"backtest" <- function(m1,rt,orig,h,xre=NULL,fixed=NULL){
# m1: is a model object
# orig: is the forecast origin
# rt: the time series
# xre: the independent variables
# h: forecast horizon
# fixed: parameter constriant
#
# The program requires the library fSeries or FinTS
regor=c(m1$arma[1],m1$arma[6],m1$arma[2])
seaor=list(order=c(m1$arma[3],m1$arma[7],m1$arma[4]),perioa=m1$arma[5])
T=length(rt)
if(orig > T)orig=T
if(h < 1) h=1
rmse=rep(0,h)
nori=T-orig
err=matrix(0,nori,h)
jlast=T-1
for (n in orig:jlast){
jcnt=n-orig+1
x=rt[1:n]
mm=arima(x,order=regor,seasonal=seaor,xreg=xre,fixed=fixed)
if (is.null(xre))
nx=NULL else nx=xre[(n+1):(n=h)]
fore=predict(mm,h,newxreg=nx)
pred=fore$pred
obsd=rt[(n+1):(n+h)]
err[jcnt,]=obsd-pred
}
for (i in 1:h){
iend=nori-i+1
tmp=err[1:iend,i]
rmse[i]=sqrt(sum(tmp^2)/iend)
}
print("RMSE of out-of-sample forecasts")
print(rmse)
backtest<-list(origin=orig,error=err,rmse=rmse)
}