A.P. Ditago, Suhartono
SWUP
SC.15
- for the first case, the residual is not correlated in three locations:
, 00
, 1
00 ,
00 ,
00 ,
00 ,
1 00
, 00
, 00
, 00
, 1
= Ω
- for the second case, the residuals are correlated in three locations:
. 00
, 1
30 ,
40 ,
30 ,
00 ,
1 20
, 40
, 20
, 00
, 1
= Ω
Step 5: Determining the dummy variable for the period of calendar variations see Table 1.
Step 6: Perform parameter estimation model for first level using the OLS method, such as Eq. 1.
Step 7: Determining the spatial weights
W
are used. Step 8: Perform parameter estimation model for second level using the OLS and GLS
method, such as Eq. 2. Step 9: Calculate the efficiency of GLS method, with form
. 100
x ˆ
SE ˆ
SE ˆ
SE
OLS GLS
OLS
β β
β
−
Step 10: This phase is done by adding up the value of out-sample forecasting of the first and second level model.
3. Results and discussion
The first step is to identify the effects of calendar variation from plot time series for a specified period according to Table 1. Plot time series of data simulation the effects of
calendar variation shown in Figure 1.
Year Month
2010 2008
2006 2004
2002 2000
1998 1996
1994 1992
1990 Jan
Jan Jan
Jan Jan
Jan Jan
Jan Jan
Jan Jan
200 150
100 50
L o
k a
s i
1
A pr 90 A pr 91
A pr 92 M ar 93
M ar 94 M ar 95
F eb 96 F eb 97
Jan 98 Jan 99
Jan 00 D ec 00
D ec 01 D ec 02
N ov 03 N ov 04
N ov 05 O ct 06
O ct 07 O ct 08
Sep 09 Sep 10
9 8
9 8
10 9
10 9
10 9
11 10
11 10
11
10 12
11 12
11 12
11 1
12 1
12 1
12 2
1 2
1 3
2 3
2 3
2 4
3 4
3 4
3 Year
Month 2010
2008 2006
2004 2002
2000 1998
1996 1994
1992 1990
Jan Jan
Jan Jan
Jan Jan
Jan Jan
Jan Jan
Jan 140
120 100
80 60
40 20
L o
k a
s i
2
A pr 90 A pr 91
A pr 92 M ar 93
M ar 94 M ar 95
F eb 96 F eb 97
Jan 98 Jan 99
Jan 00 D ec 00
D ec 01 D ec 02
N ov 03 N ov 04
Nov 05 O ct 06
O ct 07 O ct 08
Sep 09 Sep 10
9 8
9 8
10 9
10 9
10
9 11
10 11
10 11
10 12
11 12
11 12
11 1
12 1
12 1
12 2
1 2
1 3
2 3
2 3
2 4
3 4
3 4
3
Year Month
2010 2008
2006 2004
2002 2000
1998 1996
1994 1992
1990 Jan
Jan Jan
Jan Jan
Jan Jan
Jan Jan
Jan Jan
80 70
60 50
40 30
20 10
L o
k a
s i
3
A pr 90 A pr 91
A pr 92 M ar 93
M ar 94 M ar 95
F eb 96 F eb 97
Jan 98 Jan 99
Jan 00 D ec 00
D ec 01 D ec 02
N ov 03 N ov 04
N ov 05 O ct 06
O ct 07 O ct 08
S ep 09 Sep 10
9 8
9 8
10 9
10 9
10 9
11 10
11 10
11 10
12 11
12 11
12
11 1
12 1
12 1
12 2
1 2
1 3
2 3
2 3
2 4
3 4
3 4
3
Figure 1. Time series plot of simulation data with the vector AR1 model.
Two-level parameter estimates GSTARX-GLS model
SWUP
SC.16
Figure 1 shows of the time series plot of the vector AR1 model with the data containing the effects of calendar variations. A vertical dotted line was included in this plot
to emphasize the months of Eid that occurred during this period. Stage one in modeling GSTARX is to estimate of parameters the first level. Such as the following results.
