A.P. Ditago, Suhartono
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applied in the case of gross investment demand in the two companies Zellner, 1962. The results obtained are the estimated parameters by GLS for the overall model is more efficient
than the OLS parameter estimates for each model. In addition, SUR models are also applied to the spatio-temporal domains Wang Kockelman, 2007. SUR models were applied to
estimate the parameters GSTAR provide assurance that the error of the model is a multivariate white noise Wutsqa Suhartono, 2010.
Along with its development, GSTAR model can be expanded to GSTARX. In this case X is a notation for a predictor or input. Predictors can be a metric and or non-metric scale form.
For this form metric, predictor will be conducted by transfer function model, whereas for the form of non-metric conducted by dummy variables. In the case of non-metric, the variable
may be the effect of the intervention, outliers and calendar variations.
This research will be used predictor of the calendar variation model to capture of Ramadhan effects. Implementation of the GSTARX model in this study will be discussed
through a simulation study with the aim to get the right model building procedure according with the conditions of real data.
2. Materials and methods
Estimation process in GSTAR can be done with two methods of estimation, i.e., OLS and GLS. For example, applied in GSTAR1
1
for N locations, it can be written
,
i i
i i
e β
X Z
+ =
where
, ,...,
,
1 11
10 N
N i
φ φ
φ φ
= β
. In the matrix form, can be written
, 2
1 2
1
1 1
1 1
1 1
1 1
2 1
2 1
1 1
1
1 11
10 1
1 1
1 1
1
1 1
1
+
− −
− −
=
T e
e e
T e
e e
T V
T z
V z
V z
T V
T z
V z
V z
T z
z z
T z
z z
N N
N N
N
N N
N N
N N
N N
N
M M
M M
L M
M O
M M
L L
M M
O M
M L
M M
M M
M L
L
M M
M
φ φ
φ φ
where
∑
≠
=
i j
j ij
i
t Z
w t
V
. Parameter estimation of β conducted using OLS method by means
of minimizing Xβ
Z e
− =
, so the estimator for
, ,...,
,
1 11
10 N
N i
φ φ
φ φ
= β
is obtained
. ˆ
1
Z X
X X
β
−
=
Whereas estimates for GLS is obtained by minimizing generalized sum of square
ε Ω
ε
1 −
, where
Xβ Y
ε −
=
, so that the equation
1 1
Xβ Y
Ω Xβ
Y ε
Ω ε
− −
=
− −
then do decrease to the parameters, so that would be obtained estimator
, ˆ
1 1
1
Y Ω
X X
Ω X
β
− −
−
=
1 where
I Σ
Ω ⊗
=
,
Two-level parameter estimates GSTARX-GLS model
SWUP
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=
NN N
N N
N
σ σ
σ σ
σ σ
σ σ
σ
L M
O M
M L
L
1 1
2 22
21 1
12 11
Σ
dan
= 1
1 1
L M
O M
M L
L I
,
in which Σ is a error variance-covariance matrix size N x N and
I
is a identity matrix of size T x T.
In this study, parameter estimation of the GSTARX model will be done in two-level. Models for first level, i.e., the regression models with calendar variation:
, ,
, 1
, ,
, t
i t
g t
g t
i
u D
D f
Y +
+ =
−
β
2 where
∑ ∑
− −
+ =
g g,t
g g
g,t g
t g
t g
D γ
D α
D D
f
1 1
, ,
,
is the total calendar variation effects,
g,t
D
and
1 −
g,t
D
respectively represents dummy variable for the during month Eid and one month before Eid, g indicate the number of days prior to the date of Eid,
t i
u
,
is the error component, and i is a notation for number of location. Models for second level, i.e., GSTAR models:
. 1
1 1
1 1
1
3 2
1 3
2 1
32 31
23 21
13 12
31 21
11 3
2 1
30 20
10 3
2 1
+
−
− −
+
−
− −
=
t
e t
e t
e t
u t
u t
u w
w w
w w
w t
u t
u t
u t
u t
u t
u
φ φ
φ φ
φ φ
3 From Eq. 3, the models for second level can be written in the form
, 1
1 1
3 2
1 3
2 1
33 32
31 23
22 21
13 12
11 3
2 1
+
−
− −
=
t
e t
e t
e t
u t
u t
u t
u t
u t
u
φ φ
φ φ
φ φ
φ φ
φ
where
i ii
φ φ =
, for i = 1, 2, 3 and
1 i
ij ij
w
φ φ
=
, for i, j = 1, 2, 3 where i ≠ j. Stages of simulation study carried out in two-level GSTARX models are as follows.
Step 1: Determine the effects of calendar variation during the specified period.
Table 1. Eid celebration for the period 1990 to 2010. Year
Date Year
Date Year
Date
1990 1991
1992 1993
1994 1995
1996 1997
27-28 April 16-17 April
4-5 April 25-26 March
14-15 March 3-4 March
21-22 February
9-10 February 1998
1999 2000
2001 2002
2003 2004
30-31 January 19-20 January
8-9 January 28-29 December
17-18 December 6-7 December
25-26 November 14-15 November
2005 2006
2007 2008
2009 2010
3-4 November 23-24 October
12-13 October 1-2 October
21-22 September 10-11 September
Step 2: Determine coefficient parameters of the vector AR1 models:
. 25
, 20
, 20
, 15
, 20
, 15
, 10
, 10
, 30
,
1
= Φ
Step 3: Generate residual at three locations multivariate normal distribution with a mean of zero and variance covariance matrix
Ω
. Step 4: Determining two simulation scenarios.
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
