Reconsider the sales data for a certain product given in Prob. 20.5-4. The company's management now has decided to discontinue incorporating seasonal effects into its forecasting procedure for this product because there does not appear to be a substantial seasonal pattern. Management also is concerned that exponential smoothing may not be the best forecasting method for this product and so has decided to test and compare several forecasting methods. Each method is to be applied retrospectively to the given data and then its MSE is to be calculated. The method with the smallest value of MSE will be chosen to begin forecasting. Apply this retrospective test and calculate MSE for each of the following methods. (Also obtain the forecast for the upcoming quarter with each method.)
(a) The moving-average method based on the last four quarters, so start with a forecast for the fifth quarter.
(b) The exponential smoothing method with α = 0.1. Start with a forecast for the third quarter by using the sales for the second quarter as the latest observation and the sales for the first quarter as the initial estimate.
(c) The exponential smoothing method with α = 0.3. Start as described in part (b).
(d) The exponential smoothing with trend method with α = 0.3 and ß = 0.3. Start with a forecast for the third quarter by using the sales for the second quarter as the initial estimate of the expected value of the time series (A) and the difference (sales for second quarter minus sales for first quarter) as the initial estimate of the trend of the time series (B).
(e) Compare MSE for these methods. Which one has the smallest value of MSE?
Prob. 20.5-4
A manufacturer sells a certain product in batches of 100 to wholesalers. The following table shows the quarterly sales figure for this product over the last several years.
The company incorporates seasonal effects into its forecasting of future sales. It then uses exponential smoothing (with seasonality) with a smoothing constant of α = 0.1 to make these forecasts. When starting the forecasting, it uses the average sales over the past four quarters to make the initial estimate of the seasonally adjusted constant level A for the underlying constant-level model.
(a) Suppose that the forecasting started at the beginning of 1997. Use the data for 1996 to determine the seasonal factors and then determine the forecast of sales for each quarter of 1997.
(b) Suppose that the forecasting started at the beginning of 1998. Use the data for both 1996 and 1997 to determine the seasonal factors and then determine the forecast of sales for each quarter of 1998. T (c) Suppose that the forecasting started at the beginning of 2000 Use the data for 1996 through 1999 to determine the seasonal factors and then determine the forecast of sales for each quarter of 2000.
(d) Under the assumptions of the constant-level model, the forecast obtained for any period of one year also provides the best available forecast at that time for the same period in any subsequent year. Use the results from parts (a), (b), and (c) to record the forecast of sales for Quarter 4 of 2000 when entering Quarter 4 of 1997, 1998, and 2000, respectively.
(e) Evaluate whether it is important to incorporate seasonal effects into the forecasting procedure for this particular product.
(f) Evaluate how well the constant-level assumption of the constant-level model (after incorporating seasonal effects) appears to hold for this particular product.