PROS Junta DK, Setiawan, Santi PR The simulation studies fulltext
Proceedings of the IConSSE FSM SWCU (2015), pp. SC.30–37
SC.30
ISBN: 978-602-1047-21-7
The simulation studies for Generalized Space Time Autoregressive-X
(GSTARX) model
Junta Dwi Kurniaa, Setiawanb, Santi Puteri Rahayuc
a, b, c
Department of Statistics, Institut Teknologi Sepuluh Nopember Surabaya,
Kampus ITS Sukolilo Surabaya 60111, Indonesia
Abstract
Generalized Space Time Autoregressive-X (GSTARX) is a model that involve the predictor
variable (X) introduced by Pfeifer dan Deutsch. Generalized Space Time Autoregressive
(GSTAR) is one of multivariate time series models that combine elements of time and
location or spatial data or time series. X Variable in GSTAR is a symbol that has a metric
and non-metric scale. For the case of univariate time series using the predictor X with
metric scale called the Transfer Function Model, while for non-metric scale called the
Intervention Model and Calendar Variations. The literature studies showed that studies
regarding the approach of multivariate time series by using GSTAR-X is still limited to
models involving variable X with non-metric scale, so that in this research restricted use
a variable X with a metric scale. GSTAR-X estimation method for using the Generalized
Least Square (GLS), as well as the estimation method on the model Seemingly Unrelated
Regression (SUR) that introduced by Zellner. The purpose of this research is to obtain a
parameter estimation from GSTAR-X model with simulation study. Results of the
simulation study showed that, if the residual of simulation are correlated, it will generate
a error standard of parameters estimate values are small in GSTARX-SUR model than
GSTARX-OLS so it can be said that the parameter estimation using GSTARX-SUR is more
efficient than GSATRX-OLS.
Keywords GSTARX-SUR, GSTARX-OLS, metric, predictor
1.
Introduction
GSTAR is one of multivariate time series models that involve more than one response
and correlated. GSTAR is the development of models Space Time Autoregressive (STAR)
introduced by Pfeifer & Deutsch (1980). This model is a model that combines elements with
the elements of the spatial dependency of time or location. STAR model itself is a
development of the Model Vector Autoregressive Integrated Moving Average (VARIMA), but
the VARIMA model has not been paying attention time with spatial dependencies. Therefore
developed a method that combines elements of time and location dependencies multivariate
with spatially heterogeneous elements which was then called the method GSTAR (Ruchjana,
2002)
GSTAR method involving the predictor variables called GSTARX. Variable X in GSTAR is
a symbol that has a metric and non-metric scale. For the case of univariate time series using
the predictor X with metric scale called the Transfer Function Model, while for non-metric
scale called the Intervention Model and Variations Calendar. Research on Transfer Function
Model can be seen in Wu & Tsay (2003) on the role of test statistics on a limited sample case
through simulations using Transfer Function Model. For research on intervention models
have been widely applied, one by Suhartono (2007) on the effect of the first Bali bombing
against a five-star hotel occupancy. While one study on Variation Model Calendar conducted
by Lee et al. (2010) for the sales data male Muslim clothing by adding the effect of Ramadhan.
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J.D. Kurnia, Setiawan, S.P. Rahayu
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While research has been conducted by GSTARX (Suhartono et al., 2015) concerning GSTARX
model for forecasting the data spatio temporal in the case of inflation of four cities in East
Java with X-scale non-metric, ie Eid events and factors rise in fuel prices, as well as research
by Oktanindya (2014) regarding the intervention model GSTARX and a step pulse is applied
to the case of foreign tourists forecasting. Studies of multivariate time series approach using
GSTARX is still limited to models involving variable X with non-metric scale.
GSTARX estimation method using the GLS, as well as the estimation method on the
model equations SUR. Ordinary Least Square method (OLS) can not be used for multivariate
model consisting of multiple equations that are correlated because it will produce a
estimator is less efficient, in the sense that the resulting variance would be very large.
Based on the description that has been described above, in this study will be conducted
further studies on multivariate time series model with variable X metric using GLS estimation.
The aim of this study is to obtain estimates of the model parameters GSTARX through
simulation studies.
2.
Materials and methods
2.1 Multivariate time series
Time series analysis used in data that have dependencies time where there is a
relationship between the occurrence of a period with the previous period. At the time series
analysis has the period or the same observation interval (Wei, 2006). Time series analysis
involves only a single event or a phenomenon called the univariate time series analysis, while
involving some event or phenomenon which occurs correlation or relationship between the
incidence of one another called multivariate time series analysis. Similarly in the analysis of
univariate time series, multivariate time series analysis to also pay attention to stationary
which can be seen on the plot Matrix Cross Correlation Function (MCCF) and plot Matrix
Partial Cross Correlation Function (MPCCF).
