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
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

Suppose we have a quarterly time series X1, X2, X3, . . . , Xn

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

For the series X1, 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
We can correct this problem with acentered 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

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

Combining equations (2) and (3), the series Xt** 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
Example: Quarterly product sales
  • 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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