Materials and methods PROS Junta DK, Setiawan, Santi PR The simulation studies fulltext

J.D. Kurnia, Setiawan, S.P. Rahayu SWUP 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. t B t B q p a Θ Z Φ = , Where t Z is a vector with multivariate time series, B p Φ autoregressive order p matrix, and B q 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 s t t p s k k sk s s e Z W Φ Φ Z + −       + = ∑ ∑ = = 1 1 λ GSTAR model with one order of time and spatial order for three different locations is given by t t t e Z W Φ Φ Z + − + = 1 ] [ 1 11 10 that can be presented in the form of a matrix: The simulation studies for Generalized Space Time Autoregressive-X GSTARX model SWUP SC.32           +           − − −                               +           =           1 1 1 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 β is 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 β 1 1 − − = ˆ , where I Σ Ω ⊗ = − − 1 1 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 matrix Ω . b Determining the value of coefficient parameters used in the model GSTARX 1 1 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 e X Β ω ω y + − = −1 1 . 2 Case study 2 using the order of b = 1, s = 2, r = 0 into the equation it t it e X B ω Β ω ω y + + − = −1 2 2 1 . g GSTARX-OLS model building and GLS. h Getting the model parameter estimation GSTARX-OLS and GSTARX-SUR. 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