Introducing Seasonality into the Forecasts

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Although a preliminary analysis has shown that there was no evidence of any consistent seasonal behavior, some seasonal models were evaluated over the months 49 to 60 inclusive of the test period. The Box-Jenkins AR (1) model, which was the best of that class of models for the largest number of items, 6 out of 10, was compared with a seasonal AR(1) model for the months of the test period. The nonseasonal model gave lower sums of squared errors for 6 of the 10 items and gave a lower sum of root mean squared errors over all 10 items. The results are shown in Table 8.

Table 8: A comparison of ARl and seasonal AR1 for the 10 items

Items	SSE (AR1)	SSE (seasonal AR1)	F-Statistics (larger SSE being the numerator)	RMSE (AR1)	RMSE (seasonal AR1)
SAV740	74472	75618	1.015	78.78	79.38
SAV091	5273	5395.2	1.023	20.96	21.20
SAV763	13015904	12677014	1.027	1140.87	1125.52
SAV012	2242976	1196782	1.874	432.34	315.80
REM061	2208355	1865122	1.184	428.99	394.24
SAV739	89181	93241	1.046	86.21	88.15
CUA085	941180	1042557	1.108	280.06	294.75
SAV013	3308384	2692286	1.229	525.07	473.66
CUA778	10444654	14970750	1.433	932.95	1116.94
REM037	3327895	3674954	1.104	526.62	553.40
			5 % Critical Region, F>2.69	SUM= 4452.85	SUM= 4463.04

Because of this result, no further analysis was conducted with seasonal Box-Jenkins models. It was found that the Holt-Winters seasonal model gave better forecasts for the test period than the Holt's model. There was a reduction of around 10% in the sum of the root mean squared errors over the 10 items in the sample. This does not necessarily indicate that there is after all consistent seasonality in the demand over time. It is well known that a large outlier value for a month in one of the initial years of setting up the H-W model can lead to a significant "seasonal" factor being produced for that month, which can take several years to disappear in the usual updating process. The greater number of parameters in the Holt-Winters method compared to Holt's alone can in some circumstances take out some of the outlier values and exclude their effect from the updating of the underlying level and trend. It was decided therefore to research the effect of including the Holt-Winters in place of the Holt model in the earlier analysis.

The results of these analyses are summarized in Table 9, in the same format as for Table 7. The use of the "seasonal" Holt-Winters model gives a reduction of about 10% in the sum of the root mean squared errors compared to the Holt's model. The value is thus about the same as obtained with the best of the combination of nonseasonal forecasting models. More importantly, the same reduction of 10% in the sum of the root mean squared errors occurs for every weighting method, as can be seen by comparing the entries in Tables 7 and 9 row by row. The relative rankings of the various weighting methods is very similar to that found in Table 7. The only minor difference is the relatively poorer performance of the Fixed Common Weights over the four forecasting methods. The new hybrid method again gives the best performance, saving around 10% compared with using the Holt-Winters method alone.

Table 9: Summary of results if Holt-Winters method replaces Holt's method

FORECASTS	WEIGHTS	Sum of Root Mean Squared Errors	Percentage Saving on Holt-Winters
Holt-Winters		3900
Best over Past Year		3652	6.4%
Four Common Methods	Fixed Common	3767	3.4%
Four Common Methods	Individual Fixed	3553	8.9%
	Individual Rolling	3626	7.1%
	Hybrid	3507	10.1%
Individual Three	Individual Fixed	3558	8.8%
Best Methods	Individual Rolling	3576	8.3%
	Hybrid	3527	9.6%

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