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Figure 9-4
The first 132 values of the
data set AIRPASS.TSM
and predictors of the last
12 values obtained by
direct application of
the ARAR algorithm.
9.4 Choosing a Forecasting Algorithm 329
observations in the series WINE.TSM (Figure 1.1) and compare the average mean
squared errors of the forecasts of Y131,...,Y142, we find (Problem 9.2) that an MA(12)
model fitted to the mean corrected differenced series {Yt − Yt−12} does better than
seasonal Holt–Winters (with period 12), which in turn does better than ARAR and
(not surprisingly) dramatically better than nonseasonal Holt–Winters. An interesting
empirical comparison of these and other methods applied to a variety of economic
time series is contained in Makridakis et al. (1998).
The versions of the Holt–Winters algorithms we have discussed in Sections 9.2
and 9.3 are referred to as “additive,” since the seasonal and trend components enter the
forecasting function in an additive manner. “Multiplicative” versions of the algorithms
can also be constructed to deal directly with processes of the form
Yt mtstZt, (9.4.1)
where mt, st, and Zt are trend, seasonal, and noise factors, respectively (see, e.g.,
Makridakis et al., 1983). An alternative approach (provided that Yt > 0 for all t)isto
apply the linear Holt–Winters algorithms to {ln Yt} (as in the case of WINE.TSM in
the preceding paragraph). Because of the rather general memory shortening permitted
by the ARAR algorithm, it gives reasonable results when applied directly to series
of the form (9.4.1), even without preliminary transformations. In particular, if we
consider the first 132 observations in the series AIRPASS.TSM and apply the ARAR
algorithm to predict the last 12 values in the series, we obtain (Problem 9.4) an
observed root mean squared error of 18.21. On the other hand if we use the same
data take logarithms, difference at lag 12, subtract the mean and then fit an AR(13)
model by maximum likelihood using ITSM and use it to predict the last 12 values, we
(thousands)
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