Fitting ARIMA with a drift on R

Question

I had some problems fitting an ARIMA - the data are

FTSE <-log(EuStockMarkets[,"FTSE"]) 

The following link explains the problem and gives the solution http://www.stat.pitt.edu/stoffer/tsa3/Rissues.htm#Issue2
The solution sounded effective, untill I tried some $ARIMA(p,d,q)$ other than $ARIMA(1,1,0)$: with $ARIMA(1,1,1)$, $ARIMA(2,1,1)$ and $ARIMA(1,1,2)$ the procedure (1+) and the procedure (2) give different coefficients, in particular with the procedure (1+) it says that the AR part is not stationary (indeed changing method from 'css' to 'ml' the coeffiecient of $AR$ is 1 or above).
These are the codes for $ARIMA(1,1,1)$:

dFTSE<-diff(FTSE)
model2<-Arima(dFTSE, order = c(1, 0, 1))
model2
model1pluscss<-Arima(FTSE,order=c(1,1,1), xreg=1:length(FTSE))
model1plusml<-Arima(FTSE,order=c(1,1,1), xreg=1:length(FTSE), method='ML')
model1plusml

On the other side the two procedures produce the same output also in $ARIMA(0,1,1)$ and $ARIMA(0,1,2)$. Is there a theoretical explanation that I can't see or something else?

Answer 1

The problem is with the arima() function, not the Arima() function from the forecast package. The Arima() function has an argument include.drift which is TRUE by default when $d>1$.

So your models model1pluscss and model1plusml try to include a drift term twice, and so produce some numerical problems that are manifest in different ways.

The model you want is simply

model1 <- Arima(FTSE, order=c(1,1,1))

That is not identical to your model2 because ARIMA models in R are estimated using a state space representation and the non-​​stationary components are given a diffuse prior, rather than simply differenced away. (See help(arima) in R.) This will lead to different parameter estimates.