This section deals with the identification of a multiplicative SARIMA model. The required
procedure is explained step by step, using the famous airline data of Box and Jenkins (1976, Series
G) for illustrative purposes. The date give the number of airline passengers (in
thousands) in international air travel from 1949:1 to 1960:12. In the following denotes
the original series.
The identification procedure comprises the following steps: plotting the data, possibly transforming the data, identifying the dependence order of the model, parameter estimation, and diagnostics. Generally, selecting the appropriate model for a given data set is quite difficult. But the task becomes less complicated, if the following approach is observed: one thinks first in terms of finding difference operators that produce a roughly stationary series and then in terms of finding a set of simple ARMA or multiplicative SARMA to fit the resulting residual series.
As with any data analysis, the time series has to be plotted first so that the graph can be inspected. Figure 5.5 shows the airline data of Box and Jenkins.
![]() |
The series shows a strong seasonal pattern and a definite upward trend. Furthermore,
the variability in the data grows with time. Therefore, it is necessary to transform the data
in order to stabilize the variance. Here, the natural logarithm is used for transforming the
data. The new time series is defined as follows
![]() |
Figure 5.6 displays the logarithmically transformed data . The strong
seasonal pattern and the obvious upward trend remain unchanged, but the variability is
now stabilized.
Now, the first difference of time series has to be taken in order to remove its
nonseasonal unit root, i.e. we have
. The new variable
![]() |
(5.7) |
has a nice interpretation: it gives approximately the monthly growth rate of the number of airline passengers.
The next step is plotting the sample ACF of the monthly growth rate
.
The sample ACF in Figure 5.7 displays a recurrent pattern: there are
significant peaks at the seasonal frequencies (lag 12, 24, 36, etc.) which decay slowly.
The autocorrelation coefficients of the months in between are much smaller and follow a
regular pattern. The characteristic pattern of the ACF indicates that the underlying time
series possesses a seasonal unit root. Typically, is sufficient to obtain seasonal
stationarity. Therefore, we take the seasonal difference and obtain the following time
series
![]() |
that neither incorporates an ordinary nor a seasonal unit root.
After that, the sample ACF and PACF of
has to be inspected in order to
explore the remaining dependencies in the stationary series. The autocorrelation functions
are given in Figures 5.8 and 5.9. Compared with the
characteristic pattern of the ACF of
(Figure 5.7) the pattern of
the ACF and PACF of
are far more difficult to interpret. Both ACF and
PACF show significant peaks at lag 1 and 12. Furthermore, the PACF displays autocorrelation
for many lags. Even these patterns are not that clear, we might feel that we face a seasonal
moving average and an ordinary MA(1). Another possible specification could be an ordinary
MA(12), where only the coefficients
and
are different from zero.
Thus, the identification procedure leads to two different multiplicative SARIMA
specifications. The first one is a SARIMA(0,1,1)(12,0,1,1). Using the lag-operator
this model can be written as follows:
![]() |
![]() |
![]() |
|
![]() |
![]() |
The second specification is a SARIMA(0,1,12)(12,0,1,0). This model has the following
representation:
![]() |
Note, that in the last equation all MA coefficients other than and
are zero. With the specification of the SARIMA models the identification process is finished.
We saw that modeling the seasonal ARMA after removing the nonseasonal and the seasonal unit
root was quite difficult, because the sample ACF and the PACF did not display any clear
pattern. Therefore, two different SARIMA models were identified that have to be tested in the
further analysis.