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STATISTICALMETHODSINFINANCE FLOWCHARTOFTHEPROCESS

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STATIONARYMean(Testformean0)

Variance(Takelogdifference)

ADFtestforStaonarity.

MODELIDENTIFICATIONACF,PACFplot

MODELESTIMATION

ARIMA,

GARCH.

MODELADEQUACYBoxTestfor

serialcorrelaon.

FORECASTING

1 2 3

4 5

THEPROCESSUNDERTAKEN

STATISTICALMETHODSINFINANCE FLOWCHARTOFTHEPROCESS

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•  Ttestshowsmeannotzeroof

theclosingprices.

•  p-value<2.2e-16

•  alternavehypothesis:

•  truemeanisgreaterthan0

•  Since the mean is not zero,

wedifferencethedata.

20

40

60

80

closing [2007−01−03/2012−07−06]

Last 14.14

Jan 03

2007

Jan 02

2008

Jan 02

2009

Jan 04

2010

Jan 03

2011

Jan 03

2012

STATISTICALMETHODSINFINANCE

MeanZero

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•  TheaugmentedDickey–Fuller(ADF)stasc, used in the test, is anegave number. The more

negave it is, the stronger therejecon of the hypothesis thatthereisaunitrootatsomelevelofconfidence.

•  Dickey-Fuller = -12.2629, Lag order=11,

•  p-value=0.01

•  alternavehypothesis:staonary

•  Conclusion: Since p value<0.05,reject null of non staonarity,thereforedataisstaonary

•  A model that includes a constant

andametrendisesmatedusingsample of 50 observaons andyieldsthe stascof−4.57.Thisismore negave than the tabulatedcricalvalueof−3.50,soatthe95percentlevelthenullhypothesisofaunitrootwillberejected.

−10

−5

0

5

dtest [2007−01−04/2012−07−06]

Last −0.0199999999999996

Jan 04

2007

Jan 02

2008

Jan 02

2009

Jan 04

2010

Jan 03

2011

Jan 03

2012

STATISTICALMETHODSINFINANCE

ADFTEST:Sta@onarity

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Jarque-BeraNormalalityTest

TestResults:

X-squared:58437.3676,

PVALUE:<2.2e-16

theJarque–Beratestisagoodness-of-fittest of whether sample data have the

skewness and kurtosis matching a

normaldistribuon.

Thenullhypothesisisajointhypothesis

of the skewness being zero and theexcesskurtosisbeingzero.

Hence,weconcludethatthedatadoes

nothaveanormaldistribu@on.

STATISTICALMETHODSINFINANCE NORMALITYTEST

BasicSTATS

MS.Close

nobs1385.000000NAs0.000000

Mean-0.001226

Variance0.002144

Stdev0.046308

Skewness1.393276Kurtosis31.649301

Gaussianity refers to the probability

distribu@onwithrespecttothevalue,in

thiscontexttheprobabilityofthesignal

reaching an amplitude, while the term

'white' refers to the way the signal

powerisdistributedover@meoramong

frequencies.

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STATISTICALMETHODSINFINANCE MODELIDENTIFICATION

SAMPLEACFINTERPRETATIONS

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0 10 20 30 40   −      0 .      2

      0 .      0

      0 .      2

      0 .      4

      0 .      6

      0 .      8

      1 .      0

Series: y

LAG

      A      C      F

0 10 20 30 40   −      0 .      2

      0 .      0

      0 .      2

      0 .      4

      0

 .      6

      0 .      8

      1 .      0

LAG

      P      A      C      F

STATISTICALMETHODSINFINANCE ACF,PACFPLOT

•  There is no clear indicaon of a

AR(p) or MA(q) process here, as

neithertheACFnorthePACFhasa

gradualdeclineatconsecuvelagsnordotheyhavespikesatcertain

lags.

•  Secondmethodofp,qes@ma@on.

best.order<-c(0,0,0)

best.aic<-Inf

for(iin0:4)for(jin0:3){

fit.aic<-AIC(arima(resid(arma)

,order=c(i,0,j)))

if(fit.aic<best.aic){best.order<-c(i,0,j)

best.arma<-arima(resid(arma1)

,order=best.order)

best.aic<-fit.aic}}

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•  ARIMA(1,0,1),AIC=-5.141276

•  ARIMA(3,0,1),AIC=-5.165905

•  Box-Ljung test (H0: The data are

independentlydistributed)

X-squared=41.8899,df=12,p-value=

3.476e-05

•  Sincep<0.05,werejectthenullandsay

dataisdependent.

•  Here an ARIMA p=1,3 and q=1 with

d=0(asdataisstaonary)hasbeenfied

to the data without indicaons of

removal of serial correlaons between

the square of residuals. We find arelaon between the variance of the

residuals signifying that the data was

not totally characterizedby the ARIMA

models usedhere andmore analysis is

neededtogetawhitenoiseintheend.

0 10 20 30 40

      0 .      0

      0 .      2

      0 .      4

      0 .      6

      0 .      8

      1 .      0

Series: armaresi4^2

LAG

      A      C      F

0 10 20 30 40

      0 .      0

      0 .      2

      0 .      4

      0 .      6

      0 .      8

      1 .      0

LAG

      P      A      C      F

STATISTICALMETHODSINFINANCE Fi\ngARMA

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STATISTICALMETHODSINFINANCE COMPARISONBETWEENDIFFERENTMODELS

GARCH(1,1),

t

ARMA(3,1)+GARCH(1,1),

normal

ARMA(3,1)+GARCH(1,1),

t

ARMA(3,1)+GARCH(1,1),

skewedt

AIC -4.134258 -4.040692 -4.134697 -4.133303

Ar1 0 0.55364 0.7038 0.70338

ar2 0 0.01263 0.01662 0.16594

ar3 0 -0.069487 -0.04825 -0.48111

ma1 0 -0.58672 -0.7336 -0.73270

alpha 0.13389 0.13272 0.1189 0.12188

beta 0.85514 0.85570 0.8843 0.88402

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STATISTICALMETHODSINFINANCE FINALGARCHMODEL

Rt = 0.703Rt≠1 + ‘t ≠ 0.7336‘t≠1

‘t =Ô ht÷t with

ht = 0.1189‘t≠1 + 0.8843ht

Where,ŋshouldbeWN(0,1).ButsinceourGARCHdoesnotcaptureallthe

characteris@csofthemodelitisnotinthiscase.