One model is a multivariate time series model VARMA that can generally be written
into the form of the following equation.
Φ p (B )Z (t ) = Θ q (B )a(t ) ,
Where Z(t ) is a vector with multivariate time series,
Θ q (B )
Φ p (B )
autoregressive order p matrix, and
is a polynomial moving average order q.
2.2 GSTARX models
GSTAR a generalization of STAR models. Difference between STAR models with GSTAR
is autoregression parameter in the model STAR assumed to be equal to any location, while
the autoregression parameter of GSTAR be different for each location and the difference
between the location shown in the form of weighting matrix (Borovkova et al., 2008). GSTAR
in the form of a matrix is given by
λs
p
Z (t ) = ∑ Φ s 0 + ∑ Φ sk W ( k ) Z (t − s ) + e (t )
s =1
k =1
GSTAR model with one order of time and spatial order for three different locations is given
by
Z (t ) = [Φ 10 + Φ 11 W (1) ]Z (t − 1) + e (t )
that can be presented in the form of a matrix:
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The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model
0 φ11 0
0 0
z1 (t ) φ10 0
z (t ) = 0 φ
0 + 0 φ 21 0 w21
20
2
z3 (t ) 0
0 φ30 0
0 φ31 w31
w12
0
w32
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w13 z1 (t − 1) e1 (t )
w23 z 2 (t − 1) + e2 (t )
0 z3 (t − 1) e3 (t )
To determine the order of time in the model can be used AIC criteria, whereas for the
spatial order is generally limited to only order one course because of the higher order will be
difficult to interpret (Wutsqa et al., 2010).
Weighting on GSTAR there are four, namely uniform weight, inverse distance,
normalized cross correlation and inference partial normalization of cross correlation
(Suhartono & Atok, 2006).
2.3 Parameter estimates
The OLS estimators β is are as follows:
βˆ = (X′X)-1 X′Y .
Whereas the form of parameter estimate from GLS estimator is (Park, 1967):
βˆ = (X' ΩX) −1 X' Ω −1 Y ,
where Ω −1 = Σ −1 ⊗ I so the above equation will be:
βˆ = (X' Σ −1 ⊗ IX ) −1 X' Σ −1 ⊗ IY .
2.4 Methods
GSTARX-OLS and GSTARX- SUR weighted cross correlation normalized partial
correlation. Steps for simulation study are as follows.
a) Generating the data xt and yt for 3 locations with n = 300 multivariate normal
distribution with a mean of zero and variance covariance matrix Ω .
b) Determining the value of coefficient parameters used in the model GSTARX (11) with a
stationary condition.
c) Applying steps a and b in six simulations, i.e.,
1) Simulation 1 for residual between locations is not correlated with the same
variance.
2) Simulation 2 to residual between locations does not correlate with different
variances.
3) Simulation 3 for residual between locations all correlated with the same variance.
4) Simulation 4 for residual between locations is not all correlated with the same
variance.
5) Simulation 5 for residual between locations all correlated with different variances.
6) Simulation 6 to residual between locations all correlated with different variances.
d) Evaluating order ARIMA residuals.
e) Getting series yit and xit to 3 locations.
f) Incorporating order transfer function for each simulation
1) Case study 1 using the order of (b = 1, s = 1, r = 0) into the equation
y it = (ω 0 − ω 1 Β ) X t −1 + e it .
2) Case study 2 using the order of (b = 1, s = 2, r = 0) into the equation
y it = (ω 0 − ω 1 Β + ω 2 B 2 ) X t −1 + e it .
g) GSTARX-OLS model building and GLS.
h) Getting the model parameter estimation GSTARX-OLS and GSTARX-SUR.
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J.D. Kurnia, Setiawan, S.P. Rahayu
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Z 1 (t ) φ10
Z (t ) = 0
2
Z 3 (t ) 0
0
φ 20
0
0 w11
0 + w21
φ 30 w31
w12
w22
w32
w13 φ11 0
w23 0 φ 22
0
w33 0
Z 1 (t − 1) e1 (t )
Z (t − 1) + e (t )
2
2
φ 33 Z 3 (t − 1) e3 (t )
0
0
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 (11)model which is then used to build
the model GSTARX (11) with the parameters in the following equation coefficient matrix.
0,18 0,25 0,25
Φ 1 = 0,15 0,23 0,15
0,21 0,21 0,13
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.
