COMPARISON OF THE DIFFERENCING PARAMETER ESTIMATION FROM ARFIMA MODEL BY SPECTRAL REGRESSION METHODS. By Gumgum Darmawan, Nur Iriawan, Suhartono

Transcription

1 COMPARISON OF THE DIFFERENCING PARAMETER ESTIMATION FROM ARFIMA MODEL BY SPECTRAL REGRESSION METHODS By Gumgum Darmawan, Nur Iriawan, Suharono

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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, 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, 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

12 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, 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, 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

13 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

14 THANK YOU.. Mahemaics and Saisics (ICOMS-3) 14

15 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

16 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

17 ESTIMATION OF THE DIFFERENCINGPARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(3) 1 g*(t ) IZ j j g * ( T ) 1 6 / g * ( T ) 6 / g * ( T ) j g * ( T ) 2 1 g * ( T ) 2 a I 1 T 1 2 ap 2 Z j T Mahemaics and Saisics (ICOMS-3) 17

18 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