dataAs such AI; "rovides a 'eans for 'odel selection

AI; deals ith the trade6o +eteen the goodness of ?t of the 'odel and the

co'"le8it- of the 'odel It is founded on infor'ation entro"- it oers a relative

esti'ate of the infor'ation lost hen a given 'odel is used to re"resent the

"rocess that generates the data

In the general case the AI; is

here k is the nu'+er of "ara'eters in the statistical 'odel and % is the

'a8i'iQed value of the likelihood function for the esti'ated 'odel

iven a set of candidate 'odels for the data the "referred 'odel is the one ith

the 'ini'u' AI; value


9/28


GROUP - 6

&m!irical AnalysisMCCOM&E

5irst e have considered M;C;*MD3C hich is considered to +e acting as a

signi?cant +aro'eter for the "erfor'ance of co''odities 'arket in India and

ould +e an ideal invest'ent tool in co''odities 'arket over a "eriod of ti'e

000!0000

1000001!0000200000

2!0000.00000.!000000000!0000!00000

6#0000

6=0000

60000

620000

000

20000

0000

=0000

#0000

M;C;*MD3C Inde8 "rices (Rs)

;lose 5irst Dierence

/he a+ove line "lot for dail- closing "rices of M;C;*MD3C shos that the closing

"rices of this inde8 do not re"resent stationar- data /he ?rst dierence values"lotted are nearer to rando' alk values can +e considered stationar- Also the

A;5 for actual closing "rices of M;C;*MD3C have shon signi?cant autocorrelation

+eteen the lag values and the current values

1 2 . ! = 4 # J 10

-4,B

-0,B

,B

0,B

4,B

ACF

A;5 H% %%

Autocorrelation 5unction (A;5) for ?rst dierenced M;C;*MD3C closing "rices

Dee"ak > Prasanna > Ra' Prasad > Shradha# > P a g e


10/28


GROUP - 6

Partiall- Autocorrelation 5unction (PA;5) for ?rst dierenced M;C;*MD3C closing

"rices

1 2 . ! = 4 # J 10

-4+B

-4,B

-0+B

-0,B

-+B

,B

+B

0,B

0+B

PACF

PA;5 H% %%

*n o+serving the A;5 and PA;5 gra"hs of the ?rst dierenced M;C;*MD3C closing

"rices e can inter"ret that the order of auto regression F"G is 1 hich is o+tained

fro' "artiall- auto correlated function and the order of 'oving averages FG is also

1 hich is o+tained fro' auto correlated function Prasanna > Ra' Prasad > ShradhaJ > P a g e

ARIMA %p/#/'AICalue @e." Mo#el

ARIMA (110) =.!1 7est Model

ARIMA (011) =.!1ARIMA (012) =.41=

ARIMA (111) =.41=ARIMA (210) =.41=

ARIMA (112) =.J1JARIMA (211) =.J1J

ARIMA (212) =.!12.

ARIMA (010) :A


11/28


GROUP - 6

3cono'ists consider gold "rices as a good indicator of the health of the econo'- In

the "ast it has +een o+served that investors Kock to gold hen the- are "rotecting

their invest'ents fro' either a crisis or inKation 9hen gold "rices dro" that usuall-

'eans the econo'- is health- /hats +ecause investors have left gold for other

'ore lucrative invest'ents like stocks +onds or real estate

0!000

10000

1!000

20000

2!000

.0000

.!000

6.0000062!0000

620000061!000061000006!0000000!00001000001!0000

old S"ot "rices (Rs)

S"ot Price(Rs ) 5irs t Dierence

/he line "lot for s"ot 'arket "rices of *%D in the co''odit- 'arket shos that

the s"ot "rices of the co''odit- do not re"resent stationar- data /he ?rst

dierence values "lotted are nearer to rando' alk values can +e considered

stationar- Also the A;5 for actual s"ot "rices of *%D have shon signi?cant

autocorrelation +eteen the lag values and the current values

Auto ;orrelation 5unction (A;5) for ?rst dierenced *%D s"ot "rices

1 2 . ! = 4 # J 10

-0,B

-+B

,B

+B

0,B

0+B

ACF

A;5 H% %%

Partiall- Auto ;orrelation 5unction (PA;5) for ?rst dierenced *%D s"ot "rices



12/28


GROUP - 6

1 2 . ! = 4 # J 10

-0,B

-+B

,B

+B

0,B

0+B

PACF

PA;5 H% %%

*n o+serving the A;5 and PA;5 gra"hs of the ?rst dierenced M;C;*MD3C closing

"rices e can inter"ret that the

order of auto regression F"G is 2

hich is o+tained fro' "artiall- autocorrelated function and the order of

'oving averages FG is also 2 hich

is o+tained fro' auto correlated

function Prasanna > Ra' Prasad > Shradha11 > P a g e

ARIMA

%p/#/'

AIC

alue @e." Mo#elARIMA (210) 40!J2 7est Model

ARIMA (012) 40!J2

ARIMA (111) 40!J2

ARIMA (110) 40!=J!