0
0 X 1 (t − 1) − 10,16
0
0 X 1 (t − 2) e1 (t )
z1 (t ) 0,24 0,24 0,24 z1 (t − 1) 15,12
− 10,07
14,97
0 X 2 (t − 1) + 0
0 X 2 (t − 2) + e2 (t )
z 2 (t ) = 0,16 0,23 0,16 z 2 (t − 1) + 0
z (t ) 0,22 0,22 0,14 z (t − 1) 0
− 9,94 X 3 (t − 2) e3 (t )
0
14,94 X 3 (t − 1) 0
0
3
3
0
0 X 1 (t − 1) − 10,15
0
0 X 1 (t − 2) e1 (t )
z1 (t ) 0,23 0,24 0,24 z1 (t − 1) 15,13
z
t
0
,
16
0
,
22
0
,
16
z
t
−
1
0
14
,
96
0
X
(
t
−
1
)
+
0
−
10
,
06
0 X 2 (t − 2) + e2 (t )
(
)
=
(
)
+
2
2
2
z (t ) 0,22 0,22 0,14 z (t − 1) 0
0
14,96 X 3 (t − 1) 0
0
− 9,95 X 3 (t − 2) e3 (t )
3
3
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.
0 , 24
0 , 23
z 1 (t )
0 , 23
z 2 (t ) = 0 ,16
0 , 22
z (t )
0 , 22
3
0
− 10 ,13
0
+
− 10 , 05
0
0
−
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0 , 24 z 1 ( t − 1 )
0
0
15 ,13
0 ,16 z 2 (t − 1 ) +
0
14 , 97
0
0 ,13 z 3 (t − 1 )
0
0
14 , 94
0
0
− 20 , 09
X 1 (t − 2 )
0
0
− 20 , 06
X 2 (t − 2 ) +
9 , 93 X 3 ( t − 2 )
0
0
X 1 (t
X 2 (t
X (t
3
0
− 1)
− 1) +
− 1 )
X
0
X
− 20 , 04 X
(t − 3 )
e 1 (t )
(
3
)
+
−
t
e 2 (t )
2
3 (t − 3 )
e 3 (t )
1
The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model
0 , 24
z 1 (t )
0 , 23
0 , 22
z 2 (t ) = 0 ,16
z (t )
0 , 22
0 , 22
3
0
− 10 ,15
+
− 10 , 06
0
−
0
0
0 , 24 z 1 ( t − 1 )
15 ,13
0
14
0 ,16 z 2 (t − 1 ) +
0
0 ,13 z 3 (t − 1 )
0
− 10 ,15
X 1 (t − 2 )
0
0
X 2 (t − 2 ) +
9 , 95 X 3 ( t − 2 )
0
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X 1 (t − 1)
X 2 (t − 1) +
0
14 , 96 X 3 ( t − 1 )
0
0
X 1 (t − 3 )
e 1 (t )
− 10 , 06
0
X 2 ( t − 3 ) + e 2 (t )
e (t )
− 9 , 95 X 3 ( t − 3 )
0
3
0
, 96
0
0
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.
OLS
GLS
Parameter
estimasi
SE
estimasi
SE
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
Case study 1
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
OLS
GLS
Parameter
estimasi
SE
estimasi
SE
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
Case study 2
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
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J.D. Kurnia, Setiawan, S.P. Rahayu
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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.
0 , 304
0 , 304 z 1 ( t − 1 )
0
0
z 1 (t )
0 ,160
15 ,160
X 1 ( t − 1)
0 , 278
0 ,159 z 2 (t − 1 ) +
0
14 , 980
0
z 2 (t ) = 0 ,159
X 2 (t − 1) +
z (t )
0 , 229
0 , 229
0 ,130 z 3 (t − 1 )
0
0
15 , 010 X 3 ( t − 1 )
3
0
0
− 9 , 880
X 1 (t − 2 )
e 1 (t )
+
− 9 , 960
0
0
X 2 ( t − 2 ) + e 2 (t )
e (t )
− 9 , 920 X 3 ( t − 2 )
0
0
3
0 , 317
0 , 317
z 1 (t )
0 ,136
0 , 267
0 ,165
z 2 (t ) = 0 ,165
z (t )
0 , 210
0 , 210
0 ,171
3
−
9
,
960
0
0
+
− 9 , 990
0
0
− 9 , 970
0
0
z 1 ( t − 1 ) 15 , 060
0
z 2 (t − 1 ) +
z (t − 1 )
0
3
X 1 ( t − 2 ) e 1 (t )
X 2 ( t − 2 ) + e 2 (t )
X 3 ( t − 2 ) e 3 (t )
0
14 , 997
0
X 1 ( t − 1)
X 2 (t − 1) +
15 , 040 X 3 ( t − 1 )
0
0
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.
