COMPARISON OF THE DIFFERENCING PARAMETER ESTIMATION FROM ARFIMA MODEL BY SPECTRAL REGRESSION METHODS By Gumgum Darmawan, Nur Iriawan, Suharono
INTRODUCTION (1) TIME SERIES MODELS BASED ON VALUE OF DIFFERENCING PARAMATER (d) ARMA d = 0 ARIMA d 0, d = INTEGER ARFIMA d= REAL Shor Memory Nonsaionary Shor Memory Long Memory Mahemaics and Saisics (ICOMS-3) 2
Long Memory Processes Long range dependence or memory long means ha observaions far away from h eacoher are sill srongly correlaed. The correlaion of long memory processes is decay slowly as lag daa increase ha is wih a hiperbollic rae. Mahemaics and Saisics (ICOMS-3) 3
INTRODUCTION(2) Granger and Joyeux(1980) Hosking(1981) Sowell (1992) -MLE- Geweke and Porer-Hudak(1983) - Mehod- Beran(1994) -NLS- Reisen(1994) - Mehod- Robinson(1995) - Mehod- Hurvich and Ray(1995) - Mehod- Velasco(1999b) - Mehod- Mahemaics and Saisics (ICOMS-3) 4
INTRODUCTION(3) COMPARISON OF REGRESSION SPECTRAL METHODS Lopes,Olberman and Reisen(2004) -Non saionary ARFIMA- Mehod is he bes Lopes and Nunes(2006) -ARFIMA(0,d,0)- Mehod is he bes Mahemaics and Saisics (ICOMS-3) 5
GOAL OF RESEARCH Comparing accuracy of specral regression esimaion mehods of erencing he diff parameer d) ( from saionary ARFIMA Model by Simulaion Sudy. Mahemaics and Saisics (ICOMS-3) 6
ARFIMA MODEL(1) An ARFIMA(p,d,q) model can be defined as foll ows: d ( B ) 1 B z = index of observaion ( = 1, 2,..., T) d = differencing parameer (rea l number) = mean of obervaion 2 a ~ NID 0, Mahemaics and Saisics (ICOMS-3) 7
ARFIMA MODEL(2) ( B) 1 2 p 1B2B.. pb is polinomial AR(p) ( B ) 1 B 2 q 1 2 q is polinomial MA(q) fracional differencing d d d k operaor 1 B k k 0 Mahemaics and Saisics (ICOMS-3) 8
Addiive Oulier in Time Series Daa Addiive Oulier is an even ha effecs a series for one ime period only Z X X X A O Τ A O Τ Τ I AO: ˆ 1, Τ AΤ Mahemaics and Saisics (ICOMS-3) 9
DIAGRAM *Generae Daa 1 : ARFIMA(1,d,0) and ARFIMA(0,d,1) d =0.2 and 0.4 *Generae Daa 2 : 1)ARFIMA(1,d,0) and ARFIMA(0,d,1) d =0.2 and 0.4 2) Add 5 addiive ouliers Esimae parameer d by 1. mehod 2. mehod 3. mehod 4. mehod 5. mehod Deermine Mean dan SD esimae From hese mehods Mahemaics and Saisics (ICOMS-3) 10
COMPARISON RESULT (1) d = 0.2 Wihou Ouliers d = 0,2 0.2 d = 0,2 0.2 d = 0,2 0.2 d = 0,2 0.2 - - - - - - - - ARFIMA(1,d,0), T= 300 ARFIMA(0,d,1), T =300 ARFIMA(1,d,0),T =1000 ARFIMA(0,d,1),T =1000 Wih Ouliers d = 0,2 0.2 d = 0,2 0.2 d = 0,2 0.2 d = 0,2 0.2 - - - - - - - - ARFIMA(1,d,0), T= 305 ARFIMA(0,d,1), T= 305 ARFIMA(1,d,0),T =1005 ARFIMA(0,d,1), T =1005 Mahemaics and Saisics (ICOMS-3) 11
COMPARISON RESULT (2) d = 0.4 Wihou Ouliers d = 0,4 0.4 d = 0,4 0.4 d = 0,4 0.4 d = 0,4 0.4 - - - - - - - - ARFIMA(1,d,0), T= 300 ARFIMA(0,d,1), T =300 ARFIMA(1,d,0),T =1000 ARFIMA(0,d,1),T =1000 Wih Ouliers d = 0,4 0.4 d = 0,4 0.4 d = 0,4 0.4 d = 0,4 0.4 - - - - - - - - ARFIMA(1,d,0), T= 305 ARFIMA(0,d,1), T= 305 ARFIMA(1,d,0),T =1005 ARFIMA(0,d,1), T =1005 Mahemaics and Saisics (ICOMS-3) 12
CONCLUSION 1) mehod shows he bes ormance perf in esimaing he differencing rameer pa wih value d = 0.2 of ARFIMA model for bohclean daa and daa wih oulier. 2) Esimaion of specral regress ion mehods are beer o be implemened odeling for m ARFIMA(1,d,0) daa han for m ARFIMA(0,d,1) daa. Mahemaics and Saisics (ICOMS-3) 13
THANK YOU.. Mahemaics and Saisics (ICOMS-3) 14
ESTIMATION OF THE DIFFERENCING PARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(1) 1. Consruc specral densiy funcion ( SDF) of ARFIMA model 2 2d 2 q exp( i ) a fz 2sin, 2, 2 exp( i ) 2 2. Take logarihms of SDF from ARFIMA model 2 f W j ln f 3 Z j ln f 0 dln 1 exp(i ) ln W j f 0 where W 2 j j, j 1,2,., T / 2 T Mahemaics and Saisics (ICOMS-3) 15
ESTIMATION OF THE DIFFERENCING PARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(2) 3. Add naural logarihm of periodogram o equaion (3) above Z j W 2 f W I j Z j i j 4 f 0 W f Z j 4. Deermine he periodogram specral mehods based on regression, r, ln I ln f 0 dln1 exp ln ln 1 g(t ) I Z j 0 2 j cos(. ), 2 1 Mahemaics and Saisics (ICOMS-3) 16
ESTIMATION OF THE DIFFERENCINGPARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(3) 1 g*(t ) IZ j 0 0 2 j 2 1 2 3 g * ( T ) 1 6 / g * ( T ) 6 / g * ( T ) 0 2 3 j g * ( T ) 2 1 g * ( T ) 2 a I 1 T 1 2 ap 2 Z j T 1 2 0 0 Mahemaics and Saisics (ICOMS-3) 17
ESTIMATION OF THE DIFFERENCINGPARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(4) 1 2 ap ( ) 1 cos 2 T 5 Esimae d by Ordinary Leas Square Me hod. Where, j Z j j j 2 Y lni, X Mahemaics and Saisics (ICOMS-3) 18