Often a time series which has a
strong seasonal pattern is not
satisfactorily stationarized by a seasonal difference alone, and
hence the seasonal random walk model
(which predicts the seasonal difference to be constant) will not
give a good fit. For example, the seasonal
difference of the deflated auto sales series
looks more like a random walk than a stationary noise pattern.
However, if we look at the first
difference of the seasonal difference
of deflated auto sales, we see a pattern that looks more-or-less
like stationary noise with a *mean value of zero*:

(Even if the mean value of a twice-differenced series is not exactly zero, we normally assume it is zero for forecasting purposes: otherwise we would be assuming a trend-in-the-trend, which would be dangerous to extrapolate very far.)

For monthly data, whose seasonal period is 12, the first difference of the seasonal difference at period t is (Y(t) - Y(t-12)) - (Y(t-1) - Y(t-13). Applying the zero-mean forecasting model to this series yields the equation:

Rearranging terms to put Y(t) by itself on the left, we obtain:

For example, if it is now September '96 and we are using this equation to predict the value of Y in October '96, we would compute:

In other words, October's forecast equals September's value plus the September-to-October change observed last year. Equivalently, we can rewrite this as:

which says that October's forecast equals last October's value plus the year-to-year change we observed last month. (These preceding two equations are mathematically identical: we've just rearranged terms on the right-hand-side.)

This forecasting model will be
called the *seasonal random trend*
model, because it assumes that the seasonal trend (difference)
observed this month is a random step away from the trend that
was observed last month, where the steps are assumed to have mean
zero. To see this, rewrite the equation in terms of seasonal
differences:

In other words, the expected seasonal difference this month is
the same as the seasonal difference observed last month. Now compare
this behavior with that of the seasonal random
walk model:
the seasonal random walk model assumes that the expected values
of all future seasonal differences are equal to the *average
seasonal difference* calculated over the whole history of the
time series. In contrast, the seasonal random trend model assumes
that the expected values of all future seasonal differences are
equal to the *most recently observed seasonal difference*.
Moreover, the seasonal random trend model assumes that the actual
seasonal differences will be undergoing a zero-growth random
walk--rather
than fluctuating around some constant mean value--so their values
will become very uncertain in the distant future.

The seasonal random walk model
and the seasonal random trend model
both predict that next year's seasonal cycle will have exactly
the same shape (i.e., the same relative month-to-month changes)
as this year's seasonal cycle. The difference between them is
in their *trend* projections: the seasonal random walk model
assumes that the future trend will equal the *average*
year-to-year
trend observed in the past, whereas the seasonal random trend
model assumes that the future trend will equal the *most recent*
year-to-year trend.

Returning to our example, if
our last recorded value was for September
'96, the seasonal random trend model will predict October's value
to satisfy Y(Oct'96) - Y(Oct'95) = Y(Sep'96) - Y(Sep'95). If we
then bootstrap the model one month into the future to predict
November's value, the model will predict that Y(Nov'96) - Y(Nov'95)
= Y(Oct'96) - Y(Oct'95) ...but this is exactly equal to Y(Sep'96)
- Y(Sep'95) again, because the October '96 forecast must be used
in place of an actual value. Thus, all future *seasonal*
differences, as predicted from time origin September '96, will
be identical to the September-to-September difference.

The seasonal random trend model
is a special case of an ARIMA
model in which there is one order of non-seasonal differencing,
one order of seasonal differencing, and no constant or other
parameters--i.e.,
an "ARIMA(0,1,0)x(0,1,0) model." In Statgraphics, you
would specify a seasonal random trend model by choosing **ARIMA**
as the model type and then selecting:

**Differencing: Nonseasonal Order = 1, Seasonal Order = 1****AR, MA, SAR, SMA = 0****Constant = off**

If we apply the seasonal random trend model to the deflated auto sales data, using data up to November '91, we obtain the following picture:

Notice that the one-step-ahead predictions (up to November '91) respond very quickly to cyclical upturns and downturns in the data, unlike those of the seasonal random walk model. Also notice that the predictions for future seasonal cycles have exactly the same shape as the last observed seasonal cycle (the one ending in November '91). However, the long-term forecasts march off with a downward trend equal to the downward trend that was observed from November '90 to November '91. The confidence limits for the long-term forecasts also diverge very rapidly because of the assumption that the actual future trend will be randomly changing.

If we recompute the forecasts using data up to January '92, we see a very different picture in the long-term forecasts:

The upward trend between January '91 and January '92 now causes the long-term forecasts to shoot off upward! Thus, we see that the seasonal random trend model is much more responsive than the seasonal random walk model to sudden shifts in the data. This serves it well when forecasting one period ahead, but renders it rather unstable for purposes of forecasting many periods ahead.

If you are thinking at this
point that it probably would be better
to do some amount of *smoothing* when estimating the seasonal
pattern and/or the long-term trend in the forecasts, you are right.
(By "smoothing" I mean that you might want to average
over the last few season's data when estimating the seasonal pattern
and/or the trend.) You can smooth the trend estimate by adding
**MA=1** to the parameter specifications, and you can smooth
the estimate of the seasonal pattern by setting** SMA=1**.
Adding both of these terms will yield an "ARIMA(0,1,1)x(0,1,1)
model," which is probably the most commonly used ARIMA model
for seasonal data. (For the technically curious, setting MA=1
adds a multiple of *the one-month-prior forecast error* to
the right-hand-side of the forecasting equation, while setting
SMA=1 adds a multiple of the *12-month-prior error*, and
adding both terms together also causes a multiple of the 13-month-prior
error to be included. The resulting model is a kind of "seasonal
linear exponential smoothing.") The forecasts generated by
this model from time origin November '91 indeed show a smoother
seasonal pattern, a more conservative trend estimate, and narrower
confidence intervals:

The preceding qualitative observations are confirmed by the model comparison report for the seasonal random walk, seasonal random trend, and more elaborate ARIMA models fitted to the deflated auto sales data. The seasonal random trend model outperforms the seasonal random walk model within the estimation and validation periods (i.e., for all one-step-ahead forecasts), and the more elaborate models with additional ARIMA parameters improve on the simpler models without those parameters. (Return to top of page.)