An apparently different problem, but in actually very close to
parameter estimation, is that of forecasting. We consider a
situation where there is a data set available on both and
for elements
to
. We can not only estimate the relationship
between
and
. With this estimation, we can use it to
forecast or predict the value of the variable
for any given
value of
. Suppose that
is a known value of the
regressor, and we are interested in predicting
, the
value of
associated with
.
It is evident that, in general, if takes the value
,
the predicted value of
, is given by:
The conditional mean of the predictor of given
is
Thus,
is an unbiased conditional predictor of
.
Because and
are estimated with imprecision,
is also subject to error. To take account of this, we
compute the variance and confidence interval for the point
predictor. The prediction error is:
Clearly the expected prediction error is zero. The variance of
is
We see that
is a linear combination of normally
distributed variables. Thus, it is also normally distributed. and
so
By inserting the sample estimate
for
,
We can construct a prediction interval for in the usual
way, we derive a
per cent forecast interval for
where
is the critical value from the
distribution with
degrees of freedom.
We implement the following experiment using the following
Quantlet. We generate a sample of the following data
generating process:
, the vector of explanatory
variables is
. First of all, we estimate
and
, then we obtain predictions for several values of
.
In this program, the vector of takes values from
to
for the estimation, after this we want to calculate a interval
prediction for
. This procedure gives the Figure
1.11
The sample given in the previous section is that for predicting a
point. We also like the variance of the mean predictor. The
variance of the prediction error for the mean
is
We see that
is a linear combination of normally
distributed variables. Thus, it is also normally distributed. By
inserting the sample estimate
for
The
per cent confidence interval of the mean
forecast is given by
where
is the critical value from the
distribution with
degrees of freedom.