Determination of space weight

3. Model Identification

As in time series modeling, the first step is identifying a tentative model which is characterized by spatial and time order. Spatial order is generally restricted on order 1, since the higher order is difficult to be interpreted. Approving the method in VAR and VARMA models, the time autoregressive order is determined by using the Akaike Information Criterion AIC see [13] AICi = 2 2 ˆ ln T i k i Σ + 2 where k is the number of parameters in the model and T is the number of observation. The autoregressive order of GSTAR model is p such that AICp = min AIC i p i ≤ ≤ . This value p can be obtained by performing SAS program using PROCSTATESPACE.

4. Parameters Estimation

4.1 Determination of space weight

Determination of space weight by using the normalisation result of cross- correlation between locations at the appropriate time lag is firstly proposed by Suhartono and Atok see [10,11]. They demonstrated by simulation study that the method well performed on GSTAR1 1 model. In general, cross-correlation between two variables or location i and j at the time lag k, ], , [ corr k t Z t Z j i − defined as see [1, 14] , j i ij ij k k σ σ γ ρ = … , 2 , 1 , ± ± = k 3 where k ij γ is cross-covarians between observation in location i and j at the time lag k, i σ and j σ is standard deviation of observation in location i and j. The estimated of cross-correlation in sample data is ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − − − = ∑ ∑ ∑ = = + = n t j j n t i i n k t j j i i ij Z t Z Z t Z Z k t Z Z t Z k r 1 2 1 2 1 ] [ ] [ ] ][ [ . 4 Then, determination of space weight could be done by normalisation of the the cross-correlation between locations at the appropriate time lag. This process generally yields space weight for GSTARp 1 model, i.e. , | | ij ij ik k i r k w r k ≠ = ∑ where j i ≠ , k = 1, …, p 5 and this weight satisfies 1 | | 1 = ∑ ≠ j ij w Space weights by using the normalisation of the cross-correlation between locations at the appropriate time lag give all form possibilities of the relationship between locations. Hence, there is no strict constraint about the weight values that must depend on distance between locations. This weight also gives flexibility on the sign and size of the relationship between locations. 4.2. Autoregressive Parameter Estimation of GSTAR p λ 1 λ 2,..., λ p model Recall GSTAR p λ 1 λ 2,..., λ p model 1 1 2 2 1 s p i k k k i sk i i iN N i s k Z t w Z t s w Z t s w Z t s t λ ϕ = = = − + − + + − + ∑∑ e 6 for , 1,..., t p p T = + , i = 1, 2, ..., N, where 1 ij w = for i = j and zero otherwise. It is assumed that t e ∼ White Noise 0, ∑ , where et = 1 e t , 2 e t , …, N e t Least square estimator of autoregressive parameter has been dirived by Borovkova et.al [3]. They define new notations N k k i ij j j i V t w Z t ≠ = ∑ for k ≥ 1 and i i V t Z t = , ,..., i i i Z p Z T ′ = Y , ,..., i i i e p e T ′ = u 1 1 1 1 1 1 1 1 p p i i i i i i i i i V p V p V V V T V T V T p V T p λ λ λ λ ⎛ ⎞ − − ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ − − − − ⎝ ⎠ X and 1 2 10 1 20 2 ,..., , ,..., ,..., ,..., p i i i i i i i p λ λ λ φ φ φ φ φ φ ′ = β Equation 6 can be expressed for all location simultaneously as linear model = + Y X β u , 7 where 1 ,..., N ′ ′ ′ = Y Y Y , 1 ,..., N = X X X , 1 ,..., N ′ ′ ′ = β β β , 1 ,..., N ′ ′ ′ = u u u Thus, least square estimator ˆ T β is of the form 1 ˆ T − ′ ′ = β X X X u 8 Asymptotic normality of the estimator ˆ T β will be explained below. Write 1 N diag ,..., M M M = where i1 ip diag ,..., M M M i = , for i = 1, 2, ... , N and s = 1, 2, ..., p 1 1 1 1 1 , 1 , 1 , 1 , 1 , 1 1 s s s s i i i i i i N is i i i i i iN w w w w w w w w − + − + ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ M λ λ λ λ 9 Define covariance matrix 1 1 1 2 1 2 p p p p p − − + ⎡ ⎤ ⎢ ⎥ − + ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ − − ⎣ ⎦ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ 10 where [ ] s E t t s ′ = + Γ Z Z . Under some conditions Borovkova et.al. [3] have shown asymptotic multivariate normality of least square estimator in GSTAR model, i.e. ˆ T T − β β , d d ⎯⎯→ 0 C 11 where d = 1 p d p N λ λ = + + + , and 1 1 p p p − − ′ ′ ′ = ⊗ ⊗ ⊗ C M I Γ M M Γ M M I Γ M ∑ , M is defined in 9 and p Γ is defined in 10. The parameter ∑ can be estimated by 1 ˆ ˆ ˆ 1 T T t p t t t t T p = ′ = − − − + ∑ Z Z Z Z ∑ where 1 1 ˆ ˆ ˆ s p k s sk s k t t s t s λ = = = − + − ∑ ∑ Z Φ Z Φ W Z The elements of matrix p Γ can be estimated 1 ˆ 1 T s T t s t t s T s − = ′ = − − + ∑ Γ Z Z and ˆ ˆ T T s s ′ − = Γ Γ , for s ≥ 0. Cosequently the estimate of p Γ is obtained. It is also proven in [3] that ˆ T ∑ dan ˆ p Γ are consistent estimators for ∑ and p Γ respectively. The hypothesis to test the significantcy of parameter in GSTAR model can be written as H : r = R β against H 1 : r ≠ R β . 12 By using property 11 and the facts that ˆ T ∑ dan ˆ p Γ are consistent estimators for ∑ and p Γ , and modifying Wald Statistic on linear model from White [15] we develope Wald satistic for the hypothesis 12 .Let m = rank R, if H : r = R β holds, under some conditions, i ˆ T T r − R β , d d ⎯⎯→ 13 where d = 1 p d p N λ λ = + + + , and = ′ RC R ii The Wald statistic 1 ˆ ˆ ˆ T T T r r − ′ = − − W R β R β 2 d m ⎯⎯→χ 14 where ˆ = ˆ ′ RC R

5. Check Diagnostic