0
0
0 , 303 0 , 303 z 1 ( t − 1 ) 15 ,149
z 1 (t ) 0 ,163
X 1 ( t − 1)
(
)
(
)
+
=
−
0
14
,
970
0
1
0
,
163
0
,
270
0
,
163
z
t
z
t
2
X 2 ( t − 1) +
2
z (t ) 0 , 231 0 , 231 0 ,129 z (t − 1 ) 0
0
15 , 014 X 3 ( t − 1)
3
3
0
0
0
0
X 1 ( t − 3 ) e 1 (t )
− 9 , 891
X 1 ( t − 2 ) − 19 , 917
0
19
,
822
0
(
2
)
+
−
−
0
0
+
− 10 , 031
X
t
X 2 ( t − 3 ) + e 2 (t )
2
0
0
− 19 , 950 X 3 ( t − 3 ) e 3 (t )
0
0
− 9 , 946 X 3 ( t − 2 )
0
0 X 1 ( t − 1)
z1 (t ) 0 ,137 0 ,317 0 ,317 z1 ( t − 1) 15 ,058
14 ,999
0 X 2 (t − 1) +
z 2 (t ) = 0 ,168 0 , 260 0 ,168 z 2 (t − 1) + 0
z (t ) 0 , 211 0 , 211 0 ,171 z (t − 1) 0
0
15 ,030 X 3 ( t − 1)
3
3
0
0
0
0
− 9 ,959
X 1 ( t − 2 ) − 19 ,999
X 1 ( t − 3 ) e1 (t )
+
− 10 , 017
− 19 ,915
0
0
0
0
X 2 (t − 2 ) +
X 2 ( t − 3) + e 2 (t )
− 9 ,947 X 3 ( t − 2 )
− 20 ,024 X 3 ( t − 3) e 3 (t )
0
0
0
0
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
The results of simulation studies with a transfer function GSTARX Model with X is a
metric variable can be concluded, that if the residuals from the simulated data are not
mutually correlated between locations, the model GSTARX-OLS and GSTARX-SUR will
generate a standard error of parameter estimate value are the same. However, if the residual
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The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model
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of simulation data are correlated, it will generate a standard error of parameter estimate
values are small in GSTARX-SUR Model compared with GSTARX-OLS Model. So it can be said
that the parameter estimation using GSTARX-SUR Model more efficient (smaller standard
errors) compared with GSTARX-OLS for residual cases of simulation data are correlated.
Case
study 1
Case
study 1
Table 2. Efficiency value of GSTARX-SUR model in simulation 3.
OLS
GLS
Efisiensi GLS
Parameter
(%)
estimasi
SE
estimasi
SE
psi10
0,160
0,066
0,136
0,055
17,015
psi20
0,278
0,079
0,267
0,061
22,742
psi30
0,130
0,074
0,171
0,059
20,416
psi11
0,608
0,077
0,634
0,066
14,414
psi21
0,318
0,086
0,330
0,069
19,374
psi31
0,457
0,074
0,420
0,062
16,736
w10
15,160
0,080
15,060
0,056
30,000
w20
14,980
0,074
14,997
0,050
32,432
w30
15,010
0,072
15,040
0,050
30,556
w11
-9,880
0,080
-9,960
0,057
28,750
w21
-9,960
0,074
-9,990
0,050
32,432
w31
-9,920
0,072
-9,970
0,049
31,944
OLS
GLS
Efisiensi GLS
Parameter
(%)
estimasi
SE
estimasi
SE
psi10
0,163
0,066
0,137
0,055
16,970
psi20
0,270
0,079
0,260
0,061
22,748
psi30
0,129
0,074
0,171
0,059
20,430
psi11
0,606
0,077
0,634
0,066
14,382
psi21
0,325
0,085
0,336
0,069
19,343
psi31
0,461
0,075
0,422
0,062
16,772
w10
15,149
0,080
15,058
0,057
28,750
w20
14,970
0,074
14,999
0,050
32,432
w30
15,014
0,072
15,030
0,050
30,556
w11
-9,891
0,081
-9,959
0,057
29,630
w21
-10,031
0,079
-10,017
0,052
34,177
w31
-9,946
0,083
-9,947
0,056
32,530
w12
-19,917
0,080
-19,999
0,057
28,750
w22
-19,822
0,074
-19,915
0,050
32,432
w32
-19,950
0,072
-20,024
0,049
31,944
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Suhartono, Wahyuningrum, S.R., Setiawan, & Akbar, M.S. (2015). GSTARX-GLS model for spatio
temporal data forecasting. Proceedings of Malaysian Journal of Mathematical Science, Malaysia.
Wei, W.W.S. (2006). Time series analysis: Univariate and multivariate methods. United State of
America: Addison-Wesley Publishing Co..
Wu, C.S., & Tsay, R.S. (2003). Forecasting with leading indicators revisited. Journal of Forecasting,
22(8), 603–617.
Wutsqa, D.U., Suhartono, & Sutijo, B. (2010). Generalized space-time autoregressive modeling.
Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Application (ICMSA
2010), University Tunku Abdul Rahman, Malaysia.
Zellner, A. (1962). An efficient method of estimating seemingly unrelated regression and test of
aggregation bias. Journal of the American Statistical Association, 57, 348–368.
SWUP
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ISBN: 978-602-1047-21-7
The simulation studies for Generalized Space Time Autoregressive-X
(GSTARX) model
Junta Dwi Kurniaa, Setiawanb, Santi Puteri Rahayuc
a, b, c
Department of Statistics, Institut Teknologi Sepuluh Nopember Surabaya,
Kampus ITS Sukolilo Surabaya 60111, Indonesia
Abstract
Generalized Space Time Autoregressive-X (GSTARX) is a model that involve the predictor
variable (X) introduced by Pfeifer dan Deutsch. Generalized Space Time Autoregressive
(GSTAR) is one of multivariate time series models that combine elements of time and
location or spatial data or time series. X Variable in GSTAR is a symbol that has a metric
and non-metric scale. For the case of univariate time series using the predictor X with
metric scale called the Transfer Function Model, while for non-metric scale called the
Intervention Model and Calendar Variations. The literature studies showed that studies
regarding the approach of multivariate time series by using GSTAR-X is still limited to
models involving variable X with non-metric scale, so that in this research restricted use
a variable X with a metric scale. GSTAR-X estimation method for using the Generalized
Least Square (GLS), as well as the estimation method on the model Seemingly Unrelated
Regression (SUR) that introduced by Zellner. The purpose of this research is to obtain a
parameter estimation from GSTAR-X model with simulation study. Results of the
simulation study showed that, if the residual of simulation are correlated, it will generate
a error standard of parameters estimate values are small in GSTARX-SUR model than
GSTARX-OLS so it can be said that the parameter estimation using GSTARX-SUR is more
efficient than GSATRX-OLS.
Keywords GSTARX-SUR, GSTARX-OLS, metric, predictor
1.
Introduction
GSTAR is one of multivariate time series models that involve more than one response
and correlated. GSTAR is the development of models Space Time Autoregressive (STAR)
introduced by Pfeifer & Deutsch (1980). This model is a model that combines elements with
the elements of the spatial dependency of time or location. STAR model itself is a
development of the Model Vector Autoregressive Integrated Moving Average (VARIMA), but
the VARIMA model has not been paying attention time with spatial dependencies. Therefore
developed a method that combines elements of time and location dependencies multivariate
with spatially heterogeneous elements which was then called the method GSTAR (Ruchjana,
2002)
GSTAR method involving the predictor variables called GSTARX. Variable X in GSTAR is
a symbol that has a metric and non-metric scale. For the case of univariate time series using
the predictor X with metric scale called the Transfer Function Model, while for non-metric
scale called the Intervention Model and Variations Calendar. Research on Transfer Function
Model can be seen in Wu & Tsay (2003) on the role of test statistics on a limited sample case
through simulations using Transfer Function Model. For research on intervention models
have been widely applied, one by Suhartono (2007) on the effect of the first Bali bombing
against a five-star hotel occupancy. While one study on Variation Model Calendar conducted
by Lee et al. (2010) for the sales data male Muslim clothing by adding the effect of Ramadhan.
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J.D. Kurnia, Setiawan, S.P. Rahayu
SC.31
While research has been conducted by GSTARX (Suhartono et al., 2015) concerning GSTARX
model for forecasting the data spatio temporal in the case of inflation of four cities in East
Java with X-scale non-metric, ie Eid events and factors rise in fuel prices, as well as research
by Oktanindya (2014) regarding the intervention model GSTARX and a step pulse is applied
to the case of foreign tourists forecasting. Studies of multivariate time series approach using
GSTARX is still limited to models involving variable X with non-metric scale.
GSTARX estimation method using the GLS, as well as the estimation method on the
model equations SUR. Ordinary Least Square method (OLS) can not be used for multivariate
model consisting of multiple equations that are correlated because it will produce a
estimator is less efficient, in the sense that the resulting variance would be very large.
Based on the description that has been described above, in this study will be conducted
further studies on multivariate time series model with variable X metric using GLS estimation.
The aim of this study is to obtain estimates of the model parameters GSTARX through
simulation studies.
2.
