^{1}

^{2}

^{2}

^{1}

^{2}

This paper provides an application of generalized space-time autoregressive (GSTAR) model on GDP data in West European countries. Preliminary model is identified by space-time ACF and space-time PACF of the sample, and model parameters are estimated using the least square method. The forecast performance is evaluated using the mean of squared forecast errors (MSFEs) based on the last ten actual data. It is found that the preliminary model is GSTAR(2;1,1). As a comparison, the estimation and the forecast performance are also applied to the GSTAR(1;1) model which has fewer parameter. The results showed that the ASFE of GSTAR(2;1,1) is smaller than that of the order (1;1). However, the

Space-time data are frequently found in many areas of research, for example, monthly tea production from some plants, yearly housing price at capital cities, and yearly per capita GDP (gross domestic product) of several countries in some region. The generalized space-time autoregressive model of order

The term of generalization is associated with the model parameters. When a parameter matrix is diagonal, the GSTAR model is the same as space time autoregressive (STAR) model given by Martin and Oeppen [

When

This paper is presented as follows. In Section

Let

Model parameters

From (

Then, the LS estimator for parameter matrix

For order

Then, the LS estimator for parameter matrix

Under the stationary assumption, the LS estimator for the GSTAR parameters is a consistent estimator [_{4}. Spatial weight matrix and model parameters that used in the simulation, respectively, were

The LS estimator vector

# =============================================================

# [FUNCTION]: OLS estimation for GSTAR(p;L1,…,Lp) models

# =============================================================

# 3 dimension zeros matrix

# –––––––––––––––-

zeros <-function(m,n,p){

W<-rep(0, m*n*p)

dim(W)<-c(m, n, p)

W}

# “vec” operator

vec<-function(X){

a<-dim(X)

Y<- t(X[1, ])

for (i in 2:a[1]){

Y<- cbind(Y,t(X[i, ]))}

t(Y)}

#––––––––––

# Inverse of matrix

#––––––––––

inv<-function(X){

if(dim(X)[1]! = dim(X)[2]) stop(“THE MATRIX MUST BE SYMMETRIC!!!”)

else{

if (det(X)==0) stop("THE MATRIX IS SINGULAR!!!")

else { n<-dim(X)[1]

solve(X)

# –––––––––––––––––––––––––––––––

# Construction of vector Zi, for each i=1,…,N

# –––––––––––––––––––––––––––

# Construction of vector Zi, for each i=1,…,N

# –––––––––––––––––––––––––––

# suppose x = c(p,L1,..., Lp) represent the model order

Zi<-function(Zt, x){

N<-dim(Zt)[1] #number of sites

T<-dim(Zt)[2]-1 #number of time periods

p<-x [

Zi<-matrix(0, T-p+1, N)

for (i in 1: N)

Zi[,i]<-Zt[i,(p+1):(T+1)]

Zi}

# –––––––––––––––––––––––––––––––-

# Construction of matrix Xi, for each i = 1,…,N

# ––––––––––––––––––––––-

Xi<-function(Zt,x)

{N<-dim(Zt)[1] #number of sites

T<-dim(Zt)[2]-1 #number of time periods

p<-x[1]

La<-x[2: length(x)]

r<-lmd+1 # where lmd = the greatest order for weight matrices

WZ<-zeros(N, T, r)

for (k in 1: r)

WZ[,, k]<-W [,, k]%*%Zt[, 1: T]

Xi<-zeros((T-p+1),sum(La+1), N)

if (p==1)

{ for (i in 1:N){

TR<-WZ[i,p:T,1:(La[1]+1)]

Xi[,, i]<-TR

if (p>=2){

for (i in 1:N){

TR<-WZ[i,p:T,1:(La[1]+1)]

for (s in 2:p)

TR<-cbind(TR,WZ[i,(p-s+1):(T-s+1),1:(La[s]+1)])

Xi[,, i]<-TR

Xi}

# –––––––––––––––––––––––––––––––-

# OLS parameter of GSTAR model

# ––––––––––––––

gstar<-function(Zt,x){

p<-x[1]

La<-x[2:length(x)]

r<-lmd+1 # where lmd = the greatest order for weight matrices

N<-dim(Zt)[1] #number of sites

T<-dim(Zt)[2]-1 #number of time periods

Xi<-Xi(Zt,x)

Zi<-Zi(Zt,x)

coef.OLS<-matrix(0,sum(La+1),N)

col.name<-array(0,N)

for (i in 1:N){

coef.OLS[, i]<-inv(t(Xi[,, i])%*%Xi[,,i])%*%t(Xi[,, i])%*%Zi[,i]

col.name[i]<-paste("site",i)}

colnames(coef.OLS)<-col.name

round(coef.OLS,4)}

# –––––––––––––––––––––––––––––––-

# Residuals of GSTAR model

# ––––––––––––––

# (1). To find the LS estimates only, for example GSTAR(2;1,1), use

# the command:

# > gstar(Zt,c(2,1,1))

# where Zt is data matrix.

# (2). To find the estimates, prediction values, and residuals

# vector respectively, call the function by the following

# commands:

# > as.2<-res(Zt,c(2,1,1))

# > as.2$coef

# > as.2$pred

# > as.2$res

# –––––––––––––––––––––––––––––––-

res<-function(Zt,x){

coef<-gstar(Zt,x)

