Hi Dwayne,
PROC ESM does not provide a way to output the backcast values used to obtain the Starting Level and Trend values for the Linear exponential smoothing model, however, the smoothing calculations for the backcasts are essentially the same as the smoothing calculations for the "forwards casting". These smoothing calculations are described in the following section of the SAS/ETS User's Guide:
Equations for the Smoothing Models:
https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_tffordet_sect021.htm&docsetVersion=15.1&locale=en
Please note, however, that for backcasting, the sign of the initial trend parameter is reversed, since the series is sorted in reverse order. Following, please find a description of the methodology used to compute the backcasts for MODEL=LINEAR in PROC ESM:
1) Add a time trend variable to the original data set, for example, "t", which is simply the observation number 1, 2, ..., n.
2) Obtain the initial Level and Slope values:
Initial Level is the value of the last observation in the data set
Initial Slope is the negative of the slope coefficient from regressing the time series variable on a time trend variable (ie. 1, 2,...n)
3) Sort the data, descending, by the time trend variable.
4) Backcast using the reverse-ordered data:
level [1] is: alpha*y[1] + (1-alpha)*(initial level + initial trend)
trend [1] is: gamma*(level[1] - initial level) + (1-gamma)*initial trend
5) Continue with smoothing computations through reverse ordered series using the equations noted in the above link.
6) To obtain the Starting Level and Trend values from the backcasts, go to the last observation from step 5)
Starting_level = Level[n] + Trend[n] (from step 5)
Starting_trend = -Trend[n] (from step 5);
I hope this description helps, Dwayne!
DW
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