Journal of Probability and Statistics / 2012 / Article / Alg 1

Research Article

Application of Generalized Space-Time Autoregressive Model on GDP Data in West European Countries

Algorithm 1

R codes for least square estimation of GSTAR model.
# =============================================================
# [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 [1]
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$res
# –––––––––––––––––––––––––––––––-
res<-function(Zt,x){
coef<-gstar(Zt,x)
Xi<-Xi(Zt,x)
p<-x[1]
La<-x[2:length(x)]
N<-dim(Zt)[1] #number of sites
T<-dim(Zt)[2]-1 #number of time periods
Z.OLS<-matrix(0,T-p+1,N)
res.OLS<-matrix(0,N,T-p+1)
if (p==1){
for (i in 1:N){
if (La[1]!=0)Z.OLS[,i]<-Xi[,,i]%*%coef[,i]
else Z.OLS[,i]<-Xi[,,i]*coef[,i]
res.OLS[i,]<-t(Z.OLS[,i]-Zt[i,(p+1):(T+1)])}
}
if (p!=1){
for (i in 1:N){
Z.OLS[,i]<-Xi[,,i]%*%coef[,i]
res.OLS[i,]<-t(Z.OLS[,i]-Zt[i,(p+1):(T+1)]) } }
az<-new.env()
az$Xi<-Xi # matrix Xi
az$coef<-coef
az$pred<-t(Z.OLS)
az$res<-res.OLS
ax<-as.list(az)}
# –––––––––––––––––––––––––––––––-

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