Materials and methods
2.1 Multivariate time series
Time series analysis used in data that have dependencies time where there is a
relationship between the occurrence of a period with the previous period. At the time series
analysis has the period or the same observation interval (Wei, 2006). Time series analysis
involves only a single event or a phenomenon called the univariate time series analysis, while
involving some event or phenomenon which occurs correlation or relationship between the
incidence of one another called multivariate time series analysis. Similarly in the analysis of
univariate time series, multivariate time series analysis to also pay attention to stationary
which can be seen on the plot Matrix Cross Correlation Function (MCCF) and plot Matrix
Partial Cross Correlation Function (MPCCF).
One model is a multivariate time series model VARMA that can generally be written
into the form of the following equation.
Φ p (B )Z (t ) = Θ q (B )a(t ) ,
Where Z(t ) is a vector with multivariate time series,
Θ q (B )
Φ p (B )
autoregressive order p matrix, and
is a polynomial moving average order q.
2.2 GSTARX models
GSTAR a generalization of STAR models. Difference between STAR models with GSTAR
is autoregression parameter in the model STAR assumed to be equal to any location, while
the autoregression parameter of GSTAR be different for each location and the difference
between the location shown in the form of weighting matrix (Borovkova et al., 2008). GSTAR
in the form of a matrix is given by
λs
p
Z (t ) = ∑ Φ s 0 + ∑ Φ sk W ( k ) Z (t − s ) + e (t )
s =1
k =1
GSTAR model with one order of time and spatial order for three different locations is given
by
Z (t ) = [Φ 10 + Φ 11 W (1) ]Z (t − 1) + e (t )
that can be presented in the form of a matrix:
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The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model
0 φ11 0
0 0
z1 (t ) φ10 0
z (t ) = 0 φ
0 + 0 φ 21 0 w21
20
2
z3 (t ) 0
0 φ30 0
0 φ31 w31
w12
0
w32
SC.32
w13 z1 (t − 1) e1 (t )
w23 z 2 (t − 1) + e2 (t )
0 z3 (t − 1) e3 (t )
To determine the order of time in the model can be used AIC criteria, whereas for the
spatial order is generally limited to only order one course because of the higher order will be
difficult to interpret (Wutsqa et al., 2010).
Weighting on GSTAR there are four, namely uniform weight, inverse distance,
normalized cross correlation and inference partial normalization of cross correlation
(Suhartono & Atok, 2006).
2.3 Parameter estimates
The OLS estimators β is are as follows:
βˆ = (X′X)-1 X′Y .
Whereas the form of parameter estimate from GLS estimator is (Park, 1967):
βˆ = (X' ΩX) −1 X' Ω −1 Y ,
where Ω −1 = Σ −1 ⊗ I so the above equation will be:
βˆ = (X' Σ −1 ⊗ IX ) −1 X' Σ −1 ⊗ IY .
2.4 Methods
GSTARX-OLS and GSTARX- SUR weighted cross correlation normalized partial
correlation. Steps for simulation study are as follows.
a) Generating the data xt and yt for 3 locations with n = 300 multivariate normal
distribution with a mean of zero and variance covariance matrix Ω .
b) Determining the value of coefficient parameters used in the model GSTARX (11) with a
stationary condition.
c) Applying steps a and b in six simulations, i.e.,
1) Simulation 1 for residual between locations is not correlated with the same
variance.
2) Simulation 2 to residual between locations does not correlate with different
variances.
3) Simulation 3 for residual between locations all correlated with the same variance.
4) Simulation 4 for residual between locations is not all correlated with the same
variance.
5) Simulation 5 for residual between locations all correlated with different variances.
6) Simulation 6 to residual between locations all correlated with different variances.
d) Evaluating order ARIMA residuals.
e) Getting series yit and xit to 3 locations.
f) Incorporating order transfer function for each simulation
1) Case study 1 using the order of (b = 1, s = 1, r = 0) into the equation
y it = (ω 0 − ω 1 Β ) X t −1 + e it .
2) Case study 2 using the order of (b = 1, s = 2, r = 0) into the equation
y it = (ω 0 − ω 1 Β + ω 2 B 2 ) X t −1 + e it .
g) GSTARX-OLS model building and GLS.
h) Getting the model parameter estimation GSTARX-OLS and GSTARX-SUR.
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J.D. Kurnia, Setiawan, S.P. Rahayu
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Z 1 (t ) φ10
Z (t ) = 0
2
Z 3 (t ) 0
0
φ 20
0
0 w11
0 + w21
φ 30 w31
w12
w22
w32
w13 φ11 0
w23 0 φ 22
0
w33 0
Z 1 (t − 1) e1 (t )
Z (t − 1) + e (t )
2
2
φ 33 Z 3 (t − 1) e3 (t )
0
0
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 (11)model which is then used to build
the model GSTARX (11) with the parameters in the following equation coefficient matrix.
