 ARIMA models for time series forecasting

Seasonal differencing in ARIMA models

The seasonal difference of a time series is the series of changes from one season to the next. For monthly data, in which there are 12 periods in a season, the seasonal difference of Y at period t is Yt - Yt-12. If the seasonal difference of Y is stationary white noise (independently and identically distributed values with no autocorrelation), then Y is described by a seasonal random walk model: each value is a random step away from the value that occurred exactly one season ago.

Seasonal differencing is a crude form of additive seasonal adjustment: the "index" which is subtracted from each value of the time series is simply the value that was observed in the same season one year earlier. Seasonal differencing therefore usually removes the gross features of seasonality from a series, as well as most of the trend.

Seasonal differencing and other aspects of seasonal ARIMA modeling to be discussed later will be illustrated by the U.S. retail auto sales data, deflated to units of 1990 dollars, that was introduced on the seasonal adjustment page. Here is a plot of that series, which is called "AUTOSALE/CPI" in the Statgraphics file, for the period from January 1970 to May 1994: Here is a plot of the seasonal difference. Notice that little remains of the original seasonal pattern or trend, although it now looks a bit like a random walk rather than pure noise. First difference of seasonal difference: In the preceding two graphs, we see that the first difference of AUTOSALE/CPI is far from random (it is still strongly seasonal), and the seasonal difference is far from stationary (it resembles a random walk). In this case, it appears that both kinds of differencing are needed to render the series stationary and to account for the gross pattern of seasonality. The first difference of the seasonal difference of a monthly time series Y at period t is equal to (Yt - Yt-12) - (Yt-1 - Yt-13). Equivalently, it is equal to (Yt - Yt-1) - (Yt-12 - Yt-13). This is the amount by which the change from the previous period to the current period is different from the change that was observed exactly one year earlier. Thus, for example, the first difference of the seasonal difference in September 1995 is equal to the August-to-September change in 1995 minus the August-to-September change in 1994. If the first difference of the seasonal difference of Y is pure noise, then Y is described by a seasonal random trend model.

Here is a plot of the first difference of the seasonal difference of AUTOSALE/CPI. Note that it now appears stationary without obvious signs of seasonality. (We should look at an autocorrelation plot to be sure that no seasonal pattern remains, but at least the gross seasonal pattern has been eliminated.) The following spreadsheet shows how the seasonal difference and first difference of the seasonal difference are calculated in this example: 