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1.6 Testing the Estimated Noise Sequence 35
components of the series are required and whether or not it appears that the data
contain a seasonal component that does not vary with time. The program ITSM
allows two options under the Transform menu:
1. “classical decomposition,” in which trend and/or seasonal components are estimated
and subtracted from the data to generate a noise sequence, and
2. “differencing,” in which trend and/or seasonal components are removed from the
data by repeated differencing at one or more lags in order to generate a noise
sequence.
A third option is to use the Regression menu, possibly after applying a Box–Cox
transformation. Using this option we can (see Example 1.3.6)
3. fit a sum of harmonics and a polynomial trend to generate a noise sequence that
consists of the residuals from the regression.
In the next section we shall examine some techniques for deciding whether or not the
noise sequence so generated differs significantly from iid noise. If the noise sequence
does have sample autocorrelations significantly different from zero, then we can take
advantage of this serial dependence to forecast future noise values in terms of past
values by modeling the noise as a stationary time series.
1.6 Testing the Estimated Noise Sequence
The objective of the data transformations described in Section 1.5 is to produce a
series with no apparent deviations from stationarity, and in particular with no apparent
trend or seasonality. Assuming that this has been done, the next step is to model the
estimated noise sequence (i.e., the residuals obtained either by differencing the data
or by estimating and subtracting the trend and seasonal components). If there is no
dependence among between these residuals, then we can regard them as observations
of independent random variables, and there is no further modeling to be done except to
estimate their mean and variance. However, if there is significant dependence among
the residuals, then we need to look for a more complex stationary time series model
for the noise that accounts for the dependence. This will be to our advantage, since
dependence means in particular that past observations of the noise sequence can assist
in predicting future values.
In this section we examine some simple tests for checking the hypothesis that
the residuals from Section 1.5 are observed values of independent and identically
distributed random variables. If they are, then our work is done. If not, then we must
use the theory of stationary processes to be developed in later chapters to find a more
appropriate model.