0,18 0,25 0,25
Φ 1 = 0,15 0,23 0,15
0,21 0,21 0,13
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.
0
0 X 1 (t − 1) − 10,16
0
0 X 1 (t − 2) e1 (t )
z1 (t ) 0,24 0,24 0,24 z1 (t − 1) 15,12
− 10,07
14,97
0 X 2 (t − 1) + 0
0 X 2 (t − 2) + e2 (t )
z 2 (t ) = 0,16 0,23 0,16 z 2 (t − 1) + 0
z (t ) 0,22 0,22 0,14 z (t − 1) 0
− 9,94 X 3 (t − 2) e3 (t )
0
14,94 X 3 (t − 1) 0
0
3
3
0
0 X 1 (t − 1) − 10,15
0
0 X 1 (t − 2) e1 (t )
z1 (t ) 0,23 0,24 0,24 z1 (t − 1) 15,13
z
t
0
,
16
0
,
22
0
,
16
z
t
−
1
0
14
,
96
0
X
(
t
−
1
)
+
0
−
10
,
06
0 X 2 (t − 2) + e2 (t )
(
)
=
(
)
+
2
2
2
z (t ) 0,22 0,22 0,14 z (t − 1) 0
0
14,96 X 3 (t − 1) 0
0
− 9,95 X 3 (t − 2) e3 (t )
3
3
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.
0 , 24
0 , 23
z 1 (t )
0 , 23
z 2 (t ) = 0 ,16
0 , 22
z (t )
0 , 22
3
0
− 10 ,13
0
+
− 10 , 05
0
0
−
SWUP
0 , 24 z 1 ( t − 1 )
0
0
15 ,13
0 ,16 z 2 (t − 1 ) +
0
14 , 97
0
0 ,13 z 3 (t − 1 )
0
0
14 , 94
0
0
− 20 , 09
X 1 (t − 2 )
0
0
− 20 , 06
X 2 (t − 2 ) +
9 , 93 X 3 ( t − 2 )
0
0
X 1 (t
X 2 (t
X (t
3
0
− 1)
− 1) +
− 1 )
X
0
X
− 20 , 04 X
(t − 3 )
e 1 (t )
(
3
)
+
−
t
e 2 (t )
2
3 (t − 3 )
e 3 (t )
1
The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model
0 , 24
z 1 (t )
0 , 23
0 , 22
z 2 (t ) = 0 ,16
z (t )
0 , 22
0 , 22
3
0
− 10 ,15
+
− 10 , 06
0
−
0
0
0 , 24 z 1 ( t − 1 )
15 ,13
0
14
0 ,16 z 2 (t − 1 ) +
0
0 ,13 z 3 (t − 1 )
0
− 10 ,15
X 1 (t − 2 )
0
0
X 2 (t − 2 ) +
9 , 95 X 3 ( t − 2 )
0
SC.34
X 1 (t − 1)
X 2 (t − 1) +
0
14 , 96 X 3 ( t − 1 )
0
0
X 1 (t − 3 )
e 1 (t )
− 10 , 06
0
X 2 ( t − 3 ) + e 2 (t )
e (t )
− 9 , 95 X 3 ( t − 3 )
0
3
0
, 96
0
0
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.
OLS
GLS
Parameter
estimasi
SE
estimasi
SE
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
Case study 1
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
OLS
GLS
Parameter
estimasi
SE
estimasi
SE
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
Case study 2
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
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J.D. Kurnia, Setiawan, S.P. Rahayu
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.
0 , 304
0 , 304 z 1 ( t − 1 )
0
0
z 1 (t )
0 ,160
15 ,160
X 1 ( t − 1)
0 , 278
0 ,159 z 2 (t − 1 ) +
0
14 , 980
0
z 2 (t ) = 0 ,159
X 2 (t − 1) +
z (t )
0 , 229
0 , 229
0 ,130 z 3 (t − 1 )
0
0
15 , 010 X 3 ( t − 1 )
3
0
0
− 9 , 880
X 1 (t − 2 )
e 1 (t )
+
− 9 , 960
0
0
X 2 ( t − 2 ) + e 2 (t )
e (t )
− 9 , 920 X 3 ( t − 2 )
0
0
3
0 , 317
0 , 317
z 1 (t )
0 ,136
0 , 267
0 ,165
z 2 (t ) = 0 ,165
z (t )
0 , 210
0 , 210
0 ,171
3
−
9
,
960
0
0
+
− 9 , 990
0
0
− 9 , 970
0
0
z 1 ( t − 1 ) 15 , 060
0
z 2 (t − 1 ) +
z (t − 1 )
0
3
X 1 ( t − 2 ) e 1 (t )
X 2 ( t − 2 ) + e 2 (t )
X 3 ( t − 2 ) e 3 (t )
0
14 , 997
0
X 1 ( t − 1)
X 2 (t − 1) +
15 , 040 X 3 ( t − 1 )
0
0
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.
