FORECAST <data>,<series>,<predictor>,L
makes an automatic forecast of a time series by using a variant
of the Holt-Winters' seasonal forecast procedure.
(See for example, B.Abraham and J.Ledolter (1983). Statistical Methods
for Forecasting, Wiley.)
The period s of the series is given by PERIOD=s.
If the PERIOD specification is omitted, FORECAST tries to judge s
from the data.
The active part of the observations in <data> is used for the
estimation of the level, slope and seasonal components. The predicted
values of <series> are saved in <predictor> for the estimation period
plus one complete period (s observations) ahead. Number of forecast
values may be changed by the AHEAD=<#_of_values> specification.
A '-' as <predictor> rejects saving of forecast values.
L (optional) is the first edit line for the results.
The type of the model is selected by the specification C=<type>.
Default is C=1 (additive seasonal). Another alternative is C=0
(multiplicative seasonal). In case s=1 both models correspond to
Holt's double exponential smoothing procedure.
Also values 0<C<1 may be tested, but then the time required for
the estimation is much longer.
The three smoothing coefficients are selected by minimizing the
mean square error of the one-step-ahead forecast errors. The initial
values of the level, slope and seasonal parameters are obtained
first by backforecasting on the data.
However, fixed values for the smoothing coefficients can also given
by a specification of the form PAR=a1,a2,a3 where
a1=level coefficient, a2=seasonal coefficient and a3=slope coefficient.
Each of them must be in the interval (0,1). For stable components
the smoothing coefficients should be close to zero.
After estimation of the smoothing coefficients the outliers in data
may be rejected on the basis of the forecast errors. Only one (the
worst) outlier is rejected at a time by replacing the data value
by the current forecast and the whole estimation process is repeated.
The OUTLIERS=n,k specification gives n as the maximum number of
such rejections and k as the treshold for an outlier. An observation
is rejected (smoothed), if its forecast error exceeds k times the
standard error. Default is OUTLIERS=3,2.5. By OUTLIERS=0 no outliers
are considered.
In addition to the forecast also the components of the series may be
saved in the original data (file) by entering certain masks.
The components and their mask symbols are:
Trend (level) T
Slope B
Seasonal S
Example:
....................................................................................................
40 *DATA X: 1 2 5 4 2 3 4 5 3 4 5 6 4 5 6 7 5 6 5 10 6 7 8 9 END
41 *
42 *FORECAST X,X,-,43 / No specifications are given in this case
43 *Holt-Winters' Additive Seasonal Forecast: Data X, Variable X
44 *Period=4 obs. (judged from data) Estimation on observations 1-24
45 *Outliers:19,20,3 (+more to be found)
46 *MSE=0.025814 a(level)=-0.000 a(seasonal)=0.552 a(slope)=0.856
47 *Autocorrelations of residuals: r1=+0.00 r2=-0.04 r3=-0.05 r4=+0.04
48 *Obs.# Forecast
49 * 25 6.9678
50 * 26 7.9678
51 * 27 9.0452
52 * 28 9.9920
53 *
....................................................................................................
T = More information on time series analysis
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