COMPARISON OF DIF F E RE NCING PARAME T E R
E ST IMAT ION F ROM NONST AT IONE R ARF IMA
MODE L BY GPH ME T H OD WIT H COSIN E T APE RING
By
INTRODUCTION (1)
TIME SERIES MODELS
BASED ON VALUE OF DIFFERENCING PARAMATER (d)
ARMA
d = 0
d
≠
0, d = INTEGER
ARIMA
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
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
GOAL OF RE SE ARCH
Comparing accuracy of GPH estimation
methods with cosine tapering of the
differencing parameter (d) and forecasting
result from nonstationary ARFIMA Model
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
t
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
1 2 32
0 . 5
1
(
)
1
c o s
2
2
1
1
(
)
1
c o s
2
1
2
(
)
0 . 5 4
0 . 4 6
c o s
1
t
t a p
t
T
t
t a p
t
T
t
t a p
t
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
Mean and Standard deviation of
parameter estimation
d
0, 199 0, 783 0, 176 0, 794 0, 167 0, 803 d=0, 8 0, 187 0, 687 0, 176 0, 677 0, 172 0, 695 d=0, 7 0, 162 0, 592 0, 174 0, 583 0, 175 0, 598 d=0, 6
ARFIMA(0,d, 1)
0, 202 0, 834 0, 172 0, 815 0, 179 0, 803 d=0, 8 0, 197 0, 716 0, 170 0, 705 0, 178 0, 712 d=0, 7 0, 168 0, 6111 0, 173 0. 602 0, 172 0, 611 d=0, 6
ARFIMA(1,d, 0) 1000 0, 191 0, 787 0, 201 0, 790 0, 203 0, 791 d=0, 8 0, 187 0, 696 0, 204 0, 685 0, 198 0, 687 d=0, 7 0, 192 0, 585 0, 205 0, 570 0, 201 0, 581 d=0, 6
ARFIMA(0,d, 1)
0, 189 0, 830 0, 193 0, 833 0, 198 0, 840 d=0, 8 0, 192 0, 728 0, 208 0, 731 0, 196 0, 717 d=0, 7 0, 195 0, 617 0, 200 0, 620 0, 195 0, 624 d=0, 6
ARFIMA(1,d, 0) 600
Hamming
d sd(d) Hanning
d sd(d) Cosine Bell
d sd(d)
GPH Est imat ion Met hod Wit h Cosine Taper ARFIMA Model Dat a
MSE of Forecasting (
h
= 10)
1. 622 1. 633 1. 633 d=0, 8 1. 215 1. 219 1. 219 d=0, 7 1. 258 1. 262 1. 262 d=0, 6ARFIMA(0,d, 1)
1. 346 1. 352 1. 352 d=0, 8 1. 317 1. 323 1. 323 d=0, 7 1. 360 1. 367 1. 367 d=0, 6
ARFIMA(1,d, 0) 600 1. 229 1. 233 1. 232 d=0, 8 1. 251 1. 255 1. 254 d=0, 7 1. 251 1. 254 1. 254 d=0, 6
ARFIMA(0,d, 1)
1. 336 1. 342 1. 342 d=0, 8 1. 316 1. 322 1. 322 d=0, 7 1. 311 1. 317 1. 316 d=0, 6
ARFIMA(1,d, 0) 300
Hamming Hanning
Cosine Bell
Cosine Taper ARFIMA Model Dat a
CONCLUSION
1)
GPH method with Cosine Bell Ta
pering 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
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
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
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)
GPHta
T 1
2
Z
j
T 1
2
t
j
t 0
t 0
1
I
tap
2
tap t
1 2 32
0.5
1
( )
1
cos
2
2
1
1
( )
1
cos
2
1
2
( )
0.54
0.46
cos
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
j
Z
j
j
j
