[R] true time series lags behind fitted values in arima model

[Benedikt Gehr]

Hi

I am fitting an arima model to some time series X. When I was comparing 
the fitted values of the model to the true time series I realized that 
the true time series lags one time step behind the fitted values of the 
arima model. And this is the case for any model. When I did a simple 
linear regression using lm to check, I also find the same results, that 
the true series lags behind the fitted values. I don't understand this, 
what am I doing wrong? Below I copied some of the code to demonstrate 
the issue.

## Analysis using arima()

X<-c(6.841067, 6.978443, 6.984755, 7.007225, 7.161198, 7.169790, 
7.251534,  # the true time series
    7.336429, 7.356600, 7.413271, 7.404165, 7.480869, 7.498686, 7.429809,
    7.302747,  7.168251, 7.124798, 7.094881, 7.119132, 7.049250, 6.961049,
    7.013442,  6.915243, 6.758036, 6.665078, 6.730523, 6.702005, 6.905522,
    7.005191, 7.308986)

model100<-arima(X,order=c(1,0,0),include.mean=T,method="ML")  # the 
arima model
resid100<-residuals(model100)
Xfit100<-X-resid100 # the fitted values
ts.plot(cbind(Xfit100,X),col=c(2,4),main="ARIMA(1,0,0)")  # plot the 
true ts vs the fitted values
legend(20,7.5,c("true values","fitted 
values"),pch=c(16,16),col=c("blue","red"))

## Same analysis using lm()

Y<-X[2:30]  # create X(t)-> a time series without the first value
X1<-X[1:29] # create the X(t-1)

time<-seq(1978,2006)
model1<-lm(Y~X1)
summary(model1)
arima.intercept<-model1[[1]][[1]]/(1-model1[[1]][[2]]);arima.intercept
model100[[1]][[2]]
plot(Y~time,type="b")  # plot the two series
points(fitted(model1)~time,type="b",col="green")
legend(1995,7.5,c("true values","fitted 
values"),pch=c(16,16),col=c("black","green"))


Thank you very much for the help

best wishes

Benedikt