Comparison Of The Differencing Parameter Estimation From Arfima Model By Spectral Regression Methods.

COMPARISON OF TH E DIFFE RE NCING

PARAME TE R E STIMATION F ROM ARFIMA

MODE L BY SPE CTRAL RE GRE SSION

ME TH ODS

By

INTRODUCTION (1)

TIME SERIES MODELS

BASED ON VALUE OF DIFFERENCING PARAMATER (d)

ARMA

d = 0

_d

_≠

_{0, d =}

ARIMA

_INTEGER

ARFIMA

d=

REAL

Long Memory Processes



Long

range

dependence

or

memory

long

means

that

observations

far

away

from

h

eac

other

are

still

strongly

correlated.



The correlation of long memory processes is decay slowly as lag

data increase that is with a hiperbollic rate.

INTRODUCTION(2)

Granger

and Joyeux(1980)

Hosking(1981)

Sowell (1992)

-MLE-Geweke and

Porter-Hudak(1983)

-GPH

Method-Reisen(1994)

-SPR

Method-Robinson(1995)

-GPHTr

Method-Hurvich and Ray(1995)

- GPHTa

Method-Velasco(1999b)

-MGPH

Method-Beran(1994)

-NLS-INTRODUCTION(3)

COMPARISON OF

REGRESSION SPECTRAL

METHODS

Lopes,Olberman

and Reisen(2004)

-Non stationary

ARFIMA-SPR Method is the best

Lopes

and Nunes(2006)

-ARFIMA(0,d,0)-GPH Method is the best

GOAL OF RE SE ARCH

Comparing accuracy of

spectral regression

estimation

methods

of

erencing

the

diff

parameter

d)

(

from

stationary

ARFIMA

ARFIMA MODE L(1)



An ARFIMA(p,d,q) model can be defined as foll ows:



t = index of observation (t = 1, 2, . . ., T)



d = differencing parameter (rea l number)





= mean of obervation

  



 

(

B

)

d

1

_t

B

z













 

2

t

ARFIMA MODE L(2)



is polinomial AR(p)



is polinomial MA(q)



fractional differencing

operator

p

p

2

2

1

B

B

..

B

1

)

B

(



















2

q

1

2

q

( B )

1

B























d

_d

 

 

k

k

0

d

1

B

k





Additive Outlier in Time Series Data



t

_{ }

t

t

A O

t

A O

t

X

t

Z

X

t

X

I









_

_



Τ



Τ

Τ

1,

ˆ

A

AO :



_Τ



_Τ





Additive Outlier is an event that effects a series for one time

period only

DIAGRAM

*Generate

Data 1 :

ARFIMA(1,d,0)

and

ARFIMA(0,d,1)

d =0.2 and 0.4

*Generate

Data 2 :

1)ARFIMA(1,d,0)

and

ARFIMA(0,d,1)

d =0.2 and 0.4

2) Add 5 additive outliers

Estimate parameter d by

1. GPH

method

2. SPR method

3. GPHTr method

4. GPHTa method

5. MGPH method

Determine Mean dan SD estimate

From these methods

COMPARISON RESULT (1)

d

_{= 0.2}

d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2 d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2

ARFIMA(1,d,0), T= 300

ARFIMA(0,d,1), T =300

d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2 d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2

Without Outliers

With Outliers

d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2 d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2

ARFIMA(0,d,1),T =1000

ARFIMA(1,d,0),T =1000

d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2 d = 0 ,2 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.2

COMPARISON RESULT (2)

d

_{= 0.4}

ARFIMA(1,d,0), T= 300

ARFIMA(0,d,1), T =300

Without Outliers

With Outliers

ARFIMA(0,d,1),T =1000

ARFIMA(1,d,0),T =1000

d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4 d = 0 ,4 MGPH GPHTa GPHTr SPR GPH 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0.4

CONCLUSION

1)

GPH method shows the best perf

ormance in

estimating

the

differencing

rameter

pa

with

value d = 0.2 of ARFIMA model for both

clean

data and data with outlier.

2)

Estimation of spectral regress

ion methods are

better

to

be

implemented

odeling

for

m

ARFIMA(1,d,0)

data

than

for

m

E STIMATION OF THE DIFFE RE NCING PARAME TE R OF

ARFIMA MODE L BY SPE CTRAL RE GRE SSION ME THOD(1)

1. Construct spectral density function ( SDF) of ARFIMA model

2. Take logarithms of SDF from ARFIMA model

 

_



_







 

2

2 _2d

2 q a Z

exp(

i

)

f

2sin

,

,

2

exp(

i

)

2











_{ }

_



 





_

_

_

_



_

_

_

_

 



_

_

_

_

 

 

 

_{ }

 

 

3

w

here

2

_W

W

j

W

j

Z

j

j

f

ln f

ln f

0

d ln 1

exp(i

)

ln

f

0

2

j

,

j

1,2,...., T / 2

T



















_

_





_

_







E STIMATION OF THE DIFFE RE NCING PARAME TE R OF

ARFIMA MODE L BY SPE CTRAL RE GRE SSION ME THOD(2)

3. Add natural logarithm of periodogram to equation

(3) above

4.

