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