ARIMA (011) 40!=J!

ARIMA (112) 40!#J#

ARIMA (211) 40!#J#

ARIMA (212) 40=102

ARIMA (010) :A


13/28


GROUP - 6

0

10000

20000

.0000

0000

!0000

=0000

40000

6#00000

6=00000

600000

6200000

000

200000

00000

=00000

Silver S"ot Prices (Rs)

1st Di

old and silver follo an al'ost si'ilar "attern and historicall- the- have

'aintained a ratio that has Kuctuated idel- +eteen 1! and 100 since the 1J40s

Si'ilar to old "rices Silver "rices also rise hen senti'ents on the econo'- and

the ?nancial 'arkets are +earish or there is uncertaint- over future trends

/he a+ove line "lot for s"ot 'arket "rices of SI%$3R in the co''odit- 'arket shos

that the s"ot "rices of the co''odit- do not re"resent stationar- data /he ?rst

dierence values "lotted are nearer to rando' alk values can +e considered

stationar- Also the A;5 for actual s"ot "rices of SI%$3R have shon signi?cantautocorrelation +eteen the lag values and the current values

Auto ;orrelation 5unction (A;5) for ?rst dierenced SI%$3R s"ot "rices

1 2 . ! = 4 # J 10

-0+B

-0,B

-+B

,B

+B

0,B

0+B

ACF

A;5 H% %%



14/28


GROUP - 6

Partiall- Auto ;orrelation 5unction (PA;5) for ?rst dierenced SI%$3R s"ot "rices

1 2 . ! = 4 # J 10

-0+B

-0,B

-+B

,B

+B

0,B

0+B

PACF

PA;5 H% %%

/hough fro' the a+ove A;5and PA;5 gra"hs for the ?rst

dierenced values see' to

i'"l- that the value of " is #

and is # /his is +ecause of

the o+served signi?cant

correlation of current values

ith #th lag in A;5 and PA;5


15/28


GROUP - 6

000

100000

200000

.00000

00000

!00000

=00000

400000

6.00

6200

6100

0

100

200

.00

;:C :I5/E Prices (Rs)

;lose 1st Di

/he a+ove line "lot for dail- closing "rices of ;:C :ift- shos that the closing "rices

of this inde8 do not re"resent stationar- data /he ?rst dierence values "lotted are

nearer to rando' alk values can +e considered stationar- Also the A;5 for actual

closing "rices of ;:C :ift- have shon signi?cant autocorrelation +eteen the lag

values and the current values

Autocorrelation 5unction (A;5) for ?rst dierenced ;:C :ift- closing "rices

1 2 . ! = 4 # J 10

-0+B

-0,B

-+B

,B

+B

0,B

0+B

ACF

A;5 H% %%

Partiall- Autocorrelation 5unction (PA;5) for ?rst dierenced ;:C :ift- closing

"rices



16/28


GROUP - 6

1 2 . ! = 4 # J 10

-0+B

-0,B

-+B

,B

+B

0,B

0+B

PACF

PA;5 H% %%

As fro' the a+ove A;5 and PA;5

gra"hs for the ?rst dierenced

values do not directl- tell an-thinga+out the "ossi+le values for " and

the actual values of ARIMA 'odel

"ara'eters are deter'ined +ased

on the AI; values +elo

AI; /est 6 9e kno that the 'odel

ith loest value AI; is the +est

'odel In this case e o+serve that

ARIMA(110) is the +est 'odel ith loest AI; value

MA& to assess the 'orecasting Accuracy /he a+ove ta+le shos the forecasting accurac- of

ARIMA 'odels that e have a""lied on M;C;*MD3C

*%D s"ot "rices SI%$3R s"ot "rices and ;:C :ift-

using error 'easure Mean Average 3rror (MA3) It is

o+served fro' the ta+le that MA3 does e8ist in each

of the forecasts and thus there is ala-s an error

ele'ent hen ARIMA 'odel is used to "redict future

values

MA(& to assess the 'orecasting (ower /he a+ove ta+le shos the forecasting "oer of

ARIMA 'odels that e have a""lied on M;C;*MD3C

*%D s"ot "rices SI%$3R s"ot "rices and ;:C :ift-

using error 'easure Mean Average Percentage 3rror

(MAP3) It is o+served fro' the ta+le that MAP3 is

least in ARIMA (1 1 0) 'odel for the M;C;*MD3C

Dee"ak > Prasanna > Ra' Prasad > Shradha1! > P a g e

ARIMA%p/#/'

AICalue @e." Mo#el

ARIMA (110) ###= 7est ModelARIMA (011) ###=

ARIMA (012) #!0#JARIMA (111) #!0#J

ARIMA (210) #!0#JARIMA (112) #!2J.

