Example 16.7

Exponential Smoothing

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COCACOLA.XLS The data in this spreadsheet represents quarterly

sales for Coca Cola from the first quarter of 1986 through the second quarter of 1996.

As we might expect there has been an upward trend in sales during this period and there is also a fairly regular seasonal pattern as shown in the time series plot of sales.

Sales in warmer quarters, 2 and 3, are consistently higher than in the colder quarters, 1 and 4.

How well can Winter’s method track this upward tend and seasonal pattern?

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Time Series Plot of Coca Cola Sales

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Seasonality Seasonality if defined as the consistent month-to-

month (or quarter-to-quarter) differences that occur each year.

The easiest way to check if there is seasonality in a time series is to look at a plot of the times series to see if it has a regular pattern of up and/or downs in particular months or quarters.

There are basically two extrapolation methods for dealing with seasonality: – We can use a model that takes seasonality into account or;

– We can deseasonalize the data, forecast the data, and then adjust the forecasts for seasonality.

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Seasonality -- continued Winters’ model is of the first type. It attacks

seasonality directly.

Seasonality models are usually classified as additive or multiplicative.

– An additive model finds seasonal indexes, one for each month, that we add to the monthly average to get a particular month’s value.

– A multiplicative model also finds seasonal indexes, but we multiply the monthly average by these indexes to get a particular month’s value.

Either model can be used but multiplicative models are somewhat easier to interpret.

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Winter’s Model of Seasonality

Winters’ model is very similar to Holt’s model - it has level and trend terms and corresponding smoothing constants alpha and beta - but it also has seasonal indexes and a corresponding smoothing constant.

The new smoothing constant controls how quickly the method reacts to perceived changes in the pattern of seasonality.

If the constant is small, the method reacts slowly; if the constant is large, it reacts more quickly.

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Using Winters’ Method To produce the output from Winters’ method with

StatPro we proceed exactly as with the other exponential methods.

In particular, we fill out the second main dialog box as shown below.

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Portion of Output from Winters’ Method

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The Output The optimal smoothing constants (those that

minimize RMSE) are 1.0, 0.0 and 0.244. Intuitively, these mean react right away to changes in level, never react to changes in trend, and react fairly slowly to changes in the seasonal pattern.

If we ignore seasonality, the series is trending upward at a rate of 67.107 per quarter.

The seasonal pattern stays constant throughout this 10-year period.

The forecast series tracks the actual series quite well.

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Plot of the Forecasts from Winters’ Method

The plot indicates that Winters’ method clearly picks up the seasonal pattern and the upward trend and projects both of these into the future.

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In Conclusion

Some analysts would suggest using more “typical” values for the constants such as alpha=beta=0.2 and 0.5 for the seasonality constant.

To see how these smoothing constants would affect the results, we can simply substitute their values into the range B6:B8.

The summary measures get worse, yet the plot still indicates a very good fit.