J.D. Kurnia, Setiawan, S.P. Rahayu
SWUP
SC.33
+
−
− −
+
=
1 1
1
3 2
1 3
2 1
33 22
11 33
32 31
23 22
21 13
12 11
30 20
10 3
2 1
t e
t e
t e
t Z
t Z
t Z
w w
w w
w w
w w
w t
Z t
Z t
Z φ
φ φ
φ φ
φ
i Comparing the results of model parameter estimation GSTARX-ols and GSTARX-SUR.
3. Results and discussion
Study of simulation in this study using the VAR 1
1
model which is then used to build the model GSTARX 1
1
with the parameters in the following equation coefficient matrix.
= 13
, 21
, 21
, 15
, 23
, 15
, 25
, 25
, 18
,
1
Φ
As described in the previous chapter that stage simulation studies conducted through six ways with each simulation consisted of two case studies. The first case study using the
order of the transfer function b = 1, s = 1, r = 0 and b = 1, s = 2, r = 0. For the simulation study used a matrix of partial normalization of cross correlation weighting. Results of the simulation
study 1 case study 1 with a residual value of between locations are not mutually correlated, the value of the partial normalization of cross correlation weighting worth valid and
comparable on all parameters which means a partial amount of the cross-correlation between the second and third location to the first location is equally great in the lag-1 , and
the value of the partial cross-correlation between the first and third location to the second location is equally great in the lag-1, as well as the value of the partial cross-correlation
between the first and the second location to a third location is equally great in the lag-1.
It is therefore appropriate weighting to simulate one second case study is uniform weighting. The weighting value used to form the residual become GSTARX model parameter
estimation in order to obtain results using the method of OLS and SUR in the following equations.
+
−
− −
− −
− +
− −
−
+
− −
−
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1
2 2
2 94
, 9
07 ,
10 16
, 10
1 1
1 94
, 14
97 ,
14 12
, 15
1 1
1 14
, 22
, 22
, 16
, 23
, 16
, 24
, 24
, 24
,
+
−
− −
− −
− +
− −
−
+
− −
−
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1
2 2
2 95
, 9
06 ,
10 15
, 10
1 1
1 96
, 14
96 ,
14 13
, 15
1 1
1 14
, 22
, 22
, 16
, 22
, 16
, 24
, 24
, 23
,
For the first simulation case study 2 also produces a weighted value of the partial normalized cross correlation is valid and comparable, therefore, be used to obtain a uniform
weighted residual value and the resulting value of the parameter estimates in the following equations.
+
−
− −
− −
− +
− −
−
−
− −
+ +
− −
−
+
− −
−
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1 3
2 1
3 3
3 04
, 20
06 ,
20 09
, 20
2 2
2 93
, 9
05 ,
10 13
, 10
1 1
1 94
, 14
97 ,
14 13
, 15
1 1
1 13
, 22
, 22
, 16
, 23
, 16
, 24
, 24
, 23
,
The simulation studies for Generalized Space Time Autoregressive-X GSTARX model
SWUP
SC.34
+
−
− −
− −
− +
− −
−
−
− −
+ +
− −
−
+
− −
−
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1 3
2 1
3 3
3 95
, 9
06 ,
10 15
, 10
2 2
2 95
, 9
06 ,
10 15
, 10
1 1
1 96
, 14
96 ,
14 13
, 15
1 1
1 13
, 22
, 22
, 16
, 22
, 16
, 24
, 24
, 23
,
Estimation of parameters in simulation 1 case study 1 and 2 by using the Estimation Method OLS and SUR generating parameter values estimated by OLS can be said to be not
much different or produce nearly all of the same value by using the GLS estimation method, as well as the resulting standard errors OLS and SUR. This means GSTARX-OLS model is as
good as GSTARX-SUR in cases where residual data between locations are not mutually correlated. The same thing is shown in simulation 2 case studies 1 and 2 where the residual
between locations is not correlated to produce standard error estimation parameters with the same value, which means GSTARX-OLS model is as good as GSTARX-SUR. The comparison
of standard errors between GLS and OLS in simulation 1 is presented in Table 1.
Table 1. Comparison standard error of OLS and GLS in simulation 1.