ARIMA (211) #!2J.ARIMA (212) #!J4

ARIMA (010) :A

IN&ICE!DCommo#i"ie.

MAE

M;C;*MD3C 2J2#

*%D 2J24

SI%$3R ##2=1

;:C :I5/E 1...

IN&ICE!DCommo#i"ie.

MAPE

M;C;*MD3C 04244N

*%D 0#1#1N

SI%$3R 1J#.1N

;:C :I5/E 22044N


17/28


GROUP - 6

"rices So it is e8"eriential that this 'odel has "erfor'ed +etter than all the other

ARIMA 'odels Also it can +e o+served that the MAP3 is highest in the case ;:C

:ift- hich is ver- 'uch e8"ected as

the volatilit- of euit- stocks is higher

than that of the co''odit- 'arket

Since ARIMA is a linear 'odel it can

+e used to "redict co''odit- "rices

than to "redict the euit- 'arket

/his is "roved ith the MAP3 anal-sis

hich shos highest MAP3 for ;:C

:ift-

Prasanna > Ra' Prasad > Shradha1= > P a g e

IN&ICE!DCommo#i"ie.

&Fp-)alue

M;C;*MD3C =`

00001

12 0000

*%D = 0J1#

12 0!4.

SI%$3R = 0=1.

12 00==;:C :I5/E = 0#14

12 0!20

IN&ICE!DCommo#i"ie.

Lag !core C P-alue Pre.en"

M;C;*MD3C 1 2#44! .#1= =#36= /RH3

2 !1401 !JJ1= #36JJ /RH3

*%D 1 #=#J .#1= 43610# /RH3

2 J!4#=# !JJ1= 13620# /RH3

SI%$3R 1 !02=2 .#1= 136111 /RH3

2 JJJ00. !JJ1= 136214 /RH3

;:C :I5/E 1 244=0 .#1= !36J! /RH3 2 #014= !JJ1= 361#. /RH3


18/28


GROUP - 6

this e have a""lied the AR;< test on all the data values that e used in this stud-

"ro,ect /he results ere "resented in the a+ove ta+le 5ro' the a+ove ta+le it can

+e o+served all the ti'e6series data do have conditional heteroscedasticit- and

hence and AR;


19/28


GROUP - 6

References

OURNAL PAPER!

P :ason ;ha"ter 11 FStationar- and non6stationar- ti'e seriesG Dr /ri"ath- : (2011) FA ;o'"arison of Arti?cial :eural :etork (A::) Model

Auto Regressive Integrated Moving Average (ARIMA) Model for 5orecasting Indian

Stock MarketG International ournal of ;onte'"orar- 7usiness Studies /ina akaa et al 2011 F3lectricit- "rice forecasting – ARIMA 'odel a""roachG LsuQsanna


20/28


GROUP - 6

HE@!ITE!

C%S/A/ Statistical and Data Anal-sis Softare for 3C;3% 8lstatco' S"ider ?nancials – :u'C% su""ort e+site s"ider?nancialco' *nline coursed on ?nancial econo'etrics htt"sonlinecoursesscience"suedu

!TAN&AR& TET @OO!

ohn ; Prasanna > Ra' Prasad > Shradha1J > P a g e

https://onlinecourses.science.psu.edu/https://onlinecourses.science.psu.edu/


21/28


GROUP - 6

A!!endicesMAE an# MAPE calcula"ion.

