Postscript version of these notes
STAT 804
Lecture 18 Notes
Forecast standard errors
You should remind yourself that the computations of conditional
expectations we have made used the fact that the 's and
'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:
We will compute this ignoring the estimation of the parameters
and then discuss how much that might have cost us.
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:
The variance of this forecast is
so that
the forecast standard error is just
.
For forecasts further ahead in time we have
and
Subtracting we see that
so that we may calculate forecast standard errors recursively.
As
we can check that the forecast variance converges
to
which is simply the variance of individual s. When you forecast a
stationary series far into the future the forecast error is just the
standard deviation of the series.
Turn now to a general ARMA(). Rewrite the process as the infinite
order AR
to see that again, ignoring the truncation of the infinite sum in the
forecast we have
so that the one step ahead forecast standard error is again
.
Parallel to the AR(1) argument we see that
The errors on the right hand side are not independent of one another so that
computation of the variance requires either computation of the covariances or
recognition of the fact that the right hand side is a linear combination of
.
A simpler approach is to write the process as an infinite order MA:
for suitable coefficients . Now if we treat conditioning on the data
as being effectively equivalent to conditioning on all for we
are effectively conditioning on
for all . This means that
and the forecast error is just
so that the forecast standard error is
Again as
this converges to .
Finally consider forecasting the ARIMA() process
where is ARMA().
The forecast errors in can clearly be written as a linear combination of
forecast errors for permitting the forecast error in 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 is just
The forecast error is
whose standard deviation is
. Notice that the forecast standard
error grows to infinity as
. For a general ARIMA()
we have
and
which can be combined with the expression above for the forecast error for an ARMA()
to compute standard errors.
Software
The S-Plus function arima.forecast can do the forecasting.
Comments
I have ignored the effects of parameter estimation throughout. In ordinary least squares
when we predict the corresponding to a new we get a forecast standard error
of
which is
The procedure used here corresponds to ignoring the term
which is
the variance of the fitted value. Typically this value is rather smaller than the 1 to
which it is added. In a 1 sample problem for instance it is simply . Generally
the major component of forecast error is the standard error of the noise and the
effect of parameter estimation is unimportant.
In regression we sometimes compute perdiction intervals
The multiplier is adjusted to make the coverage probability
close to a desired coverage
probability such as 0.95. If the errors are normal then we can get
by taking
. When the
errors are not normal, however, the error in is dominated by
which is not normal so that the coverage probability can
be radically different from the nominal. Moreover, there is no particular
theoretical justification for the use of critical points. However, even
for non-normal errors the prediction standard error is a useful summary of
the accuracy of a prediction.
Richard Lockhart
2001-09-30