Data concepts

Multiplicative adjustment: Consider the graph of U.S. total retail sales of automobiles from January 1970 to May 1998, in units of billions of dollars, as reported at the time by the U.S. Bureau of Economic Analysis:

Much of the trend is merely due to inflation.  The values can be deflated, i.e., converted to units of constant rather than nominal dollars, by dividing them by a suitable price index scaled to a value of 1.0 in whatever year is desired as the base year.  Here’s the result of dividing by the U.S. consumer price index (CPI) scaled to 1.0 in 1990, which converts the units to billions of 1990 dollars:

(The data can be found in this Excel file, and it is also analyzed in further detail in the pages on seasonal ARIMA models on this site.)  There is still a general upward trend, and the increasing amplitude of seasonal variations is suggestive of a multiplicative seasonal pattern: the seasonal effect expresses itself in percentage terms, so the absolute magnitude of the seasonal variations increases as the series grows over time. Such a pattern can be removed by multiplicative seasonal adjustment, which is accomplished by dividing each value of the time series by a seasonal index (a number in the vicinity of 1.0) that represents the percentage of normal typically observed in that season.

For example, if December's sales are typically 130% of the normal monthly value (based on historical data), then each December's sales would be seasonally adjusted by dividing by 1.3. Similarly, if January's sales are typically only 90% of normal, then each January's sales would be seasonally adjusted by dividing by 0.9. Thus, December's value would be adjusted downward while January's would be adjusted upward, correcting for the anticipated seasonal effect. Depending on how they were estimated from the data, the seasonal indices might remain the same from one year to the next, or they might vary slowly with time.

The seasonal indices computed by the Seasonal Decomposition procedure in Statgraphics are constant over time, and are computed via the so-called "ratio-to-moving average method." (For an explanation of this method, see the slides on forecasting with seasonal adjustment and the notes on spreadsheet implementation of seasonal adjustment.) Here are the multiplicative seasonal indices for auto sales as computed by the Seasonal Decomposition procedure in Statgraphics:

Finally, here is the seasonally adjusted version of deflated auto sales that is obtained by dividing each month's value by its estimated seasonal index:

Notice that the pronounced seasonal pattern is gone, and what remains are the trend and cyclical components of the data, plus random noise.