MCCOM&E

&a"ePre#ic"e#

alue.Ac"ualalue. Error

Percen"ageError

1122201. 0214! 0=!! .40 104!N

112.201. 021=# 0=!! .44 1044N112!201. 021=J 0124. #J= 022.N

112=201. 021=J 0.!21 1.!2 0..!N1124201. 021=J 002#. 1##= 041N

112#201. 021=J .J4!4 !J! 11!=N

112J201. 021=J .JJ1# .021 04!4N

MEAN AERAGE ERROR %MAE' 4J4K

MEAN AERAGE PERCENTAGE ERROR %MAPE' ,4KB

GOL&

&a"e Pre#ic"e# alue.Ac"ualalue. Error Percen"age Error

1121201. .0J00!= .0#!#00 2!= 01.#N

1122201. .0#J#!. .0J#400 ##4 02#=N

112!201. .0#J4.2 .0.#00 !J.2 1#10N

112=201. .0#J42J .0#J#00 041 0002N

1124201. .0#J42J .04!000 142J 04JN

112#201. .0#J42# .0.4J00 !1#2# 140=N

112J201. .0#J42# .0JJ00 .J#2# 1.0=N

MEAN AERAGE ERROR %MAE' 42J4

MEAN AERAGE PERCENTAGE ERROR %MAPE' ,K0KB

!ILER

&a"ePre#ic"e#

alue.Ac"ualalue. Error

Percen"ageError

1121201. !!241J J400 !#01J 12J1N

1122201. !!24.! J0=00 =21.! 1.#N

112!201. !!24. 2#=00 121. 2#0.N112=201. !!24. !0!000 44. 10=0N1124201. !!24. #.00 =J.. 1!=N

112#201. !!24. 11100 11=. .211N

112J201. !!24. .4J00 11#. 2!##N

MEAN AERAGE ERROR %MAE' KK460

MEAN AERAGE PERCENTAGE ERROR %MAPE' 0JK1B



22/28


GROUP - 6

CN NIFT*

&a"e Pre#ic"e# alue.Ac"ualalue. Error Percen"age Error

1121201. =2000 !JJJ0! 20J! .1=N

1122201. =2000 !JJ!! 20#!! .4#N

112!201. =2000 =11!.! ##=! 1!0N

112=201. =2000 =0!J10 1J0 2.J1N

1124201. =2000 =0!410 1=J0 22!N

112#201. =2000 =0J1#! 1121! 1#1N

112J201. =2000 =14=10 24J0 0!2N

MEAN AERAGE ERROR %MAE' 01121 MEAN AERAGE PERCENTAGE ERROR %MAPE' 44,KB

Pre#ic"ion. an# Re.i#ual.

Dec11 A"r12 ul12 *ct12 an1. Ma-1. Aug1. :ov1. Mar1

.200

.00

.=00

.#00

000

200

00

=00

#00

ARIMA %MCCOM&E'

;lose ARIMA (;lose) $alidation

Prediction %oer +ound (J!N) H""er +ound (J!N)

&a"e

Clo.e

MCCOM&E



23/28


GROUP - 6

6#00

6=00

600

6200

0

200

00

=00

#00

Re.i#ual. %MCCOM&E'

&a"e

Re.i#ual

GOL& .po" price.


2!000

2=000

24000

2#000

2J000

.0000

.1000

.2000

..000

.000

ARIMA %GOL& !po" Price%R.''

S "ot Price(Rs ) ARIMA (S "ot Price(Rs )) $alidation


&a"e

!po" Price%R.'



24/28


25/28


GROUP - 6

6#000

6=000

6000

62000

0

2000

000

=000

Re.i#ual. %!ILER'

&a"e

Re.i#ual

CN Ni7"


!00

!000

!!00

=000

=!00

4000

ARIMA %CN Ni7"'

;lose ARIMA (;lose) $alidation


&a"e

Clo.e



26/28


GROUP - 6

6.00

6200

6100

0

100

200

.00

Re.i#ual. %CN Ni7"'

&a"e

Re.i#ual

Normali" "e." an# H3i"e noi.e "e." re.ul".

MCCOM&E

:or'alit- test and hite noise tests(Residuals)

Statistic D5 $alue"6

value

arue67era 21.1J

#4

`0000

1

7o86Pierce = .!=#4

`0000

1

%,ung67o8 = .=044

`0000

1

Mc%eod6%i = =J2J

`0000

1

7o86Pierce 12 .=121 0000

Dee"ak > Prasanna > Ra' Prasad > Shradha2! > P a g e


27/28


GROUP - 6

%,ung67o8 12 .=!2. 0000

Mc%eod6%i 12 #1##=

`0000

1

GOL&:or'alit- test and hite noise tests(Residuals)


value

arue67era 2.2.24

2=`

00001

7o86Pierce = 1JJ= 0J20%,ung67o8 = 2020 0J1#

Mc%eod6%i = =#1 0.417o86Pierce 12 10240 0!J2

%,ung67o8 12 10J2 0!4.

Mc%eod6%i 12 1.!1 024J

!ILER:or'alit- test and hite noise tests(Residuals)


value

arue67era 24.0J

##`

00001

7o86Pierce = 1 0=21%,ung67o8 = 4= 0=1.

Mc%eod6%i = 2=110 00007o86Pierce 12 1J=#2 004.

%,ung67o8 12 20040 00==

Mc%eod6%i 12 .0!1 0002

CN NIFT* :or'alit- test and hite noise tests

Dee"ak > Prasanna > Ra' Prasad > Shradha2= > P a g e


28/28


GROUP - 6

(Residuals)


value

arue67era 2 41202`

00001

7o86Pierce = 2J00 0#21%,ung67o8 = 2J.4 0#14

Mc%eod6%i = 1#14= 000=7o86Pierce 12 10#! 0!2

%,ung67o8 12 1110# 0!20

Mc%eod6%i 12 4J.!2`

00001

Dee"ak > Prasanna > Ra' Prasad > Shradha

term paper_group 6

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