Forecasting and Forecast Errors
First, we use the words
forecasting
and
predicting
interchangeably.
The word
forecasting
is often reserved for
predicting
values of the dependent variable beyond the current
time period or beyond the range of the sample of data.
But here, we assume these two words
mean the same thing:
estimating a value for the dependent variable given a specific value for
the independent variable
.
There are two things that we can forecast: (1) the expected value of the dependent
variable 
0
[
]
E Y X
; or (2) an actual value of the dependent variable 
0
Y X
.
We have just one
estimator that we can use:
0
0
1
0
ˆ
Y
b
b X
=
+
.
In forecasting, there will be errors that occur, and
the magnitude of those errors will logically be affected by the population value we are
forecasting or predicting and the level of the independent variable for which we are forecasting.
If we are to forecast, we will be interested in the sampling properties (distributions) for these two
forecast errors.
Let’s review/clarify three related concepts:
(1) the sampling distribution for the estimator
ˆ
Y
; (2) the sampling properties (a sampling distribution) of the errors in forecasting
0
[
]
E Y X
;
and (3) the sampling properties (a sampling distribution) of the errors in forecasting
0
Y X
.
Just
as the distribution of the disturbance (
u
) also is the distribution of the dependent variable (
Y
)
1
, so
too is the
sampling distribution
of our first type of forecast error equivalent to the
sampling
distribution
of our estimator (
ˆ
Y
).
The second forecast error has a different set of sampling
properties.
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 Spring '08
 STAFF
 Normal Distribution, Variance, Forecast error, Cov

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