COMPARISON OF DIFFERENCING PARAMETER ESTIMATION FROM NONSTATIONER ARFIMA MODEL BY GPH METHOD WITH COSINE TAPERING. By Gumgum Darmawan

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1 COMPARISON OF DIFFERENCING PARAMETER ESTIMATION FROM NONSTATIONER ARFIMA MODEL BY GPH METHOD WITH COSINE TAPERING By Gumgum Darmawan

2 INTRODUCTION (1) TIME SERIES MODELS BASED ON VALUE OF DIFFERENCING PARAMATER (d) ARMA d = 0 ARIMA d 0, d = INTEGER ARFIMA d= REAL Short Memory Nonstationary Short Memory Long Memory and Technology (ISSTEC)

3 Long Memory Processes Long range dependence or memory long means that observations far away from h eacother are still strongly correlated. The correlation of long memory processes is decay slowly as lag data increase that is with a hiperbollic rate. and Technology (ISSTEC)

4 INTRODUCTION(2) Granger and Joyeux(1980) Hosking(1981) Sowell (1992) -MLE- Geweke and Porter-Hudak(1983) -GPH Method- Beran(1994) -NLS- Reisen(1994) -SPR Method- Robinson(1995) -GPHTr Method- Hurvich and Ray(1995) - GPHTa Method- Velasco(1999b) -MGPH Method- and Technology (ISSTEC)

5 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 and Technology (ISSTEC)

6 GOAL OF RESEARCH Comparing accuracy of GPH estimation methods with cosine tapering of the differencing parameter (d) and forecasting result from nonstationary ARFIMA Model by Simulation Study. and Technology (ISSTEC)

7 ARFIMA MODEL(1) An ARFIMA(p,d,q) model can be defined as foll ows: d ( B ) 1 B z t t t = index of observation (t = 1, 2,..., T) d = differencing parameter (rea l number) = mean of obervation 2 a t ~ NID 0, and Technology (ISSTEC)

8 ARFIMA MODEL(2) 2 p is polinomial AR(p) ( B) 11B 2B.. pb ( B ) 1 B 2 q 1 2 q is polinomial MA(q) d d k fractional differencing 1 B operator t a p t a p t a p k 0 d k 1 2 t 0. 5 ( ) t 1 c o s 2 T 1 2 t 1 ( ) t 1 c o s 2 T 1 2 t ( ) t c o s T 1 and Technology (ISSTEC)

9 DIAGRAM *Generate Data 1 : ARFIMA(1,d,0) and ARFIMA(0,d,1) d =0.6, 0.7 and 0.8 N = 600 and 1000 Estimate parameter d by GPH Method 1. Cosine Bell Taper 2. Hanning Taper 3. Hamming Taper Determine Mean dan SD estimate From these methods and Technology (ISSTEC)

10 Mean and Standard deviation of parameter estimation d GPH Estimation Method With Cosine Taper T ARFIMA Model Data Cosine Bell d sd(d) Hanning d sd(d) Hamming d sd(d) d=0,6 0,624 0,195 0,620 0,200 0,617 0,195 ARFIMA(1,d,0) d=0,7 0,717 0,196 0,731 0,208 0,728 0, d=0,8 d=0,6 0,840 0,581 0,198 0,201 0,833 0,570 0,193 0,205 0,830 0,585 0,189 0,192 ARFIMA(0,d,1) d=0,7 0,687 0,198 0,685 0,204 0,696 0,187 d=0,8 0,791 0,203 0,790 0,201 0,787 0,191 d=0,6 0,611 0, ,173 0,6111 0,168 ARFIMA(1,d,0) d=0,7 0,712 0,178 0,705 0,170 0,716 0, d=0,8 d=0,6 0,803 0,598 0,179 0,175 0,815 0,583 0,172 0,174 0,834 0,592 0,202 0,162 ARFIMA(0,d,1) d=0,7 0,695 0,172 0,677 0,176 0,687 0,187 d=0,8 0,803 0,167 0,794 0,176 0,783 0,199 and Technology (ISSTEC)

11 MSE of Forecasting ( h = 10) Cosine Taper T ARFIMA Model Data Cosine Bell Hanning Hamming d=0, ARFIMA(1,d,0) d=0, d=0,8 d=0, ARFIMA(0,d,1) d=0, d=0, d=0, ARFIMA(1,d,0) d=0, d=0,8 d=0, ARFIMA(0,d,1) d=0, d=0, and Technology (ISSTEC)

12 CONCLUSION 1) GPH method with Cosine Bell Tapering shows the best performance in ting estima the differencing parameter of ARFIMA(0,d,1) 2) From ARFIMA(1,d,0) data, GPH method with Hanning Taper is the best estimator all of method. 3) From forecasting result, Mean square Error (MSE) of GPH method with Hammi ng taper has the least value of all data types. and Technology (ISSTEC)

13 THANK YOU.. and Technology (ISSTEC)

14 ESTIMATION OF THE DIFFERENCING PARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(1) 1. Construct spectral density function ( SDF) of ARFIMA model 2 2d 2 q exp( i ) a fz 2sin, 2, 2 exp( i ) 2 2. Take logarithms 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 and Technology (ISSTEC)

15 ESTIMATION OF THE DIFFERENCING PARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(2) 3. Add natural logarithm of periodogram to equation (3) above Z j W 2 f W I j Z j i j 4 f 0 W f Z j 4. Determine the periodogram spectral methods based on regression GPH ln I ln f 0 dln1 exp ln ln 1 g(t ) I Z j 0 t 2 j cos(t. ), 2 t1 and Technology (ISSTEC)

16 ESTIMATION OF THE DIFFERENCINGPARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(3) GPHta I 1 T 1 2 tap t 2 Z j T 1 2 t j t 0 t t 0.5 tap1 ( t) 1 cos 2 T tap tap t 1 ( t) 1 cos 2 T 1 2 t ( t) cos T 1 tap and Technology (ISSTEC)

17 ESTIMATION OF THE DIFFERENCINGPARAMETER OF ARFIMA MODEL BY SPECTRAL REGRESSION METHOD(4) 5 Estimate d by Ordinary Least Square Me thod. Where, j Z j j j 2 Y lni, X and Technology (ISSTEC)