SWUP
The simulation studies for Generalized Space Time Autoregressive-X
(GSTARX) model
Junta Dwi Kurniaa, Setiawanb, Santi Puteri Rahayuc
a, b, cDepartment 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)
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.
( ) ( )
B t q( ) ( )
B tp Z Θ a
Φ = ,
Where Z
( )
t is a vector with multivariate time series, Φp( )B autoregressive order p matrix, and ( )Bq
Θ 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
( )t (t s) ( )t
p
s k
k sk s
s
e Z W
Φ Φ
Z − +
+
=
∑
∑
=1 =1 ) ( 0
λ
GSTAR model with one order of time and spatial order for three different locations is given by
( )t Φ Φ W Zt e( )t
Z =[ + (1)] ( −1)+
SWUP
+
− − −
+
=
) (
) (
) (
) 1 (
) 1 (
) 1 (
0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
) (
) (
) (
3 2 1
3 2 1
32 31
23 21
13 12
31 21 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
t z
t z
t z
φ φ φ
φ φ φ
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 βis are as follows:
ˆ ′ -1 ′ β= (X X) X Y.
Whereas the form of parameter estimate from GLS estimator is (Park, 1967):
Y
Ω
X'
ΩX) (X'
βˆ= −1 −1 ,
where Ω−1 =Σ−1 ⊗I
so the above equation will be:
IY
Σ
X' IX
Σ
(X'
β= −1⊗ −1 −1⊗ )
ˆ .
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 matrixΩ.
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
it t
it ω ω Β X e
y =( 0 − 1 ) −1+ .
2) Case study 2 using the order of (b = 1, s = 2, r = 0) into the equation
it t
it ω ω Β ω B X e
y = − + 2 −1+
2 1
0 )
( .
g) GSTARX-OLS model building and GLS.
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.
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.
( ) 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.
SWUP ( )
( ) ( )
( ) ( )
( ) ( ) ( )
+
− − −
− −
−
+
− − −
− −
−
+
+
− − −
+
− − −
=
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 0
0
0 06 , 10 0
0 0
15 , 10
) 2 (
) 2 (
) 2 (
95 , 9 0
0
0 06 , 10 0
0 0
15 , 10
) 1 (
) 1 (
) 1 (
96 , 14 0 0
0 96 , 14 0
0 0
13 , 15
1 1 ) 1 (
13 , 0 22 , 0 22 , 0
16 , 0 22 , 0 16 , 0
24 , 0 24 , 0 23 , 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.
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
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.
( ) weighted residual value and the resulting value of the parameter estimates in the following equations.
( )
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
SWUP
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.
Table 2. Efficiency value of GSTARX-SUR model in simulation 3.
Parameter OLS GLS Efisiensi GLS (%) estimasi SE estimasi SE
Case study 1
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
Parameter OLS GLS Efisiensi GLS (%) estimasi SE estimasi SE
Case study 1
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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