Compute forecasts and 90% forecast intervals for ARIMA(p,1,q) models Consider the two models (ARIMA(1,1,0) and (ARIMA(0,1,1)):

 A: For the ARIMA(0,1,1) model in part (b), since you only know $r_{49}$ and $r_{50}$ you need to do finite history forecasting, that is, you want the expectation and variance of $r_{51},\dots, r_{54}$ conditional on $r_{49}$ and $r_{50}$ only.  This differs form the infinite history forecast which depends on all the past for models with a moving average part.  The most efficient way is to compute this is usually via Kalman filtering and forecasting. Using functions for this in the stats R-package, we find
# construct state space representation
mod <- makeARIMA(theta=-.43, phi=0, Delta = 1)
# run the kalman filter given the two observations
mod <- KalmanRun(c(33.4, 33.9), mod, update=TRUE)
# put final state estimate into new model object
mod <- attr(mod,"mod")
# compute forecasts via the Kalman filter
> KalmanForecast(4,mod)
$pred
[1] 33.71854 33.71854 33.71854 33.71854

$var
[1] 1.028853 1.353753 1.678653 2.003553

From this you can easily get 90% forecast limits.  Note that the 1-step ahead forecast variance is larger than the white noise variance $\sigma_a^2=1$ (it would be equal to $\sigma_a^2$ only for an infinite history forecast).
