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5.6 Adjusting for Seasonal Variations

Time-series forecasting such as that in the example of Midwestern Manufacturing involves looking at the trend of data over a series of time observations. Sometimes, however, recurring variations at certain seasons of the year make a seasonal adjustment in the trend line forecast necessary. Demand for coal and fuel oil, for example, usually peaks during cold winter months. Demand for golf clubs or suntan lotion may be highest in summer. Analyzing data in monthly or quarterly terms usually makes it easy to spot seasonal patterns. A seasonal index is often used in multiplicative time-series forecasting models to make an adjustment in the forecast when a seasonal component exists. An alternative is to use an additive model such as a regression model that will be introduced in a later section.

Seasonal Indices

A seasonal index indicates how a particular season (e.g., month or quarter) compares with an average season. An index of 1 for a season would indicate that the season is average. If the index is higher than 1, the values of the time series in that season tend to be higher than average. If the index is lower than 1, the values of the time series in that season tend to be lower than average.

Seasonal indices are used with multiplicative forecasting models in two ways. First, each observation in a time series is divided by the appropriate seasonal index to remove the impact of seasonality. The resulting values are called deseasonalized data. Using these deseasonalized values, forecasts for future values can be developed using a variety of forecasting techniques. Once the forecasts of future deseasonalized values have been developed, they are multiplied by the seasonal indices to develop the final forecasts, which now include variations due to seasonality.

There are two ways to compute seasonal indices. The first method, based on an overall average, is easy and may be used when there is no trend present in the data. The second method, based on a centered-moving-average approach, is more difficult, but it must be used when there is trend present.

Calculating Seasonal Indices with No Trend

When no trend is present, the index can be found by dividing the average value for a particular season by the average of all the data. Thus, an index of 1 means the season is average. For example, if the average sales in January were 120 and the average sales in all months were 200, the seasonal index for January would be 120/200 = 0.60, so January is below average. The next example illustrates how to compute seasonal indices from historical data and to use these in forecasting future values.

Monthly sales of one brand of telephone answering machine at Eichler Supplies are shown in Table 5.8 for the two most recent years. The average demand in each month is computed, and these values are divided by the overall average (94) to find the seasonal index for each month. We then use the seasonal indices from Table 5.8 to adjust future forecasts. For example, suppose we expected the third year’s annual demand for answering machines to be 1,200 units, which is 100 per month. We would not forecast each month to have a demand of 100, but we would adjust these based on the seasonal indices as follows:

A table shows the average seasonal indices. — Table 5.8 Answering Machine Sales and Seasonal Indices

Table 5.8 Full Alternative Text

Jan.	$\frac{1, 200}{12} \times 0.957 = 96$ $\frac{1, 200}{12} \times 0.957 = 96$	July	$\frac{1, 200}{12} \times 1.117 = 112$ $\frac{1, 200}{12} \times 1.117 = 112$
Feb.	$\frac{1, 200}{12} \times 0.851 = 85$ $\frac{1, 200}{12} \times 0.851 = 85$	Aug.	$\frac{1, 200}{12} \times 1.064 = 106$ $\frac{1, 200}{12} \times 1.064 = 106$
Mar.	$\frac{1, 200}{12} \times 0.904 = 90$ $\frac{1, 200}{12} \times 0.904 = 90$	Sept.	$\frac{1, 200}{12} \times 0.957 = 96$ $\frac{1, 200}{12} \times 0.957 = 96$
Apr.	$\frac{1, 200}{12} \times 1.064 = 106$ $\frac{1, 200}{12} \times 1.064 = 106$	Oct.	$\frac{1, 200}{12} \times 0.851 = 85$ $\frac{1, 200}{12} \times 0.851 = 85$
May	$\frac{1, 200}{12} \times 1.309 = 131$ $\frac{1, 200}{12} \times 1.309 = 131$	Nov.	$\frac{1, 200}{12} \times 0.851 = 85$ $\frac{1, 200}{12} \times 0.851 = 85$
June	$\frac{1, 200}{12} \times 1.223 = 122$ $\frac{1, 200}{12} \times 1.223 = 122$	Dec.	$\frac{1, 200}{12} \times 0.851 = 85$ $\frac{1, 200}{12} \times 0.851 = 85$

Calculating Seasonal Indices with Trend

When both trend and seasonal components are present in a time series, a change from one month to the next could be due to a trend, to a seasonal variation, or simply to random fluctuations. To help with this problem, the seasonal indices should be computed using a centered moving average (CMA) approach whenever trend is present. Using this approach prevents a variation due to trend from being incorrectly interpreted as a variation due to the season. Consider the following example.