Determine

the

periodogram based

on

regression

spectral methods

GPH, GPHtr,MGPH

 

 

 

 

 

_{i j}

 

_{ }

 

_{ }

 

4

f

I

2

_W

_j

_Z

_j

ln I

_Z

_j

ln f

_W

0

dln1

exp

ln

ln

f

_W

0

_f

j

Z

















_











_



_









_



















 



g( T )



 

Z

j

0

t

j

t 1

1

I

2

cos( t.

)

,

2











 





E STIMATION OF THE DIFFE RE NCING

PARAME TER OF

ARFIMA MODE L BY SPE CTRAL RE GRE SSION ME THOD(3)

SPR

GPHta

 



g*( T )



 

Z

j

0 0

t

t

j

t 1

1

I

2

2



 

 





 







_



 



 

2



3

3

t

g * ( T )

1

6

t / g * ( T )

6

t / g * ( T )

0

t

2

j

g * ( T )

2

1

g * ( T )

2



_{ }

 



_



_







_

_

_

_

_



_

_



 

_{ }

_{ }

T 1

 

 

2

Z

j

_{T 1}

₂

t

t 0

1

I

2

tap t











_



_



E STIMATION OF THE DIFFE RE NCING

PARAME TER OF

ARFIMA MODE L BY

SPE CTRAL RE GRE SSION

ME THOD(4)

5 Estimate d by Ordinary Least Square Me thod.

Where,





2

0.5

1

( )

1

cos

2

t

tap t

T















_

_



_

_



_

_







 

2

j

Z

j

j

j

The 3-th International Conference on

Mathematics and Statistics (ICOMS-3) 13

CONCLUSION

1)

GPH method shows the best perf

ormance in

estimating

the

differencing

rameter

pa

with

value d = 0.2 of ARFIMA model for both

clean

data and data with outlier.

2)

Estimation of spectral regress

ion methods are

better

to

be

implemented

odeling

for

m

ARFIMA(1,d,0)

data

than

for

m

The 3-th International Conference on

Mathematics and Statistics (ICOMS-3) 15

E STIMATION OF THE DIFFE RE NCING PARAME TE R OF

ARFIMA MODE L BY SPE CTRAL RE GRE SSION ME THOD(1)

1. Construct spectral density function ( SDF) of ARFIMA model

2. Take logarithms of SDF from ARFIMA model

 

_



_







 

2

2 _2d

2 q a Z

exp(

i

)

f

2sin

,

,

2

exp(

i

)

2











_{ }

_



 





_

_

_

_



_

_

_

_

 



_

_

_

_

 

 

 

_{ }

 

 

3

w

here

2

_W

W

j

W

j

Z

j

j

f

ln f

ln f

0

d ln 1

exp(i

)

ln

f

0

2

j

,

j

1,2,...., T / 2

T



















_

_





_

_











E STIMATION OF THE DIFFE RE NCING PARAME TE R OF

ARFIMA MODE L BY SPE CTRAL RE GRE SSION ME THOD(2)

3. Add natural logarithm of periodogram to equation

(3) above

4.

Determine

the

periodogram based

on

regression

spectral methods

GPH, GPHtr,MGPH

 

 

 

 

 

_{i j}

 

_{ }

 

_{ }

 

4

f

I

2

_W

_j

_Z

_j

ln I

_Z

_j

ln f

_W

0

dln1

exp

ln

ln

f

_W

0

_f

j

Z

















_











_



_









_



















 



g( T )



 

Z

j

0

t

j

t 1

1

I

2

cos( t.

)

,

2











 





The 3-th International Conference on

Mathematics and Statistics (ICOMS-3) 17

E STIMATION OF THE DIFFE RE NCING

PARAME TER OF

ARFIMA MODE L BY SPE CTRAL RE GRE SSION ME THOD(3)

SPR

GPHta

 



g*( T )



 

Z

j

0 0

t

t

j

t 1

1

I

2

2



 

 





 







_



 



 

2



3

3

t

g * ( T )

1

6

t / g * ( T )

6

t / g * ( T )

0

t

2

j

g * ( T )

2

1

g * ( T )

2



_{ }

 



_



_







_

_

_

_

_



_

_



 

_{ }

_{ }

T 1

 

 

2

Z

j

_{T 1}

₂

t

t 0

t 0

1

I

2

tap t











_





_





E STIMATION OF THE DIFFE RE NCING

PARAME TER OF

ARFIMA MODE L BY

SPE CTRAL RE GRE SSION

ME THOD(4)

5 Estimate d by Ordinary Least Square Me thod.

Where,





2

0.5

1

( )

1

cos

2

t

tap t

T















_

_



_

_



_

_







 

2

j

Z

j

j

j