0
0
0 , 303 0 , 303 z 1 ( t − 1 ) 15 ,149
z 1 (t ) 0 ,163
X 1 ( t − 1)
(
)
(
)
+
=
−
0
14
,
970
0
1
0
,
163
0
,
270
0
,
163
z
t
z
t
2
X 2 ( t − 1) +
2
z (t ) 0 , 231 0 , 231 0 ,129 z (t − 1 ) 0
0
15 , 014 X 3 ( t − 1)
3
3
0
0
0
0
X 1 ( t − 3 ) e 1 (t )
− 9 , 891
X 1 ( t − 2 ) − 19 , 917
0
19
,
822
0
(
2
)
+
−
−
0
0
+
− 10 , 031
X
t
X 2 ( t − 3 ) + e 2 (t )
2
0
0
− 19 , 950 X 3 ( t − 3 ) e 3 (t )
0
0
− 9 , 946 X 3 ( t − 2 )
0
0 X 1 ( t − 1)
z1 (t ) 0 ,137 0 ,317 0 ,317 z1 ( t − 1) 15 ,058
14 ,999
0 X 2 (t − 1) +
z 2 (t ) = 0 ,168 0 , 260 0 ,168 z 2 (t − 1) + 0
z (t ) 0 , 211 0 , 211 0 ,171 z (t − 1) 0
0
15 ,030 X 3 ( t − 1)
3
3
0
0
0
0
− 9 ,959
X 1 ( t − 2 ) − 19 ,999
X 1 ( t − 3 ) e1 (t )
+
− 10 , 017
− 19 ,915
0
0
0
0
X 2 (t − 2 ) +
X 2 ( t − 3) + e 2 (t )
− 9 ,947 X 3 ( t − 2 )
− 20 ,024 X 3 ( t − 3) e 3 (t )
0
0
0
0
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
The results of simulation studies with a transfer function GSTARX Model with X is a
metric variable can be concluded, that if the residuals from the simulated data are not
mutually correlated between locations, the model GSTARX-OLS and GSTARX-SUR will
generate a standard error of parameter estimate value are the same. However, if the residual
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The simulation studies for Generalized Space Time Autoregressive-X (GSTARX) model
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of simulation data are correlated, it will generate a standard error of parameter estimate
values are small in GSTARX-SUR Model compared with GSTARX-OLS Model. So it can be said
that the parameter estimation using GSTARX-SUR Model more efficient (smaller standard
errors) compared with GSTARX-OLS for residual cases of simulation data are correlated.
Case
study 1
Case
study 1
Table 2. Efficiency value of GSTARX-SUR model in simulation 3.
OLS
GLS
Efisiensi GLS
Parameter
(%)
estimasi
SE
estimasi
SE
psi10
0,160
0,066
0,136
0,055
17,015
psi20
0,278
0,079
0,267
0,061
22,742
psi30
0,130
0,074
0,171
0,059
20,416
psi11
0,608
0,077
0,634
0,066
14,414
psi21
0,318
0,086
0,330
0,069
19,374
psi31
0,457
0,074
0,420
0,062
16,736
w10
15,160
0,080
15,060
0,056
30,000
w20
14,980
0,074
14,997
0,050
32,432
w30
15,010
0,072
15,040
0,050
30,556
w11
-9,880
0,080
-9,960
0,057
28,750
w21
-9,960
0,074
-9,990
0,050
32,432
w31
-9,920
0,072
-9,970
0,049
31,944
OLS
GLS
Efisiensi GLS
Parameter
(%)
estimasi
SE
estimasi
SE
psi10
0,163
0,066
0,137
0,055
16,970
psi20
0,270
0,079
0,260
0,061
22,748
psi30
0,129
0,074
0,171
0,059
20,430
psi11
0,606
0,077
0,634
0,066
14,382
psi21
0,325
0,085
0,336
0,069
19,343
psi31
0,461
0,075
0,422
0,062
16,772
w10
15,149
0,080
15,058
0,057
28,750
w20
14,970
0,074
14,999
0,050
32,432
w30
15,014
0,072
15,030
0,050
30,556
w11
-9,891
0,081
-9,959
0,057
29,630
w21
-10,031
0,079
-10,017
0,052
34,177
w31
-9,946
0,083
-9,947
0,056
32,530
w12
-19,917
0,080
-19,999
0,057
28,750
w22
-19,822
0,074
-19,915
0,050
32,432
w32
-19,950
0,072
-20,024
0,049
31,944
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