Example 16.6 Regression-Based Trend Models
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b EXXON.XLS n The file contains the data on quarterly sales for the period from 1986 through the second quarter of n A times series chart of these sales is shown on the next slide shows that there is some evidence of an upward trend in the early years, but that there is no obvious trend during the 1990s. n Does a simple exponential smoothing model track these data well? How do the forecasts depend on the smoothing constant ?
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution n We will use StatPro to implement the simple exponential smoothing model. n We do this again with the StatPro/Forecasting menu item. We first specify that the data for validation, and we ask for 8 quarters of future forecasts. n We then fill out the next dialog box as shown on the next slide. That is, we select the exponential smoothing option in the upper left, select the Simple option, choose a smoothing constant and elect not to optimize, and specify that the data are not seasonal.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n On the next dialog sheet we ask for time series charts of the series with the forecasts superimposed and the series of forecast errors. n The results appear on the next three slides. n The heart of the method takes place in columns F, G, and H of the next slide. Column F calculates the smoothed levels, column G calculates the forecasts, and column H calculates the forecast errors as the observed values minus the forecasts.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Initialization n One exception to this schedule is in row 2. Every exponential smoothing method requires initial values, in this case the initial smoothed level in cell F2. n There is no way to calculate this value, L 1, from the equation because previous value L 0, is unknown. n Different implementations of exponential smoothing initialize in different ways.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n Note that the 8 future forecasts are all equal to the last calculated smoothed level, the one for the second quarter of 1996 in cell F43. n The fact that these remain constant is a consequence of the assumption behind simple exponential smoothing, namely, that the series is not really going anywhere. n Therefore, the last smoothed level is the best indication of future values of the series.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n The plot of Forecasts from Simple Exponential Smoothing shows that the forecast series superimposed on the original series. n We see the obvious smoothing effect of a relatively small level. n The forecasts do not track the series very well, but if the various zigzags in the original series are really random noise, then perhaps we don’t want the forecasts to track these random ups and downs too closely.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n Perhaps we instead prefer a forecast series that emphasizes the basic underlying pattern. n The Plot for Forecast Errors from the Simple Exponential Smoothing shows the time series of forecast errors. n Although these errors are sometimes quite large, they do appear to be fairly random. n We see several summary measures of the forecast errors from the Simple Exponential Smoothing Output.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n The RMSE and MAE indicate that the forecasts from this model are typically off by a magnitude of about 2300, and the MAPE indicates that this magnitude is about 7.4% of sales. This is a fairly sizable error. n One way to try to reduce it is to use a different smoothing constant. n The optimal for this example is somewhere between 0.8 and 0.9, although RMSE is relatively constant for any between these two values.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n The plot on the next slide shows the forecast series with = n Its values of RMSE, MAE and MAPE, not shown, are 1885, 1355, and 5.74%. n The forecast series now appears to track the original series very well – or does it? n A closer look shows that we are essentially forecasting each quarter’s sales value by the previous sales value.
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b
| 16.1a | 16.2 | 16.3 | 16.4 | 16.5 | 16.2a | 16.7 | 16.7a | 16.7b16.1a a a16.7b Solution -- continued n Therefore, any time the series changes much from one quarter to the next, the forecast will be way off. n There is no doubt that = 0.85 gives lower summary measures for the forecast errors, but it is possibly reacting too quickly to random noise and might not really be showing us the basic underlying pattern of sales that we see with = 0.2.