our goal will be to guess, or
forecast, XT or more generally XT+r. There are a variety
of ad hoc methods as well as a variety of statistically
derived methods. I illustrate the ad hoc methods with the
exponentially weighted moving average (EWMA). In this case
we simply take
Statistically based methods concentrate on some measure of the size of ; the mean squared prediction error is the most common.
must be some function
The mean squared prediction error can be seen by conditioning
on the data to be minimized by
When T is large the computation of these forecasts is difficult in general. There are some shortcuts, however.
Forecasting AR(p) processes
When the process is an AR the computation of the conditional
expectation is easier:
Notice the the forecast into the future uses current values where these are available and forecasts already calculated for the other X's.
Forecasting ARMA(p,q) processes
An ARMA(p,q) can be inverted to be an infinite order AR process. We could then use the method just given for the AR except that now the formula actually mentions values of Xtfor t < 0. In practice we simply truncate the series and ignore the missing terms in the forecast, assuming that the coefficients of these omitted terms are very small. Remember each term is built up out of a geometric series for with .
A more direct method goes like this:
Whenever the time index on an epsilon is T or more the conditional
expectations are 0. For T+r-i < T we need to guess the value
The same recurtion can be re-arranged to
As we discussed in the section on estimation these computed estimates of the epsilon's can be improved by backcasting the values of for negative t and then forecasting and backcasting, etc.
Forecasting ARIMA(p,d,q) series
Z=(I-B)d X and X is ARIMA(p,d,q) then we:
compute Z, forecast Z and reconstruct X by
undoing the differencing. For d=1 for example we
Forecast standard errors
You should remind yourself that the computations of conditional
expectations we have just made used the fact that the a's and
b's are constants - the true parameter values. In fact we
then replace the parameter values with estimates. The quality of
our forecasts will be summarized by the forecast standard error:
If then so that our forecast standard error is just the variance of .
Consider first the case of an AR(1) and one step ahead forecasting:
For forecasts further ahead in time we have
Turn now to a general ARMA(p,q). Rewrite the process as the infinite
Parallel to the AR(1) argument we see that
A simpler approach is to write the process as an infinite order MA:
Finally consider forecasting the ARIMA(p,d,q) process
(I-B)d X= W where W is ARMA(p,q).
The forecast errors in X can clearly be written as a linear combination of
forecast errors for W permitting the forecast error in X to be written as
a linear combination of the underlying errors
As an example consider
first the ARIMA(0,1,0) process
The forecast of
is just 0 and so the forcast of XT+r is just
The forecast error is
The S-Plus function arima.forecast can do the forecasting.
I have ignored the effects of parameter estimation throughout. In ordinary least squares
when we predict the Y corresponding to a new x we get a forecast standard error