[R] ARIMA prediction

[Chris Bowring]

Hi

This is my first post to this group, so apologies in advance if I get it wrong.

I would like to know how the prediction for arima models works in R. I
have a time series to which I fit an arima model, of varying AR and MA
orders. I then use the predict function to project it forward. I have
also written my own function to perform the prediction, but it gives
different answers to Arima.predict when the MA order is non-zero.


I use the residuals from the arima function in my custom prediction
function. I think this may be my problem. In the arima model:


x{t} = a(1)x{t-1} + a(2)x{t-2} + ... + a(p)x{t-p} + e{t} + b(1)e{t-1}
+ b(2)e{t-2} + ... + b(q)e{t-q}


I am treating the residuals (i.e. arima(....)$res)  as the e{t} terms.
This gives different answers both in the region of the simulation and
in the region of the prediction, so I'm guessing they're not what I
think they are. Indeed, after q intervals in the prediction, the
prediction proceeds as I would expect, presumably because all the
residuals that have an effect are zero by this stage.


Any help greatly appreciated - my code is below.


Thanks


Chris


--------------------------------------------------------------

The code to produce the two predictions is as follows:


AR <- 5
MA <- 3


sim <- arima.sim(list(order=c(AR,0,MA), ar=c(.1, .1, .1, .1, .1),
ma=c(.1, .1, .1)), n=100) + 50


fit <- arima(sim, order = c(AR, 0, MA))


coefs <- fit$coef
series <- sim
innov <- fit$res
pred <- 100


fit.predict <- predict(fit, n.ahead = pred)


fit.r <- c(sim, fit.predict$pred)


fit.custom <- ProjectCentralArima(AR = AR, MA = MA, d = 0, coefs =
coefs, series = sim, innov = innov, pred = pred)$ser


ProjectCentralArima function:


ProjectCentralArima <- function(AR, MA, d, coefs, series, innov, pred)
{


  if(d==0){
    series.diff <- series
  }
  else {
    series.diff <- diff(series, lag=1, differences=d)
  }


  intercept <- coefs[length(coefs)]


  for(i in 1:pred){
    temp <- intercept
    l.s <- length(series.diff)
    if(AR > 0){
      for(j in 1:AR){
        temp <- temp + coefs[j] * (series.diff[l.s - j + 1] -
intercept)
      }
    }
    if(MA > 0){
      for(j in (1:MA)){
        temp <- temp + coefs[j + AR] * innov[l.s - j + 1]
      }
    }


    innov <- c(innov, 0)
    series.diff <- c(series.diff, temp)
  }


  if(d==0){
     series.undiff <- series.diff
  }
  else {
    series.undiff <- diffinv(series.diff, lag=1, differences=d, xi =
series[1:d])
  }


  return(list(series = series.undiff, innov = innov))
}