Quarterly sales figures for Turner Industries are shown in Table 5.9. Notice that there is a definite trend, as the total is increasing each year, and there is an increase for each quarter from one year to the next as well. The seasonal component is obvious, as there is a definite drop from the fourth quarter of one year to the first quarter of the next. A similar pattern is observed in comparing the third quarters to the fourth quarters immediately following.

If a seasonal index for quarter 1 were computed using the overall average, the index would be too low and misleading, since this quarter has less trend than any of the others in the sample. If the first quarter of year 1 were omitted and replaced by the first quarter of year 4 (if it were available), the average for quarter 1 (and consequently the seasonal index for quarter 1) would be considerably higher. To derive an accurate seasonal index, we should use a CMA.

Consider quarter 3 of year 1 for the Turner Industries example. The actual sales in that quarter were 150. To determine the magnitude of the seasonal variation, we should compare this with an average quarter centered at that time period. Thus, we should have a total of four quarters (1 year of data) with an equal number of quarters before and after quarter 3 so the trend is averaged out. Thus, we need 1.5 quarters before quarter 3 and 1.5 quarters after it. To obtain the CMA, we take quarters 2, 3, and 4 of year 1, plus one-half of quarter 1 for year 1 and one-half of quarter 1 for year 2. The average will be

CMA (quarter 3 of year 1) = \frac{0.5 (108) + 125 + 150 + 141 + 0.5 (116)}{4} = 132.00

$CMA (quarter 3 of year 1) = \frac{0.5 (108) + 125 + 150 + 141 + 0.5 (116)}{4} = 132.00$

We compare the actual sales in this quarter to the CMA, and we have the following seasonal ratio:

Seasonal ratio = \frac{Sales in quarter 3}{CMA} = \frac{150}{132.00} = 1.136

$Seasonal ratio = \frac{Sales in quarter 3}{CMA} = \frac{150}{132.00} = 1.136$

Thus, sales in quarter 3 of year 1 are about 13.6% higher than an average quarter at this time. All of the CMAs and the seasonal ratios are shown in Table 5.10.

Since there are two seasonal ratios for each quarter, we average these to get the seasonal index. Thus,

\begin{array}{l} Index for quarter 1 & = & I_{1} & = & (0.851 + 0.848) / 2 & = & 0.85 \\ Index for quarter 2 & = & I_{2} & = & (0.965 + 0.960) / 2 & = & 0.96 \\ Index for quarter 3 & = & I_{3} & = & (1.136 + 1.127) / 2 & = & 1.13 \\ Index for quarter 4 & = & I_{4} & = & (1.051 + 1.063) / 2 & = & 1.06 \end{array}

$\begin{array}{l} Index for quarter 1 & = & I_{1} & = & (0.851 + 0.848) / 2 & = & 0.85 \\ Index for quarter 2 & = & I_{2} & = & (0.965 + 0.960) / 2 & = & 0.96 \\ Index for quarter 3 & = & I_{3} & = & (1.136 + 1.127) / 2 & = & 1.13 \\ Index for quarter 4 & = & I_{4} & = & (1.051 + 1.063) / 2 & = & 1.06 \end{array}$

The sum of these indices should be the number of seasons (4), since an average season should have an index of 1. In this example, the sum is 4. If the sum were not 4, an adjustment would be made. We would multiply each index by 4 and divide this by the sum of the indices.

Table 5.9 Quarterly Sales ($1,000,000s) for Turner Industries

QUARTER	YEAR 1	YEAR 2	YEAR 3	AVERAGE
1	108	116	123	115.67
2	125	134	142	133.67
3	150	159	168	159.00
4	141	152	165	152.67
Average	131.00	140.25	149.50	140.25

Table 5.9 Full Alternative Text

Table 5.10 Centered Moving Averages and Seasonal Ratios for Turner Industries

YEAR	QUARTER	SALES ($1,000,000s)	CMA	SEASONAL RATIO
1	1	108
	2	125
	3	150	132.000	1.136
	4	141	134.125	1.051
2	1	116	136.375	0.851
	2	134	138.875	0.965
	3	159	141.125	1.127
	4	152	143.000	1.063
3	1	123	145.125	0.848
	2	142	147.875	0.960
	3	168
	4	165

Table 5.10 Full Alternative Text

Steps Used to Compute Seasonal Indices Based on CMAs

Compute a CMA for each observation (where possible).
Compute Seasonal ratio = Observation/CMA for that observation.
Average seasonal ratios to get seasonal indices.
If seasonal indices do not add to the number of seasons, multiply each index by (Number of seasons)/(Sum of the indices).

The seasonal indices found using the CMAs will be used to remove the impact of seasons so that the trend is easier to identify. Later, the seasonal indices will be used again to adjust the trend forecast based on the seasonal variations.

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Table of Contents for 5.6 Adjusting for Seasonal Variations

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Table of Contents for
5.6 Adjusting for Seasonal Variations