These data show the number of gallons of gasoline sold by a gasoline distributor in Bennington, Vermont, over the past 12 weeks.
(1,000s
of gallons)
1
17
2
21
3
19
4
23
5
18
6
16
7
20
8
18
9
22
10
20
11
15
12
22
With a smoothing constant of a = 0.8, the exponential smoothing forecast equation $\hat{y}_{t+1} = \alpha y_t + (1-\alpha)\hat{y}_t$, shows that the forecast for week 13 of the gasoline sales data from the table is given by $\hat{y}_{13} = 0.8Y_{12} + 0.2\hat{y}_{12}$. However,
forecast for week 12 is given by $\hat{y}_{12} = 0.8Y_{11} + 0.2\hat{y}_{11}$. Thus, we could combine these two results to show that the forecast for week 13 can be written
$\hat{y}_{13} = 0.8Y_{12} + 0.2(0.8Y_{11} + 0.2\hat{y}_{11}) = 0.8Y_{12} + 0.16Y_{11} + 0.04\hat{y}_{11}$
(a) Making use of the fact that $\hat{y}_{11} = 0.8Y_{10} + 0.2\hat{y}_{10}$ (and similarly for $\hat{y}_{10}$ and $\hat{y}_9$), continue to expand the expression for $\hat{y}_{13}$ until it is written in terms of the past data values $Y_{12}, Y_{11}, Y_{10}, Y_9, Y_8$, and the forecast for period 8, $\hat{y}_8$. (Let yhatg to denote $\hat{y}_8$.)