- Model for location 1
. 440
, 33
548 ,
45 823
, 49
283 ,
60 224
, 71
903 ,
79 571
, 90
684 ,
98 843
, 105
702 ,
116 863
, 126
634 ,
135 766
, 138
139 ,
145 698
, 155
080 ,
165 700
, 170
075 ,
180 453
, 193
178 ,
181 097
, 175
960 ,
163 563
, 152
775 ,
141 300
, 128
849 ,
116 003
, 110
889 ,
99 302
, 86
113 ,
76 747
, 69
142 ,
64 248
, 50
016 ,
37 371
, 33
393 ,
20
, 1
1 ,
29 1
, 27
1 ,
26 1
, 24
1 ,
22 1
, 20
1 ,
18 1
, 16
1 ,
15 1
, 13
1 ,
11 1
, 9
1 ,
8 1
, 7
1 ,
5 1
, 3
1 ,
2 1
, ,
29 ,
27 ,
26 ,
24 ,
22 ,
20 ,
18 ,
16 ,
15 ,
13 ,
11 ,
9 ,
8 ,
7 ,
5 ,
3 ,
2 ,
, 1
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
u D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D Y
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
=
− −
− −
− −
− −
− −
− −
− −
− −
− −
4
- Model for location 2
. 458
, 13
428 ,
18 733
, 21
763 ,
27 403
, 33
442 ,
41 232
, 47
652 ,
52 402
, 54
992 ,
61 158
, 67
089 ,
73 810
, 76
371 ,
80 693
, 84
449 ,
91 296
, 94
745 ,
98 196
, 132
653 ,
122 330
, 117
898 ,
109 161
, 104
874 ,
95 987
, 87
555 ,
78 243
, 75
474 ,
67 875
, 59
549 ,
52 717
, 46
394 ,
44 187
, 36
291 ,
27 357
, 22
722 ,
15
, 2
1 ,
29 1
, 27
1 ,
26 1
, 24
1 ,
22 1
, 20
1 ,
18 1
, 16
1 ,
15 1
, 13
1 ,
11 1
, 9
1 ,
8 1
, 7
1 ,
5 1
, 3
1 ,
2 1
, ,
29 ,
27 ,
26 ,
24 ,
22 ,
20 ,
18 ,
16 ,
15 ,
13 ,
11 ,
9 ,
8 ,
7 ,
5 ,
3 ,
2 ,
, 2
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
u D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D Y
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
=
− −
− −
− −
− −
− −
− −
− −
− −
− −
5
- Model for location 3
. 942
, 9
558 ,
16 888
, 15
820 ,
22 282
, 27
128 ,
31 304
, 35
231 ,
37 547
, 39
706 ,
42 713
, 48
560 ,
52 652
, 53
552 ,
56 777
, 59
286 ,
64 182
, 65
318 ,
70 915
, 67
409 ,
64 822
, 62
961 ,
58 500
, 55
610 ,
51 522
, 47
505 ,
42 077
, 41
226 ,
35 430
, 31
756 ,
27 345
, 26
099 ,
24 967
, 19
600 ,
17 601
, 14
054 ,
10
, 3
1 ,
29 1
, 27
1 ,
26 1
, 24
1 ,
22 1
, 20
1 ,
18 1
, 16
1 ,
15 1
, 13
1 ,
11 1
, 9
1 ,
8 1
, 7
1 ,
5 1
, 3
1 ,
2 1
, ,
29 ,
27 ,
26 ,
24 ,
22 ,
20 ,
18 ,
16 ,
15 ,
13 ,
11 ,
9 ,
8 ,
7 ,
5 ,
3 ,
2 ,
, 3
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
t t
u D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D D
D Y
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
=
− −
− −
− −
− −
− −
− −
− −
− −
− −
6
From the estimation parameters such as the first level of Eq. 4 to Eq. 6 obtained residual models i.e.,
t i
u
,
. Furthermore, the residual models used for estimation at second level, i.e., GSTAR models with parameter between locations spatial.
Characteristics of GSTAR models is weighted with the location. Spatial weighting method used in this study is limited only by inference Partial Correlation Normalized Cross
NIPKS. This method is based on the high or low value of the partial cross correlation between locations. Statistical inference process is done by using a 95 confidence interval.
A.P. Ditago, Suhartono
SWUP
SC.17
Table 2. Estimates normalization of cross correlation inference partial data simulation case one.
Parameter Estimates
95 confidence interval Conclusion
Lower Upper
P
12
1 0.186
0.073 0.300 Valid and concurrent
P
13
1 0.185
0.072 0.298 Valid and concurrent
P
21
1 0.212
0.099 0.326 Valid and concurrent
P
23
1 0.197
0.084 0.310 Valid and concurrent
P
31
1 0.140
0.027 0.253 Valid and concurrent
P
32
1 0.267
0.154 0.380 Valid and concurrent
Based on the calculation of the amount of the cross-correlation between the location at the time to lag 1, the process of inference statistics in Table 2 shows that the confidence
interval gives the same amount the relationship. Thus, the decision obtained are valid and comparable, it showed no difference in weighting between locations. Thus, the appropriate
weighting method in this case is uniform
. 5
. 5
. 5
. 5
. 5
. 5
.