Xi<-Xi(Zt,x)

p<-x[1]

La<-x[2:length(x)]

N<-dim(Zt)[1] #number of sites

T<-dim(Zt)[2]-1 #number of time periods

Z.OLS<-matrix(0,T-p+1,N)

res.OLS<-matrix(0,N,T-p+1)

if (p==1){

for (i in 1:N){

if (La[

else Z.OLS[,i]<-Xi[,,i]*coef[,i]

res.OLS[i,]<-t(Z.OLS[,i]-Zt[i,(p+1):(T+1)])}

}

if (p!=1){

for (i in 1:N){

Z.OLS[,i]<-Xi[,,i]%*%coef[,i]

res.OLS[i,]<-t(Z.OLS[,i]-Zt[i,(p+1):(T+1)])

az<-new.env()

az$Xi<-Xi # matrix Xi

az$coef<-coef

az$pred<-t(Z.OLS)

az$res<-res.OLS

ax<-as.list(az)}

# –––––––––––––––––––––––––––––––-

and empirical mean squared error (MSE)

where

The result is presented in Table

The LS estimated values (in average) for the data generated from GSTAR(1;1) model with 1000 replications for various sample sizes

0.1799 | 0.1822 | 0.1891 | 0.1996 | 0.2004 | 0.2000 | |

0.4639 | 0.4684 | 0.4819 | 0.4971 | 0.4987 | 0.5002 | |

0.2786 | 0.2815 | 0.2942 | 0.2966 | 0.2983 | 0.2998 | |

0.1802 | 0.1828 | 0.1929 | 0.1983 | 0.1986 | 0.1998 | |

0.4000 | 0.4025 | 0.4028 | 0.3999 | 0.4002 | 0.4005 | |

0.2961 | 0.2944 | 0.2970 | 0.005 | 0.2998 | 0.2999 | |

0.4880 | 0.4906 | 0.4931 | 0.4988 | 0.5001 | 0.5002 | |

0.6989 | 0.6980 | 0.6961 | 0.7008 | 0.7002 | 0.7002 | |

MSE | 0.0279 | 0.0219 | 0.0105 | 0.0002 | 0.0001 | 0.0001 |

In this section, we apply the GSTAR model to the ratio of per capita GDP data in 16 West European countries. The data was kindly given by Maddison [

The dataset is the GDP ratio data for periods 1955–2006. It consists of 52 observations of 16 dimensional vectors. For the purpose of forecasting the data was grouped into the training data set and test data set. The training data is the first 42 observations that will be used for model building and the test data is the last ten data that will be used in forecasting performance comparison.

Clearly, the 42 observations in the training data, depicted in Figure

(a) The ratio of per capita GDP in 16 West European countries for the period 1955–2006 and (b) plot of the centralized data of the difference data.

Suppose

and the centralization of the differenced data is

where

As a preliminary model building, we set some notations used in model (

Geographical neighbourhood of order 1 and order 2.

No. | Country | Countries in the 1st neighborhood | Countries in the 2nd neighborhood |
---|---|---|---|

1 | Austria | 6,9,15 | 2,3,5,7,10,14 |

2 | Belgium | 5,6,10,16 | 1,3,8,9,13,14,15 |

3 | Denmark | 6,10,11,14 | 1,2,4,5,15,16 |

4 | Finland | 11,14 | 3,6 |

5 | France | 2,6,9,13,15,16 | 1,3,7,8,10,12,14 |

6 | Germany | 1,2,3,5,10,14,15 | 4,9,11,13,16 |

7 | Greece | 9 | 1,5,15 |

8 | Ireland | 16 | 2,5,10 |

9 | Italy | 1,5,7,15 | 2,6,13,16 |

10 | Netherland | 2,3,6,16 | 1,5,8,11,14,15 |

11 | Norway | 3,4,14 | 6,10 |

12 | Portugal | 13 | 5 |

13 | Spain | 5,12 | 2,6,9,15,16 |

14 | Sweden | 3,4,6,11 | 1,2,5,10,15 |

15 | Switzerland | 1,5,6,9 | 2,3,7,10,13,14,16 |

16 | United Kingdom | 2,5,8,10 | 3,6,9,13,15 |

Suppose

and the second order of spatial weight is

After transforming the data and constructing spatial weight matrix, the next step is identification of the model order. In STAR model-building, [

Figure

Sample space-time ACF and PACF of the differenced data.

The 64 parameters and the error variance in this model were estimated using the least square method and the result is presented in Table

Least square estimates for GSTAR(2;1,1) parameter.

