Question 13.61

13.61 Exponential smoothing forecast equation.

We have learned that the exponential smoothing forecast equation is written as

${\hat{y}}_{t} = w y_{t - 1} + (1 - w) {\hat{y}}_{t - 1}$

Show that the equation can be written as

${\hat{y}}_{t} = {\hat{y}}_{t - 1} + w e_{t - 1}$

where $e_{t - 1}$ is the residual, or prediction error, for period $t - 1$ .
Explain in words how the reexpressed equation can be interpreted.

13.61

(a) ${\hat{y}}_{t} = w y_{t - 1} + (1 - w) {\hat{y}}_{t - 1} = w y_{t - 1} + {\hat{y}}_{t - 1} - w {\hat{y}}_{t - 1} = {\hat{y}}_{t - 1} + w (y_{t - 1} - {\hat{y}}_{t - 1}) = {\hat{y}}_{t - 1} + w e_{t - 1}$ . (b) The forecasted value is equal to the previous predicted value plus a percentage of the residual of the previous value.