Workbook: HowToFindSEForecast.xls
.
This document assumes you have read Section 15.6 on “The Standard Error of the Forecast and the Standard Error of the Forecast Error.”
The formulas for the estimated SEs of Forecasted Y and the Forecast Error are somewhat difficult to work with. Fortunately there is a simple procedure that makes it relatively easy to obtain these SEs. The procedure works like this:
1.
Transform the original X variables by subtracting the value of X
F
from each observation on X .
2.
Run the regression of Y on transformed X .
3.
The slope coefficient, its estimated SE, and the RMSE will all remain the same as they were in the original regression. The intercept coefficient will equal Forecasted Y, and its reported SE is the estimated SE of
Forecasted Y .
4.
To obtain the estimated SE of the Forecast Error, use the square-root formula:
Estimated SE

Forecast Error


RMSE
2 
SE

Forecasted Y

.
This procedure is illustrated in the Bivariate sheet of HowToFindSEForecast.xls
, which you should open now. Original values of X are in column A; transformed values of X are in column D. Figure 1 displays the calculations made to obtain the SE of the Forecast Error. The top box contains the value of X
F
and
Forecasted Y as well as four values which would not be observed: the expected value of Actual Y , obtained using the true parameter values; the actual error,

F
;
Actual Y ; and the Forecast Error. These values are in bold red on the workbook.
As always,
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Forecasted Y
 b
0
 b
1
X
F
.
Results of the original regression that produced the estimates of b
0
and b
1 are in the middle on the left in Figure 1. To the right are results of the regression of the original Y on Transformed X . Note that all the regression statistics are the same with the exception of the estimate of the intercept and its SE. As advertised, the intercept estimate in the transformed regression is equal to
Forecasted Y. It can be shown that the estimated SE is the estimated SE of the
Forecast.
X
F
E(Actual Y )
40

F
Forecast Values
-5.17
Forecasted Y
210 Actual Y 204.83
Forecast Error
209.56
4.73
Regression on Original
Data
4.99
0.17
0.98
10.16
1.98
4.71
863.69
19
19135 420.94
Regression on Transformed
Data
4.99
0.17
0.98
863.69
19135
209.56
5.19
4.71
19
420.94
Estimating SE via
Formula
Estimating SE via
Transformed Regression
SE(Forecasted Y )
2
26.95
X
F
- Avg X 30.00
SD( X ) 6.06
RMSE
2
Estimated
Variance of
22.15
Average(X)
Estimated SE of
10 Forecast Error
Estimated SE of
49.10
Forecast Error 7.01
Forecast Error 7.01
Figure 1. Two Methods for Computing the SE of the Forecast Error.
Source: HowToFindSEForecast.xls
.
Figure 1 shows two routes to obtain the estimated SE of the Forecast
Error. On the left we display various steps in implementing the formula:
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SE ( ForecastEr ror )

SD
Errors

1

1
 n
( X value
 n

SD
X
X )
2
.
On the right, we show the steps needed to obtain the Estimated SE of the
Forecast Error via the transformed regression. The sheet makes clear that the two methods produce exactly the same answer.
It is very easy to extend our transformed regression method to the multiple regression case, as demonstrated by the MultipleRegression sheet in
EstimatingSEForecast.xls. In that sheet you can choose values of X
1F
, X
2F
, and
X
3F
. The transformed X
’s are simply the original
X
’s less the respective assigned forecast values.
To demonstrate that this procedure works, we used the MCSim Add-In on the
Multivariate sheet, tracking the values of the Forecast Error (cell K5) and the Estimated
SE of the Forecast Error (cell K26). The results are in the MCMultipleRegression sheet.
The relevant comparison is between the empirical average value of the Estimated SE and the empirical spread of the Forecast Error. The latter is the Monte Carlo approximation to the true SE of the Forecast Error. On average it appears that the Estimated SE of the
Forecast Error slightly underestimates the SE of the Forecast Errors. This is not surprising, because we know that the RMSE is a biased estimator of the SD of the errors.
For more on this procedure, see Wooldridge (2003), p. 203.
Reference:
Wooldridge, Jeffrey M. (2003) Introductory Econometrics: A Modern Approach. Second
Edition. Mason, Ohio: Southwestern.
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