With a smoothing constant of a = 0.2, equation (5.7) shows that the forecast for week 13 of the gasoline sales data from Table 5.1 is given by y^13 = 0.2y12 1 0.8y^12. However, the forecast for week 12 is given by y^12 = 0.2y11 1 0.8y^11. Thus, we could combine these two results to show that the forecast for week 13 can be written y^13 = 0.2y12 1 0.8 (0.2y11 1 0.8y^11 ) = 0.2y12 1 0.16y11 1 0.64y^11
a. Making use of the fact that y^11 = 0.2y10 1 0.8y^10 (and similarly for y^10 and y^9), continue to expand the expression for y^13 until it is written in terms of the past data values y12, y11, y10, y9, y8, and the forecast for period 8, y^8.
b. Refer to the coefficients or weights for the past values y12, y11, y10, y9, y8. What observation can you make about how exponential smoothing weights past data values in arriving at new forecasts? Compare this weighting pattern with the weighting pattern of the moving averages method.