Chapter 17.3, Problem 10E

With a smoothing constant of α = .2, equation (17.2) shows that the forecast for week 13 of the gasoline sales data from Table 17.1 is given by F₁₃ = .2Y₁₂ + .8F₁₂. However, the forecast for week 12 is given by F₁₂ = .2Y₁₁ + .8F₁₁. Thus, we could combine these two results to show that the forecast for week 13 can be written

     $F_{13}=.2Y_{12}+.8(.2Y_{11}+.8F_{11})=.2Y_{12}+.16Y_{11}+.64Y_{11}+.64F_{11}$

1. a. Making use of the fact that F₁₁ = .2Y₁₀ + .8F₁₀ (and similarly for F₁₀ and F₉), continue to expand the expression for F₁₃ until it is written in terms of the past data values Y₁₂, Y₁₁, Y₁₀, Y₉, Y₈, and the forecast for period 8.
2. 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.