=
W
7 For the second case, the weighting method is the same as Eq. 7. By using the weight
of locations the results of parameter estimation GSTAR1
1
model shown in Table 3.
Table 3. Comparison of parameter estimates from OLS and GLS method for the first and second case.
Case Parameter
OLS GLS
Efisiensi GLS
Estimasi SE
Estimasi SE
1
1 10
φ
0.283 0.060
0.299 0.060
0.000
1 11
φ
0.247 0.082
0.230 0.082
0.000
1 20
φ
0.175 0.061
0.169 0.061
0.000
1 21
φ
0.282 0.074
0.288 0.074
0.000
1 30
φ
0.329 0.061
0.335 0.061
0.000
1 31
φ
0.246 0.082
0.240 0.082
0.000
2
1 10
φ
0.271 0.067
0.254 0.062
6.625
1 11
φ
0.114 0.082
0.132 0.078
4.531
1 20
φ
0.161 0.064
0.176 0.062
3.416
1 21
φ
0.293 0.077
0.279 0.076
2.169
1 30
φ
0.143 0.070
0.133 0.064
8.185
1 31
φ
0.270 0.084
0.280 0.079
5.873
Table 3 shows the results of estimation GSTAR models there is a difference in the standard error of the estimation OLS with GLS method. The standard error of the GLS method
is smaller than the OLS, the difference is largely occurs on all parameters. For the second case can be stated that the parameter estimation using GLS better than OLS. This can be seen in
Two-level parameter estimates GSTARX-GLS model
SWUP
SC.18
almost all GLS efficiency coefficient is worth above five percent. In addition, comparison of the efficiency of the standard error of each parameter GSTAR model can also be shown
through the curve probability distribution function p.d.f in Figure 2.
0.5 0.4
0.3 0.2
0.1 7
6 5
4 3
2 1
psi10 D
e n
s it
y
0.3 O LS
GLS
0.4 0.3
0.2 0.1
0.0 -0.1
-0.2 5
4 3
2 1
psi11 D
e n
s it
y
0.1 O LS
GLS
0.4 0.3
0.2 0.1
0.0 7
6 5
4 3
2 1
psi20 D
e n
s it
y
0.2
OLS GLS
0.5 0.4
0.3 0.2
0.1 0.0
6 5
4 3
2 1
psi21 D
e n
s it
y
0.15
OLS GLS
0.4 0.3
0.2 0.1
0.0 7
6 5
4 3
2 1
psi30 D
e n
s it
y
0.25
O LS GLS
0.6 0.5
0.4 0.3
0.2 0.1
0.0 5
4 3
2 1
psi31 D
e n
s it
y
0.2
O LS GLS
Figure 2. Parameter distribution plot for
1 i
φ
left dan
1 1
i
φ
right with OLS and GLS
method a location 1, b location 2, dan c location 3.
Efficiency of each parameter estimation by using GLS method look more efficient than OLS method, it is marked on the shape of the curve p.d.f blue color a more narrow. In
addition, the vertical a dotted line shows the actual coefficient values of each parameter. Visually, the coefficient of parameter estimation approach with a true value. Furthermore,
for the first case GSTAR1
1
-OLS model can be written
+
−
− −
=
1
1 1
329 ,
123 ,
123 ,
141 ,
175 ,
141 ,
124 ,
124 ,
283 ,
3 2
1 3
2 1
3 2
1
t e
t e
t e
t u
t u
t u
t u
t u
t u
and GSTAR1
1
-GLS can be written
. 1
1 1
335 ,
120 ,
120 ,
144 ,
169 ,
144 ,
115 ,
115 ,
299 ,
3 2
1 3
2 1
3 2
1
+
−
− −
=
t
e t
e t
e t
u t
u t
u t
u t
u t
u
A.P. Ditago, Suhartono
SWUP
SC.19
As for the second case GSTAR1
1
-OLS model can be written
+
−
− −
=
1
1 1
143 ,
135 ,
135 ,
147 ,
161 ,
147 ,
057 ,
057 ,
271 ,
3 2
1 3
2 1
3 2
1
t e
t e
t e
t u
t u
t u
t u
t u
t u
and GSTAR1
1
-GLS can be written
. 1
1 1
133 ,
140 ,
140 ,
140 ,
176 ,
140 ,
066 ,
066 ,
254 ,
3 2
1 3
2 1
3 2
1
+
−
− −
=
t
e t
e t
e t
u t
u t
u t
u t
u t
u
4. Conclusion and remarks
