This paper presents expressions for the variance of the forecast error for arbitrary lead times for both the additive and multiplicative Holt-Winters seasonal forecasting models. It is shown that even when the smoothing constants are chosen to have values between zero and one, when the period is greater than four, the variance may not be finite for some values of the smoothing constants. In addition, the regions where the variance becomes infinite are almost the same for both models. These results are of importance for practitioners, who may choose values for the smoothing constants arbitrarily, or by searching on the unit cube for values which minimize the sum of the squared errors when fitting the model to a data set. It is also shown that the variance of the forecast error for the multiplicative model is nonstationary and periodic.
KEY WORDS Seasonal models Exponential smoothing Error analysis Variance
Winters (1 960) presented both additive and multiplicative seasonal forecasting models in his paper. Properties of the additive seasonal model (ASM) have been discussed by many authors, including Thiel and Wage (1964), Brenner, D'Esopo and Fowler (1968), McClain (1974) and McKenzie (1976). However, methods for the computation of the variance of the forecast errors for arbitrary lead times do not appear to have been previously discussed in the published literature for either of the two models. As a forecast generally requires an estimate of its uncertainty (Montgomery and Johnson, 1976), it is of importance to be able to compute the variance of the forecast error, and to understand how it changes as the forecast horizon increases. Bowerman and O'Connell (1 979) give expressions for 'an approximate confidence interval' based on an 'intuitive method' for both models, but they can be seriously in error. Sweet (1981) derived expressions for the variance of the forecast error for both of the models when forecasting is performed using a discounted least squares method with one smoothing constant. In Winters' models three smoothing constants are used. As the governing equations for the ASM are linear difference equations, the author used the method of generating functions to derive equations to be used for computing the variance. For lead time equal to one, an efficient method of computation for the ASM can be constructed using the impulse response function (see McClain, 1974