Using a centered moving average to extract the seasonal component of a time series
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Using a Centered Moving Average to Extract the Seasonal Component of a Time Series. If we are forecasting with say, quarterly time series data, a 4-period moving average should be free of seasonality since it always includes one observation for each quarter of the year.

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Using a centered moving average to extract the seasonal component of a time series l.jpg
Using a Centered Moving Average to Extract the Seasonal Component of a Time Series

If we are forecasting with say, quarterly time series data, a 4-period moving average should be free of seasonality since it always includes one observation for each quarter of the year


Slide2 l.jpg

The first value that can be calculated for this series by a 4-period MA process would use observations X1, X2, X3, and X4. Notice that our first 4-period average has a center between quarter 2 and quarter 3. Hence we will designate it X*2.5. Thus we have:

The next value is:


Slide3 l.jpg

For the series X Component of a Time Series1, X2, X3, . . . , Xn, the formula is1 :

(1)

This algorithm gives us a series that is free of seasonality. Alas, the location of the values of this series do not correspond to the original series.


We can correct this problem with a centered moving average l.jpg
We can correct this problem with a Component of a Time Seriescentered moving average

If we average adjacent pairs of X*t’s, we obtain a series that is free of seasonality and is aligned correctly with our original series


Slide5 l.jpg

To get a 4-period moving average that is centered at quarter 3 (designated by X3**), take the average of X2* and X3*:

The general formula is:

(2)


Slide6 l.jpg

Combining equations (2) and (3), the series X 3 (designated by t** can be expressed by a weighted moving average:

The seasonal index (St) can be computed by dividing Xt by Xt**. That is:


Example quarterly product sales l.jpg
Example: Quarterly product sales 3 (designated by

  • Notice we lose 2 data points at the beginning of the series, and 2 at the end.

  • For monthly data, we would lose a total of 12 data points.


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