1. 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

2. Suppose we have a quarterly time series X 1, X 2, X 3,..., X n The first value that can be calculated for this series by a 4-period MA process would use observations X 1, X 2, X 3, and X 4. 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:

3. For the series X 1, X 2, X 3,..., X n, the formula is 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. (1)

4. We can correct this problem with a centered 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

5. To get a 4-period moving average that is centered at quarter 3 (designated by X 3 **), take the average of X 2 * and X 3 *: The general formula is: (2)

6. Combining equations (2) and (3), the series X t ** can be expressed by a weighted moving average: The seasonal index (S t ) can be computed by dividing X t by X t **. That is:

7. 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.