Let us assume that the series
follows the
general
model that can be rewritten in terms of
the present and past values of
:
Our objective is to forecast a future value
given our
information set that consists of the past values
. The future value
is generated by model
(4.32), thus
Let us denote by the
-step ahead forecast of
made at origin
. It can be shown that, under reasonable weak conditions,
the optimal forecast of
is the conditional expectation of
given the information set, denoted by
.
The term optimal is used in the sense that minimizes the Mean Squared Error
(MSE). Although the conditional expectation does not have to be a linear
function of the present and past values of
, we shall consider linear
forecasts because they are fairly easy to work with. Furthermore, if the
process is normal, the Minimum MSE forecast (MMSE) is linear. Therefore,
the optimal forecast
-step ahead is:
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The -step ahead forecast error is a linear combination of the future
shocks entering the system after time
:
Given these results, if the process is normal, the
forecast
interval is:
For , the one-step ahead forecast error is
, therefore
can be interpreted as the one-step ahead
prediction error variance.
Let us consider again the general
model that can
be written as well as:
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In practice, the parameters of the
model should be
estimated, but for convenience, we assume that they are given.
Following the results of section 4.4.2, if the series follows
an
model, the
-step ahead forecast at origin
is
given by:
The expression (4.37) is called the eventual forecast
function, because it holds only for
. If
, then
the eventual forecast function holds for all
. This eventual
forecast function passes through the
values given by
.
Let us consider the
process in
deviations to the mean
:
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Let us take as an example the forecast of an process with
and
. Figure 4.12 shows the eventual
forecast function of the model considered (dotted line). It can be observed
that this function increases at the beginning until it reaches the mean
value, 3. This result, that is,
Let us consider the following
model:
If , we get the random walk model (4.20) and
the eventual forecast function takes the form:
The eventual forecast function for the random walk plus drift model (4.21) is the solution to the following difference equation:
It can be observed in figure 4.13 that the interval
forecast limits increase continuously as the forecast horizon
becomes larger. It should be taken into account that when
the process is nonstationary the limit
does not exist.
Let us consider the
model: