Springer - Read

6.2 Identification Techniques 187
6.2 Identification Techniques
decaying autocorrelation function (see also Section 6.5). Figures 6.9 and 6.10 show
the sample autocorrelation functions obtained after applying the operators 1−B +B 2
and 1 − B 6 , respectively, to the data shown in Figure 6.7. For either one of these two
differenced series, it is then not difficult to fit an ARMA model φ(B)Xt θ(B)Zt
for which the zeros of φ are well outside the unit circle. Techniques for identifying
and determining such ARMA models have already been introduced in Chapter 5. For
convenience we shall collect these together in the following sections with a number
of illustrative examples.
(a) Preliminary Transformations. The estimation methods of Chapter 5 enable us to
find, for given values of p and q, an ARMA(p, q) model to fit a given series of data.
For this procedure to be meaningful it must be at least plausible that the data are in
fact a realization of an ARMA process and in particular a realization of a stationary
process. If the data display characteristics suggesting nonstationarity (e.g., trend and
seasonality), then it may be necessary to make a transformation so as to produce a
new series that is more compatible with the assumption of stationarity.
Deviations from stationarity may be suggested by the graph of the series itself or
by the sample autocorrelation function or both.
Inspection of the graph of the series will occasionally reveal a strong dependence
of variability on the level of the series, in which case the data should first be
transformed to reduce or eliminate this dependence. For example, Figure 1.1 shows
ACF
-0.5 0.0 0.5 1.0
Figure 6-9
The sample ACF of
(1 − B + B2 )Xt with
0 10 20 30 40
{Xt } as in Figure 6.7. Lag