Parameter OLS
GLS estimasi
SE estimasi
SE
Case study 1 psi10
0.24 0.05
0.23 0.05
psi20 0.23
0.05 0.22
0.05 psi30
0.14 0.06
0.14 0.05
psi11 0.47
0.08 0.48
0.08 psi21
0.32 0.07
0.32 0.07
psi31 0.43
0.07 0.43
0.07 w10
15.12 0.06
15.13 0.06
w20 14.97
0.06 14.96
0.06 w30
14.94 0.06
14.96 0.06
w11 –10.16
0.06 –10.15
0.06 w21
–10.07 0.06
–10.06 0.06
w31 –9.94
0.06 –9.95
0.06
Parameter OLS
GLS estimasi
SE estimasi
SE
Case study 2 psi10
0.23 0.05
0.23 0.05
psi20 0.23
0.05 0.22
0.05 psi30
0.13 0.06
0.13 0.05
psi11 0.47
0.08 0.48
0.08 psi21
0.32 0.07
0.32 0.07
psi31 0.44
0.07 0.44
0.07 w10
15.13 0.06
15.13 0.06
w20 14.97
0.06 14.96
0.06 w30
14.94 0.06
14.96 0.06
w11 –10.13
0.07 –10.13
0.07 w21
–10.05 0.07
–10.05 0.07
w31 –9.93
0.06 –9.94
0.06 w12
–20.09 0.06
–20.07 0.06
w22 –20.06
0.06 –20.05
0.06 w32
–20.04 0.06
–20.05 0.06
As for the simulation of 3, 4, 5, and 6 for the residual between locations correlated produce GSTARX-SUR model is more efficient than the GSTARX-OLS because it produces a
J.D. Kurnia, Setiawan, S.P. Rahayu
SWUP
SC.35
standard error of estimate parameter values are smaller. In the third simulation case study 1 generated value weighted partial normalization of cross correlation are valid and
comparable, and therefore a uniform weighting may be applied to this case resulting parameter estimates OLS and SUR in the following equations.
+
−
− −
− −
− +
+
−
− −
+
−
− −
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1
2 2
2 920
, 9
960 ,
9 880
, 9
1 1
1 010
, 15
980 ,
14 160
, 15
1 1
1 130
, 229
, 229
, 159
, 278
, 159
, 304
, 304
, 160
,
+
−
− −
− −
− +
+
−
− −
+
−
− −
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1
2 2
2 970
, 9
990 ,
9 960
, 9
1 1
1 040
, 15
997 ,
14 060
, 15
1 1
1 171
, 210
, 210
, 165
, 267
, 165
, 317
, 317
, 136
,
Simulation 3 in the case study 2 also produces a weighted value of the partial normalized cross correlation is valid and comparable, therefore, be used to get a uniform
weighted residual value and the resulting value of the parameter estimates in the following equations.
+
−
− −
− −
− +
− −
−
−
− −
+ +
− −
−
+
− −
−
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1 3
2 1
3 3
3 950
, 19
822 ,
19 917
, 19
2 2
2 946
, 9
031 ,
10 891
, 9
1 1
1 014
, 15
970 ,
14 149
, 15
1 1
1 129
, 231
, 231
, 163
, 270
, 163
, 303
, 303
, 163
,
+
−
− −
− −
− +
− −
−
−
− −
+ +
− −
−
+
− −
−
=
t e
t e
t e
t X
t X
t X
t X
t X
t X
t X
t X
t X
t z
t z
t z
t z
t z
t z
3 2
1 3
2 1
3 2
1 3
2 1
3 2
1 3
2 1
3 3
3 024
, 20
915 ,
19 999
, 19
2 2
2 947
, 9
017 ,
10 959
, 9
1 1
1 030
, 15
999 ,
14 058
, 15
1 1
1 171
, 211
, 211
, 168
, 260
, 168
, 317
, 317
, 137
,
In the third simulation case studies 1 and 2 where the residual data between locations are correlated to produce standard error of estimate parameter values that are smaller in
GSTARX-SUR Model compared with GSTARX-OLS. This means that the model is more efficient GSTARX-SUR applied to the case where correlated residuals between sites. Value Model
GSTARX-SUR efficiency can be seen in Table 2. Simulation of 4, 5, and 6 to the same conclusion as in the simulation 3.
4. Conclusion
