How to Compute the Estimated SEs of Forecasted Y and the Forecast
Error
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 XF 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 XF 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  b0  b1 X F .
Results of the original regression that produced the estimates of b0 and b1are 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.
Forecast Values
40 F
-5.17 Forecasted Y
210 Actual Y 204.83 Forecast Error
XF
E(Actual Y )
Regression on Original
Data
4.99
10.16
0.17
1.98
0.98
4.71
863.69
19
19135
420.94
Estimating SE via
Formula
X F - Avg X
SD(X )
Average(X)
Estimated SE of
Forecast Error
30.00
6.06
10
7.01
209.56
4.73
Regression on Transformed
Data
4.99
209.56
0.17
5.19
0.98
4.71
863.69
19
19135
420.94
Estimating SE via
Transformed Regression
SE(Forecasted Y )2
26.95
RMSE2
Estimated
Variance of
Forecast Error
Estimated SE of
Forecast Error
22.15
49.10
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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2
 X)
1  (X
 .
SE ( ForecastError )  SDErrors  1    value
n  n  SD X 
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 X1F, X2F, and
X3F.
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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