Site | ||||
---|---|---|---|---|

1 | −0.083 | 0.058 | 0.045 | −0.046 |

2 | −0.092 | 0.028 | 0.404 | 0.052 |

3 | −0.218 | 0.730 | −0.113 | 0.638 |

4 | 0.566 | −0.534 | −0.262 | 0.253 |

5 | 0.152 | 0.563 | −0.030 | 0.255 |

6 | 0.176 | 0.211 | −0.052 | 0.046 |

7 | 0.106 | −0.048 | −0.018 | 0.261 |

8 | 0.332 | −0.126 | −0.114 | 0.540 |

9 | 0.271 | −0.424 | −0.217 | 0.461 |

10 | −0.026 | −0.250 | 0.097 | −0.332 |

11 | 0.549 | −0.278 | −0.166 | 0.239 |

12 | 0.442 | 0.198 | −0.254 | −0.204 |

13 | 0.376 | 0.242 | −0.054 | 0.037 |

14 | 0.390 | −0.120 | −0.457 | 0.304 |

15 | 0.042 | 0.564 | −0.250 | −0.949 |

16 | 0.293 | 0.226 | 0.119 | −0.318 |

Sample space-time autocorrelation of the residuals from the GSTAR(2;1,1) model.

Histogram and normal probability plot of the GSTAR(2,1,1) residuals (a) and (b) and plot of residuals versus fitted value (c).

For forecasting purpose, the estimated parameters in Table

GSTAR(1;1) model is the simplest model of GSTAR(

The model has an interpretation that the current observation in a certain location only depends on the immediate past observations recorded at the location of interest and at the nearest locations. The GSTAR(1;1) model has

Least square estimates for GSTAR(1;1) parameters.

Site | ||
---|---|---|

1 | −0.073 | 0.086 |

2 | −0.090 | 0.250 |

3 | −0.201 | 0.871 |

4 | 0.422 | −0.452 |

5 | 0.192 | 0.582 |

6 | 0.187 | 0.221 |

7 | 0.116 | 0.011 |

8 | 0.355 | 0.077 |

9 | 0.183 | −0.261 |

10 | 0.012 | −0.123 |

11 | 0.465 | −0.175 |

12 | 0.358 | 0.054 |

13 | 0.317 | 0.287 |

14 | 0.285 | −0.162 |

15 | 0.006 | 0.119 |

16 | 0.340 | 0.203 |

: GSTAR(1;1) residuals histogram (a), normal probability (b), and residuals versus fitted values (c).

Space-time autocorrelation of the GSTAR(1;1) residuals.

For the GDP data case, GSTAR(2;1,1) model has 64 parameters while GSTAR(1;1) has 32 parameters. Hence, it is not a surprise if the empirical MSE of GSTAR(2;1,1) is less than that of GSTAR(1;1). However, though the number of parameters of GSTAR(1,1) is half of the other one, the empirical MSE of GSTAR(1,1) is only decreasing 0.257 compared to the GSTAR(2;1,1).

The distribution of the MSE difference for each country is presented in a bubble plot (Figure

Bubble plot for the difference value of empirical MSE between GSTAR(2;1,1) and GSTAR(1;1). The bubble placed under the zero axis indicates that GSTAR(2,1,1) has a smaller value of MSE difference than that of the GSTAR(1;1) model.

For the purpose of forecasting model, it would be useful if we also consider their forecast performance. Therefore, in this section we will examine the one-step-ahead forecasting performance for each model candidate using the last ten actual data points of the per capita GDP ratio data set. Result of this section is expected to become a supplementary reference in finding the most parsimony space-time model for the case of per capita GDP ratio.

In (

The one-step-ahead forecast

To compare the forecast performance, we use mean of square forecast error (MSFE) which is defined by

To measure the performance closeness between two models, we also calculate the MSFE difference between model

The performance of

Suppose

Bubble plot for the difference of mean of square forecast error (MSFE) between GSTAR(2;1,1) and GSTAR(1;1). The bubble under the zero axis indicates that the GSTAR(2,1,1) has a smaller value of MSFE difference than that of the GSTAR(1;1) model.

The behavior of both differences for each time is displayed in Figure

Square forecast error difference (SFED) for GSTAR(2;1,1) and GSTAR(1; 1).

GSTAR modeling has been built to the ratio of per capita GDP data in West European countries. The model of order (2;1,1) model has been identified as the candidate model. However, when the forecast performance is compared with GSTAR(1;1), it is found that the performance is significantly indifferent. Due to parsimony principle, we recommend that the GSTAR(1;1) might be considered as a forecasting model.

The research of the first author is supported by the Scholarship of Sandwich-Like Program no. 1995.1/D4.4/2008 of the Ministry of National Education of Republic of Indonesia c/q Directorate General of Higher Education (DIKTI). The authors are very grateful to Professor Gopalan Nair from the School of Mathematics and Statistics, the University of Western Australia, for his supervision during the first author’s visit in 2008. Helpful comments from earlier manuscript’s reviewer are also thankfully acknowledged.